IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 5, SEPTEMBER 2000 777

An Experimental Study of Adaptive Force/PositionControl Algorithms for an Industrial Robot

Luigi Villani , Member, IEEE, Ciro Natale, Student Member, IEEE, Bruno Siciliano, Fellow, IEEE, andCarlos Canudas de Wit

Abstract—This work presents the results of an experimentalstudy of force/position control algorithms for an industrial robotwith open control architecture, equipped with a wrist force sensor.The robot’s end effector is in contact with a compliant surface ofunknown stiffness. Two types of control schemes are consideredwhich respectively achieve regulation and tracking of position onthe contact surface and force along the direction normal to the sur-face. The desired position is scaled along the constrained directionso as to fulfill the force control objective; adaptive mechanisms aredesigned for the two schemes to compute the scaling factor, whichdepends on the unknown stiffness coefficient of the environment.A number of case studies are developed throughout the paper toillustrate the effectiveness of the force/position control algorithmsunder nonideal conditions.

Index Terms—Adaptive control, force control, position control,robots, robot dynamics, robot sensing systems.

I. INTRODUCTION

T HE use of force feedback is well known to enhance theperformance of control schemes for a robot interacting

with the environment [1]. Several strategies have been proposedto achieve control of the contact force, besides control of theend-effector position [2], among which hybrid position/forcecontrol [3], inner–outer position/force control [4], and parallelforce/position control [5] are worth mentioning.

Modeling the environment constitutes an essential require-ment to design an interaction control scheme, analyze its sta-bility, and ultimately test its performance in simulation. To thispurpose, it is opportune to distinguish between the case of rigidenvironment and that of compliant environment. In the former,kinematic constraints are imposed to the robot motion whichin turn can be described by a reduced number of task coordi-nates. In the latter, instead, a relationship exists between forceand motion along certain directions which is characterized bythe compliance at the contact.

A conceptually new approach coping with the force/positioncontrol problem in the presence of a compliant environmenthas been proposed recently [6]. The control law derives from

Manuscript received April 27, 1998; revised March 1, 1999 and November 1,1999. Recommended by Associate Editor, F. Ghorbel. This work was supportedin part by MURST and ASI.

L. Villani and B. Siciliano are with the PRISMA Lab, Dipartimento di In-formatica e Sistemistica, Università de Napoli Federico II, 80125 Napoli, Italy(e-mail: lvillani@unino.it; siciliano@unina.it).

C. Natale is with the Dipartimento di Ingeneria dell’Informazione, SecondaUniversità di Napoli, 81031 Aversa, Italy (e-mail: cinatale@unina.it).

C. Canudas de Wit is with the Laboratoire d‘Automatique de Grenoble,ENSEIG-INPG, 38402 Saint Martin d’Héres, France (e-mail: canudas@lag.en-seig.inpg.fr).

Publisher Item Identifier S 1063-6536(00)05737-7.

an impedance control scheme [7] where the reference positiontrajectory is adaptively scaled so as to fulfill the force con-trol objective. This approach, originally developed for a one-degree-of-freedom linear case, has been extended in [8] to athree-degree-of-freedom task for a robot manipulator in contactwith an elastically compliant surface. Tracking of the end-ef-fector position on the surface and tracking of the contact forceis achieved with exponential stability.

On the other hand, if only regulation of position and force isdesired, a computationally lighter scheme has been proposed [9]which ensures asymptotic stability without requiring computa-tion of the manipulator dynamic model but the gravity term.

The present paper reports the results of an extensive exper-imental study of both types of force/position control schemesfor an industrial robot. A key feature of the work is the avail-ability of a PC-based open control architecture which permitsimplementing model-based torque control algorithms as well asinterfacing a wrist force sensor. The effectiveness of the algo-rithms is demonstrated when the robot’s end effector is in con-tact with compliant surfaces of regular curvature.

II. M ODELING

Consider a rigid robot manipulator whose end effector is incontact with a frictionless elastic surface. Assuming bilateralpoint contact, the surface can be modeled as the twice-differ-entiable scalar equation [10]

(1)

where denotes the (3 1) vector of end-effector position andparameterizes the surface deflection [11].The (3 1) vector of the contact force exerted by the end

effector on the environment has the form

(2)

where

(3)

is a vector orthogonal to the surface at the contact point.A simple elastic model is assumed for the scalar component

of the contact force along, i.e.,

(4)

where is the stiffness coefficient.

