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Research The International Journal of Robotics

DOI: 10.1177/027836498700600303 1987; 6; 49 The International Journal of Robotics Research

Jean-Jacques E. Slotine and Weiping Li On the Adaptive Control of Robot Manipulators

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49

On the AdaptiveControl of Robot

Manipulators

Jean-Jacques E. SlotineWeiping LiNonlinear Systems LaboratoryMassachusetts Institute of TechnologyCambridge, Massachusetts 02139

Abstract

A new adaptive robot control algorithm is derived, whichconsists of a PD feedback part and a full dynamics feedfor-ward compensation part, with the unknown manipulator andpayload parameters being estimated online. The algorithm iscomputationally simple, because of an effective exploitationof the structure of manipulator dynamics. In particular, itrequires neither feedback of joint accelerations nor inversionof the estimated inertia matrix. The algorithm can also beapplied directly in Cartesian space.

1. Introduction

Adaptive control, as a branch of systems theory, is notyet quite mature (see, for instance, Astr6m 1983;1984). Yet, the practically motivated drive to makerobot manipulators capable of handling large loads inthe presence of uncertainty on the mass properties ofthe load or its exact position in the end-effector, aswell as the old &dquo;cybernetic&dquo; ideal of developing learn-ing capabilities in machines, has spurred much re-search on adaptive control of robot manipulators (see,e.g., Hsia 1986, for a recent review). The nonlinearityof robot dynamics, however, makes them even morecomplex to analyze than the linear dynamic systemson which most of the existing adaptive control theoryhas been traditionally focused.

Several approaches have been considered. Somechoose to ignore the dynamic complexity and fit themeasured data to a second-order, linear, time-varyingmodel, using for instance a recursive least-squaresapproach (see, e.g., Koivo 1986). Others do exploit the

known structure of the system dynamics (e.g., Khoslaand Kanade 1985; Atkeson et al. 1985; Craig et al.1986), although they generally require estimation ofjoint accelerations. Another class of algorithms con-siders the &dquo;learning&dquo; of specific tasks through the useof feedforward signals (Arimoto et al. 1985; Atkeson etal. 1986), without explicitly updating the manipulatormodel itself.

In this paper a new adaptive robot control algorithmis derived, which consists of a PD feedback part and afull dynamics feedforward compensation part, withthe unknown manipulator and payload parametersbeing estimated online. The algorithm is computation-ally simple, because of an effective exploitation of theparticular structure of manipulator dynamics. As inKhosla and Kanade (1985) and Atkeson et al. (1985),we use the remark that the dependence of the systemdynamics on the unknown parameters can be madelinear in terms of a suitably selected set of robot andload parameters. However, contrary to most algo-rithms in the literature, there is no need to measurethe joint accelerations or to invert the estimated inertiamatrix.The layout of the paper is as follows: Section 2

presents our basic adaptive structure in joint space,and in Section 3 we discuss its extension to Cartesian

space control. Simulation results are presented in Sec-tion 4. Section 5 offers brief concluding remarks.

Extensive experimental results are presented in Slo-tine and Li ( 1987).

2. Adaptive Robot Controller in Joint Space

~.1. Dynamic Model of Robot Manipulators

In the absence of friction or other disturbances the

dynamics of an n-link rigid manipulator can be writtenas

This research was supported in part by a grant from the Sloan Fund.



50

where q is the n X 1 vector of joint displacements, i isthe n X 1 vector of applied joint torques (or forces),H(q) is the n X n symmetric positive definite manipu-lator inertia matrix, C(q, q)q is the n X 1 vector ofcentripetal and Coriolis torques, and G(q) is the n X 1vector of gravitational torques.Two simplifying properties should be noted about

this dynamic structure. First, as remarked by sev-eral authors (e.g., Arimoto and Miyazaki 1984;Kodistcheck 1984), the matrices H and C are not in-dependent. Specifically, given a proper definition of C,the matrix H - 2C is skew-symmetric, as shown inAppendix II. Physically, this property can be easilyunderstood: The derivative of the manipulator’s ki-netic energy qTHq must equal the power input pro-vided by the actuators and the gravitational torques:

which implies that at all times

Another important property is that the dynamic struc-ture is linear in terms of a suitably selected set ofrobot and load parameters (Khosla and Kanade 1985;Atkeson et al. 1985), as illustrated in Appendix I for atwo-link manipulator.

