observer-based sliding mode control of a robotic manipulator

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Observer-based sliding mode control of a robotic manipulator

Karel Jezernik, Boris Curk and Jože Harnik

Robotica / Volume 12 / Issue 05 / September 1994, pp 443 - 448DOI: 10.1017/S0263574700017999, Published online: 09 March 2009

Link to this article: http://journals.cambridge.org/abstract_S0263574700017999

How to cite this article:Karel Jezernik, Boris Curk and Jože Harnik (1994). Observer-based sliding mode control of a robotic manipulator. Robotica,12, pp 443-448 doi:10.1017/S0263574700017999

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Robotica (1994) volume 12, pp 443-448. © 1994 Cambridge University Press

Observer-based sliding mode control of a robotic manipulatorKarel Jezernik, Boris Curk, Joze HarnikUniversity ofMaribor, Faculty of Technical Sciences, Smetanova 17, 62000 Maribor (Slovenia)

(Received in Final Form: December 14, 1993)

SUMMARYThis paper presents a new approach for the design ofvariable structure control (VSC) of nonlinear systems.The approach is based on estimation of joint accelerationsignals with introduction of load estimation with theasymptotic observer. The control system is insensitive toparameter variations for a chosen switching hypersurfacein conditions when it is reached by the dynamic motionwith the required dynamics. The parameter insensitiveresponse provided by this control method is dem-onstrated on the model of the SCARA robot. Simulationresults confirm the validity of accurate tracking capabilityand the robust performance.

KEYWORDS: Sliding mode control; Nonlinear systems;SCARA robot.

1. INTRODUCTIONA Variable Structure Control (VSC) approach isproposed for robust and accurate trajectory tracking of arobotic manipulator with electrical actuators. Decentral-ized acceleration controllers are used to generate thelocal switching function. A PI disturbance estimator isproposed to ensure favourable performance. This novelcontroller gives a zero steady state error and enableseach joint to trace the acceleration command. Theparameter variation and disturbance insensitive responseprovided by this control method is demonstrated on amodel of a SCARA robot.

The dynamics of an n-link robot mechanism ischaracterized by a set of highly nonlinear and stronglycoupled second-order differential equations

D(q)q + C(q, q) + G(q) + F(q) = x (1)

where D(q) is the nxn inertial matrix; C{q, q), G(q)and F(q) are n vectors representing Coriolis andcentrifugal forces, the gravity loading, and the friction;q, q and q are n vectors of joint angular position,velocity and acceleration; and t is the n joint torquevector. In general, the matrices D, C, G, F are verycomplicated functions of q and q. The fundamentalmanipulator control problem is to determine thealgorithm for generating the joint torque x, which drivesthe joint position q{t) to follow closely a desired positiontrajectory qd(t). The design of a control algorithm for (1)is generally complicated due to the presence ofnonlinearity and dynamic coupling.1'2 Even in a wellstructured industrial setting, the manipulators aresubjected to structured and unstructured uncertainties.

Structured uncertainty is the case of a correct dynamicmodel with parameter uncertainty due to the imprecisionon the manipulator link properties, unknown loads,inaccuracies of the torque constants of the actuators, etc.Unstructured uncertainty corresponds to the case ofunmodelled dynamics, which results from the presence ofthe high frequency mode of the manipulator, neglectedtime-delays, nonlinear friction, etc. The computedtorque method is effective for the trajectory control ofrobotic manipulators.3 It has become widely recognizedthat the tracking performance of the method in highspeed operations is often affected by the uncertaintiesmentioned above. This is especially true for direct driverobots that have no gearing to reduce the dynamiceffects.