1063–6536/00$10.00 © 2000 IEEE

778 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 5, SEPTEMBER 2000

A suitable set of task space coordinates can be introduced byconsidering the vector

(5)

where is a two-time differentiablevector such that the Jacobian of the mapping (5) is non-singular in a domain of the robot workspace. The vectorcanbe decomposed as

(6)

where is a scalar and is a (2 1) vector; in the sequel,subscripts and are used to denote, respectively, the com-ponents of a vector in the one-dimensional force task subspaceand in the two-dimensional position task subspace.

By virtue of the force/velocity duality principle, the resultingcontact force in task coordinates is

(7)where (2) has been used, and is a constant unitvector. By taking into account (1) and (5), (4) can be rewrittenas

(8)

where the undeformed rest position of the surface has been takento be zero without loss of generality.

For a three-joint nonredundant robot manipulator, the vectorof task coordinates constitutes a set of Lagrangian coordinatesin a domain of the task space. The manipulator dynamic modelin task coordinates has the form

(9)where is the (3 3) inertia matrix, is the (3 1)vector of Coriolis and centrifugal forces, is the (3 1) vectorof gravitational forces and is the (3 1) vector of drivingforces. The vector of the joint actuating torques can be com-puted as

(10)

where is the (3 1) vector of joint variables and isthe Jacobian of the mapping which is assumed to benonsingular.

The following well-known properties of the task space dy-namic model (9) can be established.

Property 1: is a symmetric and positive definite ma-trix satisfying

(11)

where and .Property 2: There exists a choice of the matrix

such that the matrix is skew-symmetric. Fur-ther, the following inequality holds:

(12)where .

III. EXPERIMENTAL SETUP

The setup available in the PRISMA Lab consists of an indus-trial robot Comau SMART-3 S. The robot manipulator has a six-

revolute-joint anthropomorphic geometry with nonnull shoulderand elbow offsets and nonspherical wrist. The joints are actuatedby brushless motors via gear trains; shaft absolute resolvers pro-vide motor position measurements. The robot is controlled by anopen version of the C3G 9000 control unit (Fig. 1), which hasa VME-based architecture with two processing boards (RobotCPU and Servo CPU) both based on a Motorola 68 020/68882,where the latter has an additional DSP and is in charge of tra-jectory generation, inverse kinematics and joint position servocontrol. Connection of the VME bus to the ISA bus of a stan-dard PC is made possible by two Bit 3 Computer bus adapterboards, and the PC and C3G controller communicate via theshared memory available in the Robot CPU [12]; at present, aPC Pentium/233 is used. Time synchronization is implementedby interrupt signals from the C3G to the PC with data exchangeevery 1 ms. A set of C routines are available to drive the busadapter boards.

The open version of the control unit allows seven different op-erational modes, including the standard mode available on theindustrial version of the controller. To implement force/positioncontrol schemes, the operational mode number 4 has been usedin which the PC is in charge of computing the control algorithmand passing the references to the current servos through the com-munication link at ms sampling rate.

A 6-axis force/torque sensor ATI FT30-100 with force rangeof 130 N and torque range of10 N m is mounted at themanipulator’s wrist. The sensor is connected to the PC by a par-allel interface board which provides readings of six componentsof generalized force atms. An end effector has been built as astick with a sphere at the tip, both made of brass.

For the purpose of this work, only the inner three joints areconsidered and the outer three joints are mechanically braked.Three-degree-of-freedom tasks are considered, involving end-effector position and linear force.

Meaningful experiments have been conceived to quantify thevalidity of the force/position control algorithms in the next twosections. Two types of surface with regular curvature have beenconsidered; namely, a sphere and a plane. For each surface,two case studies have been chosen: one is aimed at verifyingthe capability of tracking a desired contact force profile whilemaintaining the same end-effector position; the other is aimedat testing the performance of the system when an end-effectormotion is commanded together with a contact force profile. Ex-periments involving transition from noncontact to contact withthe surface are also presented.

The experimental tests of the force/position control algo-rithms to follow have been carried out in nonideal conditionsbecause of inaccurate geometric model of the compliantsurface, uncertainty on the robot dynamic model, unmodeledcontact friction, and absence of joint velocity measurementswhich are reconstructed through numerical differentiation ofjoint position readings.

A. Sphere

The spherical surface consists of an inflated rubber ball of10-cm radius fixed to a table, where the stiffness is about 5000

VILLANI et al.: EXPERIMENTAL STUDY OF ADAPTIVE FORCE/POSITION CONTROL ALGORITHMS 779

Fig. 1. Schematic of open control architecture.