2.2. Controller Design

The controller design problem is as follows: Given thedesired trajectory qd(t), and with some or all the ma-nipulator parameters being unknown, derive a controllaw for the actuator torques and an estimation law forthe unknown parameters such that the manipulatoroutput q(t) tracks the desired trajectories after an ini-tial adaptation process.

-

We derive our adaptive controller in two steps. First,in Section 2.2.1 a simple globally stable adaptive con-troller is obtained from a Lyapunov stability analysis.The controller strongly exploits the structure of themanipulator dynamics pointed out in the previous

section. After the initial transients, however, althoughthe adaptive controller does yield zero velocity errors,it may present nonzero position errors. We solve thisproblem in Section 2.2.2 by restricting the residualtracking errors to lie on a sliding surface (see Slotine1985), thus guaranteeing asymptotic convergence ofthe tracking.

2.2. J. A Globally Stable Adaptive Controller

To derive the control algorithm and adaptation law, ,we consider the Lyapunov function candidate

where a is an m-dimensional vector containing theunknown manipulator and load parameters, and i isits estimate; Kp and r are symmetric positive definitematrices, usually diagonal; q(t) = q(t) - qd(t) is thetracking error; and à = i(t) - a denotes the parameterestimation error vector. Differentiating i~ yields

where we have used the property of skew-symmetry toeliminate the term 2 qT(H - 2C)q. Let us define thecontrol law as

where the positive definite matrix K~ may be chosento be time varying. Then

where

Choice (3) cancels the terms associated with the known



51

manipulator parameters, so only the unknown manip-ulator parameters have to be retained and estimated ini. Further, since the matrices H, C, and G are linearin terms of the manipulator parameters, we can write

where Y = Y(q, q, qa, qd) is an n X m matrix, andtherefore

This suggests choosing the adaptation law such that

that is

Note that a = a, since the unknown parameters a areconstants. The resulting expression of V is

Therefore the control law (3) and the adaptation law(5) yield a globally stable adaptive controller.

Expression (6) implies that the steady-state jointvelocity error is zero. However, it does not necessarilyguarantee that the steady-state position error is alsozero. We now modify the previous adaptive scheme inorder to solve this potential problem.

2.2.2. Elimination of Steady-State Position Errors

Undesirable steady-state position errors can be elimi-nated if we restrict them to lie on a sliding surface

where A is a constant matrix whose eigenvalues arestrictly in the right-half complex plane. Formally, weachieve this by replacing the desired trajectory qd(t) inthe above derivation by the virtual &dquo;reference trajec-tory&dquo;

Accordingly, 4d and qd are replaced by

If we define

the control law and adaptation law become

Note that the matrix Y is now a function of q, and Q,rather than 4d and 4d, We can again demonstrateglobal convergence of the tracking by now using theLyapunov function

instead of (2), which yields

instead of (6). Note that control law (8) does not con-tain a term in Kp, since the position error q is alreadyincluded in qr. Expression (11) shows that the outputerrors converge to the sliding surface

This in turn implies that q ~ 0 as t ~ 00. Thus, theadaptive controller defined by (8) and (9) is globallyasymptotically stable and guarantees zero steady-stateerror for joint positions.The previous proof of tracking convergence may

seem somewhat unorthodox to readers not familiarwith sliding control theory. Let us detail the basicfeatures. First, the vector s conveys information aboutboundedness and convergence of q and q, since thedefinition of s can also be viewed as a stable, f’-crst-orderdifferential equation in 4, with s as an input. Thus, forbounded initial conditions, boundedness of s impliesboundedness of 4 and q and, therefore, of q and q;



52

similarly, one can easily show that if s tends to 0 ast ~ 00, so do q and q. Second, the function is actu-ally a quasi-Lyapunov function, in our case simply apositive continuous function of time. Let us now detailthe proof itself. Since V is negative or zero and V islower bounded (by zero), V tends to a constant ast ~ 00 and therefore remains bounded for t E [0, 00].Given the definition (10) of V, this in turn implies,since H is uniformly positive definite (i.e., H ~ hI forsome strictly positive h), that s is bounded and, there-fore, that q and q are bounded; it also implies that a isbounded and, therefore, that i is bounded. From thesystem dynamics this then makes s bounded, and thuss is uniformly continuous on t E [0, 00]. Assuming thatthe (perhaps time-varying) matrix Ko is chosen to beuniformly continuous (as is typically the case, for in-stance, with KD constant, or with KD = ..18), Y is thenuniformly continuous on t E [0, 00]; therefore, since Vis bounded on that time interval and Fis of constantsign (T~ -- 0), V tends to zero as t ~ assuming thatK~ is uniformly positive definite (as is again the case ifKD is chosen to be constant, or if KD = AH), this im-plies from (11) that s ~ 0 as t ~ ~, and therefore thatq &horbar;~ 0 as f &horbar;~- oo.The structure of the adaptive controller given by (8)