A severe disadvantage of computed torque controlalgorithms is that perfect knowledge of the systemdynamics is required. The inability to consider the totaldynamic model for decoupling and compensation in thecontrol structure requires robustness of the feedbackcontroller to parameter variations and disturbances.These are the declared performances of a variablestructure controller in the sliding mode. Many attemptsto use VSC in robotics have been reported, includingYoung,4 Bailey,5 Wijesoma & Singh.7 Exact modelling isnot necessary, since it is sufficient that limiting of modelparameters and disturbances, on which basis the controlsignal is determined, are known. Numerous papers onthe sliding mode based robot control have selected jointtorques as inputs into the system plant as the startingpoint for the synthesis of the control law. Theoretically,an approach of this kind yields good results.

However, the avoidance of the dynamics of theformation of joint torques may cause a problem. This isparticularly true in the case of transistor inverter fed DCor AC motors which use the switching structure withdiscontinuous torque control. In this case the directimplementation of VSC with a discontinuous controllerscan result in a hierarchical cascade structure ofdiscontinuous laws.8 Since a condition for stableelectrical torque tracking is the assurance of a continuousreference torque curve, the use of this cascadedhierarchical structure results in excessive chattering.910

In order to avoid this problem, some authors havesuggested smoothing of the switching law.11-12 In this waythe discontinuous torque control law is replaced bycontinuous nonlinear control, which ensures smoothdynamic motion. However, the motion of the systemdeviates slightly from that achieved by the ideal

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444 Sliding mode control

switching function, and this results in a steady state errorwithin the boundary layer. The relationship between theboundary layer and the constraints of the real controlinput signals is complex, so the control synthesis may bedifficult. We have found an approach which considers thedynamics of torque formation in the synthesis of thecontrol to be more realistic. In the case of electricalactuators this indicates that armature voltages are controlobject inputs instead of joint torques.

One of the underlying assumptions in the design andanalysis of VSC systems is that the control can beswitched from one value to another infinitely fast. Inpractical systems, however, it is impossible to achieve thehigh switching control that is necessary for most VSCdesigns. There are several reasons for this, including thepresence of finite time delays for control computationand the limitations of physical actuators. Since it isimpossible to switch the control at an infinite rate,chattering always occurs in the sliding and steady-statemodes of a VSC system. Chattering is almost alwaysobjectionable in robotic applications. Here we suggest anew approach to the design of independent VSC jointcontrollers. Besides the joint acceleration feedbackstructure and disturbance torque estimation, eachcontroller may possibly comprise elements of computedtorque structure. The salient feature of the proposedapproach is that the disturbance torque is effectivelytreated by a computationally straightforward procedure.

2. VARIABLE STRUCTURE CONTROL DESIGNVariable structure systems consist of a set of continuoussubsystems with a proper switching logic and, as a result,control actions are discontinuous functions of systemstate and reference inputs. The design of the controlsystem will be first demonstrated for a nonlinear systemin the form:

x=f(x,t, u); xeW, ue\ (2)

with control

/ = 1 (3)?(x,t) if a,(x)>0T(x,0 if a,(x)<0

where ffT(x) = (a i (x ) , . . . , om(x)) are continuousswitching functions. The system (2) with (3) is VSS andconsists of 2m subsystems. Its structure varies on msurfaces at the state space. The goal of the synthesis is tofind m switching functions, represented in vector formsas ff(r) and control u so that the motion of the system(2,3) reaches the manifold 5 in the state space in finitetime and stays in this mode.

The physical meaning of the above statement is asfollows:• Design a switching surface a(x) = 0 to represent a

desired system dynamics (namely linear) which is oflover order than the given plant.

• Design a variable structure control u(x, t) so that anystate x outside the switching surface is driven to reach

the surface in finite time. On the switching surfacesliding mode takes place, following the desired systemdynamics. In this way, the overall VSC system isglobally asymptotically stable.Whenever one seeks to establish a bridge between

theory and applications, it not longer suffices to ensurethe conditions for the existence of the sliding mode. Inreal physical systems, such as robots and servodrives, thepresence of measuring sensors, idle times due totransistor switchings, idle times due to computercalculation and effects of unmodelled dynamics, causeundesired chattering of the control.