N/m (Fig. 2). With reference to (1), the model of the surface isdescribed by

(13)

where is the radius of the undeformed sphere andis theposition vector of the center of the sphere. Because of the geom-etry of the contact task, spherical coordinates are adopted, i.e.,the task coordinates in (6) are given by

(14)

Atan

Atan(15)

where , , are the components of the vector .Two case studies are considered for this environment; namely,

one where a time-varying force is commanded while keeping thesame contact point, and another where a time-varying positionis commanded in addition to the force.

Case Study 1:A desired force profile is assigned along thenormal to the sphere (corresponding to the task coordinate),where the edges are generated according to a linear interpo-lating polynomial with fourth-order blends and a duration of 0.8s (Fig. 3). The desired position is kept constant.

Case Study 2:A desired force profile is assigned as in CaseStudy 1. A desired end-effector path (Fig. 3) is assigned along aparallel on the sphere at latitude giving a longitude displace-ment of (corresponding to the first task coordinate of);the trajectory on the path is generated according to a fifth-orderpolynomial with null initial and final velocity and acceleration,and a duration of 6 s.

Fig. 2. Experimental setup.

(a) (b)

Fig. 3. Desired force profile (a) and end-effector path (b) in the case of contactwith the sphere.

B. Plane

The planar surface consists of a wooden table whose stiffnessis about 10000 N/m. With reference to (1), the model of thesurface is described by

(16)

where denotes the (3 1) vector of the position of the un-deformed plane and is the constant unit vector orthogonal tothe plane. Therefore, by introducing two constant unit vectors

and so as to complete a right-handed frame with, thetask coordinates in (6) are given by

(17)

(18)

where , , ,are the components of the vectors , , , , respectively.

780 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 5, SEPTEMBER 2000

As above, two case studies are considered for this environ-ment.

Case Study 3:A desired force is assigned according to theprofile in (Fig. 3). The desired position is kept constant.

Case Study 4:A desired force profile is assigned as in CaseStudy 3. A desired end-effector path is assigned alongwith adisplacement of 0.28 m; the trajectory on the path is generatedaccording to a fifth-order polynomial with null initial and finalvelocity and acceleration and a duration of 8 s.

IV. FORCE AND POSITION REGULATOR

A. Design

In this section a force/position regulator is derived from aposition regulator. The idea is that of scaling the component ofthe constantdesired position along the direction normal tothe surface in order to achieve regulation to aconstantdesiredforce [9]. Let

(19)

denote ascaledposition trajectory, where is the scaling factor.Consider the PD control law with gravity compensation and

desired force feedforward

(20)

where block diag andblock diag are positive definite matrices. It iseasy to show that control law (20) asymptotically stabilizessystem (9) around the equilibrium , , i.e.,

and .If the stiffness coefficient of the environmentwere known,

the scaling factor should be taken as

(21)

so that gives .When the stiffness coefficient is unknown, the scaling factorin (19) can be replaced with an estimate which has to be

suitably adjusted as a function of time. Let

(22)

denote the time-varying scaled position when . Then,consider the control law

(23)

The following theorem holds, whose proof is omitted forbrevity and is conceptually similar to that in [9] where the caseof scalar gains was considered.

Theorem 1: Consider system (9) under control law (23). Fur-ther, consider the adaptive law

(24)

(25)

with , , . If the matrix gains , and satisfythe inequalities

(26)

(27)

where is so that , and are definedas in (11) and (12), respectively, then the equilibrium point

, , is asymptotically stable.Remark 1: The scheme achieves force regulation thanks to

a nonlinear force PI action with force feedforward, as can berecognized by integrating (24) and folding it into (23).

B. Experiments

In order to test the performance of the above force/positioncontrol algorithm, the four case studies described in Section IIIhave been executed. It should be noticed that, even though thegoal is just to regulate the force and position to constant desiredvalues, the choice of time-varying trajectories allows analyzingthe behavior of the system not only at steady state but also duringthe transient.

The inequalities in (26) and (27) only give sufficient condi-tions for stability of the closed-loop system, with no concernabout the actual transient. Hence, the choice of the control pa-rameters in (23)–(25) has been made by referring to the lo-cally linearized system about the desired equilibrium. In partic-ular, by considering the constant block diagonal matrixblock diag that can be extracted from the inertiamatrix and neglecting the Coriolis and centrifugal forcevector , the linearized dynamics is described by

(28)

(29)

where , , are taken as diagonal matrices. Then,a pole placement technique has been adopted to find the initialguess for the control parameters so as to achieve a satisfactorydynamic behavior.