and (9) is sketched in Fig. 1. The controller consists oftwo parts. The first part consists of three feedforwardterms corresponding to inertial, centripetal and Corio-lis, and gravitational torques. The second part con-tains two terms representing PD feedback. The re-quired inputs to the controller are the desired jointposition qd, velocity 4d, and acceleration qd from thetrajectory planner, and the required measurements arethe joint position q and velocity q. Contrary to severalalgorithms in the literature (e.g., Craig et al. 1986),there is no need for measuring the joint accelerationsq or for inverting the estimated inertia matrix. Notethat if measurements of joint accelerations were indeedexplicitly available online, one could easily show(Slotine 1986) that the effect of parametric uncertaintyon performance could in principle be made arbitrarilysmall by simply increasing the value of the accelera-tion gain, without using adaptation; however, thisprocedure would be extremely sensitive to imprecisionon the joint acceleration measurement, which thenessentially would enter as a pure disturbance added to q.Note from Fig. 1 that the integral term f o 4 dt of (7)

Fig. 1. Structure of the jointspace adaptive controller.

need not be actually computed, since only qr and q~(not qr) are explicitly used in the control law. There-fore, the formal definition of qr is, in effect, equivalentto adding a feedback loop.

2.3. Discussion

In this section we discuss implementation aspects,computational efficiency, and strategies that combineadaptation on certain parameters with robustness touncertainty on others and to disturbances.

2.3.1. Implementation Aspects

Since the load is usually fixed with respect to the lastlink, it can be regarded as part of that link. In practice,the parameters of the robot itself can be measured orestimated beforehand (Khosla and Kanade 1985; At-keson et al. 1985), so only the parameters of the loadare unknown. Models of Coulomb and viscous friction

may also be included in (1), and the correspondingcoefficients can be identified similarly.Although convergence of the trajectory tracking is

guaranteed in the previous derivation, the parameterestimates themselves do not necessarily converge totheir exact values. Intuitively, to guarantee parameterconvergence, the desired trajectory must be &dquo;su~-ciently rich&dquo; so that only the true set of parameterscan yield exact tracking. A formalization of this con-cept in the context of robot control and the generationof trajectories that speed up parameter convergenceconstitute interesting research topics in themselves(Morgan and Narendra 1977; Craig et at. 1986).We stop updating a given parameter when it reaches



53

its known bounds, and we resume updating as soon asthe corresponding derivative changes signs. This intu-itively motivated procedure can easily be shown topreserve convergence of the tracking.

2.3.2. Computational DeficiencyIn the practical implementation of the previous adap-tive controller, the matrices H, C, and G may be up-dated at a low rate, whereas a high update rate is usedfor q&dquo; Q,, and s, since typically the error terms varymuch faster than the dynamic coefhcient matrices(see, e.g., Khatib 1986). Further, the matrix Y, whosecalculation is naturally coupled to the dynamics com-putation, can also be updated at the slow rate, sincethe choice of the adaptation gain matrix r is generallysuch that the adaptation process is slower than thecontrol bandwidth.

Because of the presence of q, in the second term ofcontrol law (8), however, the controller cannot beimplemented directly with fast recursive formulations,such as the Newton - Euler method, and, therefore,requires explicit computations of H, C, and G. Thesame is true of adaptation law (9). We now introducea recursive Newton-Euler method as an alternative

way of implementing the control and adaptation laws.This Newton - Euler formulation can be seen as anapproximation of the previous development, for whichnew stability conditions are derived.Assume that the second term t4, in (8) is approxi-

mated by Cq. Then we can compute the first threeterms in (8) by a recursive Newton - Euler method,based on the parameters obtained from the adaptationlaw. The resulting control torque is

which is computed through a number of operationsproportional to the number of links. Accordingly, thesame approximation is made in the calculation of thematrix Y, namely,

Let us examine the effects of these approximations.We have

with now

From (10),

Thus from (13), (15), and (16), we obtain

using the skew-symmetry of the matrix (H - 2C).Therefore, the stability of this recursive formulation ofthe adaptive controller is guaranteed as long as KD ischosen large enough (perhaps time varying) to satisfyKD > -tH.