These chatter oscillations are known to result in lowcontrol accuracy, high heat losses in electrical powercircuits and excessive wear of moving mechanical parts.These phenomena have been considered to be seriousobstacles for the application of sliding mode control. InJezernik9-10 practical experiments have shown that thechattering caused by unmodelled dynamics may beeliminated by an appropriate choice of the switchingfunction.

3. VARIABLE STRUCTURE CONTROL OFROBOT JOINTFor the development of the decentralized control schemeit is convenient to view each joint as a subsystem of theentire manipulator system, with these subsystemsinterconnected by "coupling torques" representing theinertial coupling terms and the Coriolis, centrifugal,friction and gravity terms in (1). The manipulatordynamic model (1) is then represented by a collection ofn second-order nonlinear scalar differential equations

Ji{q)qi + hi(q, q, q) = T, (5)

where the subscript / refers to the i-th element. Jt is theknown varying effective inertia at the i-th joint and isalways positive due to the positive-definiteness of D. SoJ(q) can be chosen as a constant diagonal matrix.

Equation (5) is the input-output dynamic model of thei-th joint (subsystem) with the joint torque T,(r) as theinput and the joint angle q-,{t) as the output. The term hh

given by (6), is treated as a "disturbance torque" by thei-th joint controller (i = 1 , . . . , n) and contains unknownparts of the inertial, gravity, friction, Coriolis andcentrifugal torques for the i-th joint, as well as theinertial coupling effects from the other joints

(6)Ci(q,q)+gi(q)+fXq)

The dynamics of a DC motor or a DC equivalent of anAC motor with resolved commutation and fieldgeneration, can be represented by a first orderdifferential equation

= Ui - e, (7)

where L, is the motor inductance and e, represents allvoltage drops from resistance, back EMF, and for ACmotors 'also equivalent voltages due to inexact



Sliding mode control 445

commutation, etc. Let us assume a linear connectionbetween the measurable equivalent current i, and thetorque T, (r, = Xm,i,). The controlled plant will berepresented by

I

9

_9_

X

=

0hq)-lKm

0

0

0

u

00

'nX/i

00

0_

I

q

.q.

0

00

(8)

In order to obtain smooth mechanical motion of therobot mechanism we prescribe a continuous trajectorywith values qf, qf and qf. The switching function whichdetermines the mechanical motion is chosen to be secondorder and a function of angular position, velocity andacceleration errors (for each joint) is

o, = (qf ~ qd + Kvl(qf - ?,) + Kpiiqf ~ <?.) (9)

where Kvi and Kpi are constants that determine, in thesliding mode (a, = 0), the damping and the maximumfrequency of the decentralized prescribed dynamics ofsecond order. For the practical control implementationthe measured quantities are state variables qt and q,. Theacceleration signal q, is not measurable and can beobtained by double differentiation of the angular positionq,\ but is contaminated by the measurement noise tosuch a degree that it can no longer be used.

4. ESTIMATION OF THE DISTURBANCEConsequently, the acceleration signal q, needs to bereplaced by an estimated value q, which is obtainedsimply from the differential equation of motion

T,-ft", (10)

where / is the mean inertia of the robot axes, r, is theactive measurable drive torque developed by theactivator and h, is the unknown value of the load torque.The expression (10) is inserted into the control schemeby replacing the real load torque /i, with an estimatedvalue ft",. An estimator of reduced order proposed byJezernik910 is

* > / / ( # "9/) (»)

where /, is a positive constant linked to the selecteddynamics of the asymptotic load observer. The calculatedangular acceleration signal qi is derived from (9), so thecondition for the sliding mode operation (a, = 0) of thesystem is fulfilled.

97 = qf + Kul(qf " 9/) + M 9 ? " 9,)

(13)jf = q] dv

As a result the control input is based on a modifiedswitching function which contains the estimated ac-celeration and the estimated disturbance torque

Ui={% T<1 (14)

W//////////////////////////////'.