2.3.3. Combining Adaptation with Robustness

In practice, we may simplify the algorithm by notexplicitly estimating all unknown parameters. Someparameters may have relatively minor importance inthe dynamics, in which case we may choose to makethe controller robust to the uncertainty on these pa-rameters rather than explicitly estimating them on-line. Similarly, some geometric parameters may al-ready be known with reasonable precision or may havebeen estimated through sorting devices or visual infor-mation. Further, the controller must be robust to re-sidual time-varying disturbances, such as stiction ortorque ripple.We categorize the unknown parameters a into two

groups: group a~ contains the parameters estimatedonline; group aR contains the parameters not estimatedonline. A sliding control term is then incorporatedinto the torque input (8) to account for the effects ofuncertainties on the parameters in aR and of distur-bances.

Assume, without loss of generality, that only thefirst a unknown parameters are to be actually estimated:



54

and let, correspondingly, Y = [YE YR]. Assume thatthe uncertainties on aR, as well as the disturbancetorques dt reflected to the manipulator joints, arebounded:

Add a sliding control term to torque input (8):

where the notation k sgn (s) stands for the n X 1 vectorof components ki sgn (si), with the k; yet to be speci-fied. With aE and r~ in place of a and z’ in the Lya-punov function (10), we obtain

Since

we let

where the tJi are positive constants. This yields

The system trajectories are thus guaranteed to reachsliding surface s = 0, and therefore convergence of thetracking is achieved.

Further, to avoid undesirable control chattering, wecan use saturation functions sat (siloi) in place of theswitching function sgn (si), with the 0, representingthe thicknesses of the corresponding &dquo;boundarylayers.&dquo; Similarly to Slotine (1984), s is then guaran-teed to converge to the boundary layers, with corre-

sponding small tracking errors; further, the Oi can bemodulated based on bandwidth considerations. Simi-

larly to Slotine and Coetsee (1986), parameter adapta-tion must then be stopped when the system trajec-tories are inside the boundary layers; indeed, bydefinition, disturbances and errors on aR can drive thetrajectories anywhere in the boundary layers withoutthis providing any information about the estimationerror on a~. This procedure also has the advantage ofavoiding long-term drift of the estimated parameters.Note from (18) that K~s can be eliminated from

control input (17), since the sliding control actionmakes it unnecessary; however, this term must be keptin a Newton - Euler implementation of the algorithmto compensate for the approximation of Cq, by Cq, asdiscussed earlier. It may also be retained in order toaccelerate convergence. Note that fixed-parametersliding control is obtained if none of the unknown pa-rameters is explicitly estimated (a = m).

3. Extension to Cartesian Space Control

In this section we extend the previous joint spaceadaptive controllers to task space. To this effect, for anonredundant manipulator, we simply replace thereference trajectories in (7b) and (7c) by .

and, accordingly,

so that

The same control and adaptation laws (8) and (9) arethen used, again with (10) as the Lyapunov function.Following the same derivation as before, we obtain



55

Fig. 2. Two-link manipulatorcarrying a large unknownload.

which implies convergence to

Using the kinematic relation x = Jq, we recognizeexpression (20) as the equation of the sliding surfaceac + Air = 0, which in turn guarantees that x - 0 ast ~ 00. Therefore, the previous adaptive controller isglobally stable and guarantees zero steady-state, Carte-sian space, position error.Note from (19a) and (19b) that only the desired

trajectories in Cartesian space Xd, Xd, and xd have tobe given (i.e., explicit inverse kinematics is not neces-sary). The quantities to be measured are joint posi-tions q and joint velocities q. End-effector position xand velocity x can be obtained from the direct kine-matics, and therefore do not need to be explicitly mea-sured. Also, note that the inverse Jacobian J-’ appearsin (19a) and (19b), and therefore singularity pointsshould be avoided (see Khatib 1986 for a relaxation ofthis condition).