U j

f ~ 9/) + r l F~ "

Fig. 1. The PI estimator

" 9/)

(15)

where the desired trajectories of angular position,velocity and acceleration are denoted by the superscriptd and fi{ is the estimated disturbance torque. The blockdiagram of the controller with the disturbance torqueestimator is shown in Figure 1. The asymptotic observerserves as a bypass for high frequency components,therefore the unmodelled dynamics is not exceeded.

If the load torque varies slowly (dhjdt = 0), then theswitching function erf converges to the function a, withdesired dynamics of first order. Let us present theequation (15) in form (16).

I,

A/l,- = ft, ~ ^

(16)

(17)

It is easy to see that (of = 0) and (a, = 0) defines thesame manifold if A/i, = 0. However if A/i, converges to 0then the manifold 5 ' = {(q, q, i): a* = 0} reaches themanifold S = {(y, q, i): a = 0). Let us present, that anestimator (11) assures this and let us choose theLyapunov function candidate

A/i, A/i,

2(18)

V = Aft, A/i, (19)

For the system on S' we obtain from (11,17) Ah\:

/i,- = /,(# - q,) - ft, = l,(ql - 9?" j k * > *,))A/i,-

With the substitution of (20) in (19) we get

ft,4•I i

(20)

(21)

V has always a negative value, so the system in thesliding mode on S' converges to S, where it has theprescribed dynamics of second order. However, the




dynamic of /-th subsystem in the sliding mode on Se

represents a local tracking error space system

00

-liKpi/Ji -Kpi -

~xfi +

00

-III

10l.KJl

hi

01

-Kvi -

where

Xj\ t) =

The poles of the system are

I,

(22)

(23)

(24)

(25)

(26)

The global reaching condition, that the system (8) withthe control law (14,15) and with the PI disturbanceestimator (11,13) reach the manifold S' in finite time isguaranteed if llf and U~ satisfy the condition (27)

ur + v,<urt<ur-vr, »?,>o, (27)where13 the equivalent control f/fq is defined as thecontrol voltage which assures (jf = 0,

(l{q)Kpi

(28)

5. SIMULATION RESULTSSimulations have been done to verify the proposed VSCjoint controller to compensate unstructured uncer-tainties. A two degree of freedom SCARA manipulatorwas used in the simulation.

The desired trajectory for each joint is

t>t,

^[l-cos(Qr)]

A.q2jz

K s i.0

t>t,

t>t.

(29)

(30)

(3D

The desired trajectory q\ is shown in Figure 2. Thevariation in the moment of inertia {dn{q)) from itsnominal value to triple the nominal value is presented in

qf[rad]

t[s]

-0.5

-1.0

-1.5

Fig. 2. The desired trajectory

1 . 0

o.o o. 5 1 . 0 1.5 2 . 0

Fig. 3. The varied moment of inertia.

Figure 3. We have studied the behaviour of two types ofreduced order disturbance observers,a) PI estimator

jq1 = jq1dv

b) linear disturbance observer14

(32)

(33)

(34)

(35)

(36)

(37)

The same testing procedure was used for the controllerwith the PI estimator and the linear observer. Thedisturbances, i.e. the varied load torque (ht(t)) and itsestimated vaue (/ii(f))> are presented in Figure 4 for thePI estimator and in Figure 5 for the linear observer. Theload torque varies from zero to the nominal value.Figures 6 and 7 show the computed current for the PIestimator and linear observer. The tracking errors of thePI estimator and linear observer are compared in Figure8. The nominal value of joint inertia 7, = /" o m . Poles inthe sliding mode are p , = -500, p2 = P3= -25. Theacceleration controller output was calculated every0.5 ms. The steady state error is compensated and thedynamic error is also asymptotically stable without any



Sliding mode control 447

5 . 0

Fig. 4._ PI estimator: the load torque h,(t) and its estimatedvalue hi(t).

o. 0006

o.

-0 . 0006

-0.0012

d-qj [rad]

2.0 >

Fig. 8. Tracking error: a-PI estimator, b-linear observer.

20.0

15.0

10. 0

5 . 0

Fig. 5. Linear observer: the load torque h,(t) and its estimatedvalue £|(r).

45. 0

30.0

15.0

0.

ii |

/

0

A]

a.

Ik\s

w1.

/

/

0

_J

I. 5 2 . 0 t[»]

Fig. 6. PI estimator: the computed current i'.

if [A]45.0

30.0

15.0

0|0 0.5 1.0 1.5 2.0

Fig. 7. Linear observer: the computed current i\.

restriction for the PI estimator. A major feature of thenew controller is its inherent ability to reject payloaduncertainty. VSC with the disturbance estimator is alsoable to solve efficiently the tracking tasks in high speedand direct drive robots.

6. CONCLUSIONSThe desired motion control algorithm consists of theacceleration feedback and disturbance torque estimation,and assures a good dynamic performance even in thepresence of an initial conditions mismatch, parameterperturbations and disturbances. The chattering caused byunmodelled dynamics was eliminated by the use of PIload estimator.

Due to the structural properties, the direct use of thetheory of VSC cannot solve all robotics problemsregarding insensitivity to parameter and disturbancevariations. It has been found to be necessary to completethe appended on-off controller with an asymptoticobserver to estimate the disturbance torque. In this wayit is possible to achieve the sliding mode conditions in thevicinity of the desired trajectory by introducing localconditions. Tracking errors are controlled and thenonlinear dynamic system is asymptotically stable.

References1. T.J. Tarn, A.K. Begezy, A. Isideru & Y.L. Chun,

"Nonlinear feedback in robot arm control" Proc. IEEEConf. on Decision and Control (1984) pp. 736-751.

2. A. Isidori, Nonlinear Control Systems: An Introduction(Second Edition, Springer-Verlag, Berlin, 1989).

3. J.J. Craig, Adaptive Control of Mechanical Manipulators(Addison-Wesley, Reading, Maryland, 1988).

4. K.D. Young, "Controller design for a manipulator usingtheory of variable structure systems" IEEE Trans. Sys.,Man. and Cyber. SMC-8, 101-109 (1978).

5. E. Bailey & A. Arapostathis, "Simple sliding mode controlscheme applied to robot manipulator" Int. J. Control 45,1197-1209 (1987).

6. S.W. Wijesoma, "Robust trajectory following of robotsusing computed torque structure with VSS" Int. J. Control52, 935-962 (1990).

7. S.K. Singh, "Decentralized variable structure control fortracking in non-linear systems" Int. J. Control 52, 811-831(1990).

8. H. Hashimoto, H. Yamamoto, S. Yanagisawa & F.Harashima, "Brushless servo motor control using variable




structure approach" IEEE Tr. Ind. App. 24, 160-170(1988).

9. K. Jezernik, J. Harnik & B. Curk, "Variable structurecontrol of AC servo motors used in industrial robots"Proc. First IEEE International Workshop on VariableStructure Systems and their Applications, Sarajevo (1990)pp. 139-148.

10. K. Jezernik, B. Curk & J. Harnik, "Variable structure fieldoriented control of an induction motor drive" 4thEuropean Conference on Power Electronics andApplications, Firenze, I (1991) pp. 2.161-2.166.

11. J.X. Xu, H. Hashimoto, J.J. Slotine, Y. Arai & F.Harashima, "Implementation of VSS control to roboticmanipulators-smoothing modification" IEEE Trans. Ind.Electron. 36, 321-329 (1989).

12. J.J. Slotine & S.S. Sastry, "Tracking control of non-linearsystems using sliding surfaces, with application to robotmanipulators" Int. J. Control 38, 465-492 (1983).

13. V.I. Utkin, Sliding Mode and their Applications in VariableStructure Systems (MIR Publishers, Moscow, 1978).

14. A. Sabanovic\ F. Bilalovic, "Sliding mode control of ACdrives" IEEE/AS Ann. Meet., Denver (1986) pp. 50-55.


observer-based sliding mode control of a robotic manipulator

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