4. Simulation Results

We present computer simulations using the two-linkplanar manipulator considered in Appendix I, carryinga large load of unknown mass properties (Fig. 2). The

Fig. 3. Desired joint trajec-tories for Examples 1 and 2.

two links are identical uniform beams, with actuatorsmounted at the joints. In the simulations the un-known load actually has the same geometry as thelinks but is twice as heavy. For simplicity, the parame-ters of the robot itself are assumed to be exactlyknown. The parameters to be adapted are a, ~3, E, and?7, whose true values are a = 6.7, /3 = 3.4, E = 3.0, and~ = 0. The initial estimates of the load mass propertiesassume that the load is identical to the second link.The corresponding initial parameter estimates area = 4.1; /3 = 1.9, E = 1.7, and ( = 0. In the simulationplots the estimates of the first three parameters arenormalized by the true values, and ( is normalized by3 (the true value of E), since p is itself zero.

Example 1: Comparison with conventional con-trollersThe task is to move the load from position A to

position C, as indicated in Fig. 2. Three controllers areused: (1) PD controller, (2) PD + full dynamics feed-forward compensation, and (3) adaptive controllergiven by (3) and (5). The desired joint trajectories arechosen to be fifth-order polynomials and are shown inFig. 3. The matrices Kp and KD are chosen to be iden-tical for all three controllers, with Kp = 8001 and KD =160/. The results are plotted in Fig. 4 for controller a,Fig. 5 for controller b, and Fig. 6 for controller c. Themaximum joint position errors are about 7.5 ° forcontroller a, 3 for controller b, and only about 0.5 °

for the adaptive controller. The maximum actuatortorques are smaller for the adaptive controller than for .controllers a and b. The parameter estimates do notconverge to their exact values, since the desired trajec-tory is not persistently exciting. Also, as anticipated inSection 2.2.1, the joint position errors do not exactlyconverge to zero, a problem that we now remedy usingthe development of Section 2.2.2.



56

Fig. 4. PD controller inExample 1.

Example 2: Elimination of steady-state position errorThe adaptive controller given by (7) and (8) is simu-

lated with the same parameters as in Example 1, andA = 30/. The joint position errors now converge tozero (Fig. 7). We also note that the maximum jointposition errors have been reduced to only 0.08 with-out significant increase in actuator torques.A smaller value of A is also simulated. With A = 51,

the product of KD and A is the same as Kp of con-troller c in Example 1; however, the resulting maxi-mum position errors are only 0.12°, and convergenceto zero is observed.

Example 3: Parameter convergenceIn this example the desired trajectory is chosen to be

The coefficients a; and bi are chosen to make the de-sired trajectory satisfy the initial and final conditionson position, velocity, and acceleration. The sameadaptive controller as in Example 2 is used. Althoughit may not be necessary to have six frequency compo-nents for the desired trajectory to be persistently excit-ing, this example demonstrates that sufficiently richdesired trajectories do yield convergence of the param-eter estimation (Fig. 8).

Fig. 5. PD + full dynamicsfeedforward controller inExample 1.

-

’

Example 4: Cartesian space adaptive controllerThe same task as that in previous examples is per-

formed by the adaptive Cartesian space controller ofSection 3. The desired path is now a straight line fromA to B in Fig. 2. A fifth-order polynomial is con-structed for the desired displacement along the path,which has zero velocities and accelerations at the startand the end of the path. The feedback gains and allother parameters are the same as before. The perform-ance of this controller (Fig. 9) is similar to that of thejoint space adaptive controller. The steady-state Carte-sian position errors are zero, and the maximum Carte-sian path errors in the x- and y-directions are about8 X 10-4 m.

_

Extensive experimental results (Slotine and Li 1987)confirm these simulations.

5. Concluding Remarks

It is of interest to further investigate specific choices ofthe adaptation gain matrix r that yield optimal con-vergence rates while still avoiding the excitation ofhigh-frequency unmodeled dynamics (such as struc-tural resonant modes, actuator dynamics, or samplingeffects). This may involve employing a time-varying To,based, e.g., on a Gauss-Newton algorithm. Although



57

Fig. 6. Adaptive controller(3), (5).

in principle an approach similar to that of Slotine andCoetsee (1986) could be used to this effect, we believethat in this instance it may be more effective to tryagain to take full advantage of the specific structure ofthe manipulator dynamics. This will be the object of aseparate study.

Further, in the more general context of control sys-tem design for physical nonlinear systems, we believethat the approach that consists of modifying, throughfeedback, the system’s natural energy function ratherthan its explicit expanded dynamics is worthy of fur-ther investigation in its own right.

Appendix I: Two-Link Manipulator withLarge Unknown Load

A two-link planar manipulator carrying an unknownpayload is shown in Fig. 2. The second link, with thepayload attached, can be regarded as an augmentedlink with four unknown parameters, namely, mass ma,moment of inertia le, the distance lee of its mass centerto the second joint, and the angle 5, relative to theoriginal second link. The dynamics of the manipulatorwith payload can then be written as

Fig. 7. Adaptive controllerwith steady-state positionerror eliminated.

where

where g is the acceleration of gravity, and the fourunknown parameters a, /3, E, and 17 are functions ofthe unknown physical parameters:

Conversely, the four unknown physical parameters areuniquely determined by a, (3, E, and ’1.

Appendix II: The Matrix H - 2C

We show here that, with a proper definition of thematrix C, the matrix H - 2C is skew-symmetric, thusmaking more precise the result obtained earlier fromconservation of energy.



58

Fig. 8. Showing the conver-gence of the estimates forpersistently exciting trajec-tories: (a) normalized a andÎ3; {b) normalized E and ~.

The ith element of the vector C4 is (see, e.g., Asadaand Slotine 1986)

where the Christoffel coefficients hijk verify

Thus, (A 1 ) can be written

where we used reindexing to obtain the second termon the right side. Now take

and let W = H - 2C. Then

Fig. 9. Adaptive controller inCartesian space.

Thus for all i, j

which shows the skew-symmetry of H - 2C. Althoughother choices of Cij could satisfy (A I ), they usually donot possess this skew-symmetry property.

References ’

An, C. G., Atkeson, C. G., and Hollerbach, J. M. 1986 (SanFrancisco, Calif.). Experimental determination of theeffect of feedforward control on trajectory tracking errors.IEEE Int. Conf. Robotics and Automation.

Arimoto, S., and Miyazaki, F. 1984 (Bretton Woods, N.H.).On the stability of P.I.D. feedback with sensory informa-tion. Robotics Research, eds. M. Brady and R. P. Paul.Cambridge: MIT Press.

Arimoto, S., Kawamura, S., Miyazaki, F., and Tamaki, S.1985 (Fort Lauderdale, Fla.). Learning control theory fordynamical systems. IEEE Conf. Decision and Control.

Asada, H., and Slotine, J-J. E. 1986. Robot Analysis andControl. New York: John Wiley and Sons.

Åström, K. J. 1983. Theory and applications of adaptivecontrol: A survey. Automatica 19:471.

Åström, K. J. 1984 (Las Vegas, Nev.). Interaction betweenexcitation and unmodeled dynamics in adaptive control.IEEE Conf. Decision and Control.

Atkeson, C. G., An, C. G., and Hollerbach, J. M. 1985(Gouvieux, France). Estimation of inertial parameters ofmanipulator loads and links. Robotics Research, eds. O.Faugeras and G. Giralt. Cambridge: MIT Press.



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Craig, J. J., Hsu, P., and Sastry, S. 1986 (San Francisco,Calif.). Adaptive control of mechanical manipulators.IEEE Int. Conf. Robotics and Automation.

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Hsia, T. C. 1986 (San Francisco, Calif.). Adaptive control ofrobot manipulators—A review. IEEE Int. Conf Roboticsand Automation.

Khatib, O. 1986 (Osaka, Japan). U.S.-Japan Symp. FlexibleAutomation.

Khosla, P., and Kanade, T. 1985 (Fort Lauderdale, Fla.).Parameter identification of robot dynamics. IEEE ConfDecision and Control.

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Koivo, A. J. 1986 (San Francisco, Calif.). Force-position-ve-locity control with self-tuning for robotic manipulators.IEEE Int. Conf. Robotics and Automation.

Morgan, A. P., and Narendra, K. S. 1977. On the uniformasymptotic stability of certain linear nonautonomousdifferential equations. SIAM J. Control Optim. 15.

Slotine, J. J. E. 1984. Sliding controller design for nonlinearsystems. Int. J. Control 40(2).

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