Such values have finally been adjusted as a result of exten-sive MATLAB simulations and preliminary experimental testsfor the first type of environment (the sphere), leading todiag , diag ,

, , . Note that has been set equalto the estimated value of the stiffness coefficient. Also, the ini-tial value of the variable has been set to zero, correspondingto contact with the undeformed surface.

The results for Case Study 1 are reported in Fig. 4. From thetime history of the contact force it can be seen that tracking ofthe desired value is achieved with a bounded error; a delay oc-curs, but it should be pointed out that the scheme was thoughtof as a regulator, i.e., not to track the desired force profile. As

VILLANI et al.: EXPERIMENTAL STUDY OF ADAPTIVE FORCE/POSITION CONTROL ALGORITHMS 781

expected, good force regulation is achieved, thanks to the pres-ence of an integral action on the force error. Since no integralaction on the position error is present, small errors are left atsteady state for the latitude and longitude; these are due mainlyto robot unmodeled dynamic effects, e.g., joint friction. On theother hand, the reference currents to the joint motors reveal amoderate control effort.

The results for Case Study 2 in Fig. 4 show that force andposition regulation continues to be satisfactory. Compared tothe above, however, the position error along the longitude is in-creased during the transient while that along the latitude is prac-tically the same; this is an obvious consequence of the desiredend-effector trajectory (time-varying longitude and constant lat-itude). The visible oscillations on the measured force as wellas on the position errors can be ascribed to the contact frictioncaused by end-effector rubbing on the surface. The control ef-fort remains moderate.

With reference to the second type of environment (the plane),the same control parameters as above have been chosen to testthe robustness of the scheme to contact with a stiffer surface(estimated to be twice as much as above). Notice, however, thatthe use of the task space coordinates in (17), (18) in lieu ofthose in (14), (15) introduces a factor of 100 in the elementsof which has to be accounted for by introducing the samefactor in the parameters , which then becomediag , diag . All the otherparameters are unchanged.

The results for Case Study 3 are reported in Fig. 5. From thetime history of the contact force it can be seen that tracking ofthe desired value is much better than Case Study 1, thanks toan equivalent larger bandwidth of the force control loop. Again,good force regulation is achieved. On the other hand, the posi-tion errors are small although a direct comparison with those inFig. 4 is not meaningful due to the different units.

The results for Case Study 4 in Fig. 5 reveal that, because ofthe larger bandwidth, oscillations affect the time history of thecontact force to a greater extent than Case Study 2; as above,these are induced by the time-varying end-effector trajectorythrough the contact friction.

To mitigate this phenomenon, the control parameterhasbeen reduced to and both Case Studies 3 and 4 havebeen executed again. The results in Fig. 6 evidence the presenceof a delay on the time history of the contact force for Case Study3, which is analogous to that appearing in Fig. 4. The advantagefor Case Study 4, though, is a reduction of the oscillations onthe force.

In all the above tests, the end effector has always been placedinitially in contact with the environment. Case Study 3 has beenexecuted once more to test the performance of the scheme whenthe end effector is not initially in contact with the environment,i.e., at a distance of 0.05 m from the plane. The control param-eters have been chosen as in the last experiment. The results inFig. 7 show that after an initial lapse the contact force tracksthe desired profile with a behavior similar to that in Fig. 6; suchlapse occurs during the approach phase to the surface which ismerely governed by the desired force profile, since no end-ef-fector trajectory toward the surface is commanded. On the otherhand, the position errors remain small.

Fig. 4. Performance with force and position regulator (left: Case Study 1;right: Case Study 2; dashed: desired; solid: measured).

Fig. 5. Performance with force and position regulator (left: Case Study 3;right: Case Study 4; dashed: desired; solid: measured).

782 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 5, SEPTEMBER 2000

(a) (b)

Fig. 6. Performance with force and position regulator in Case Study 3 andsmaller bandwidth for the force control loop (dashed: desired; solid: measured).

(a) (b)

Fig. 7. Performance with force and position regulator in Case Study 3 andinitial noncontact (dashed: desired; solid: measured).

V. FORCE AND POSITION TRACKING CONTROLLER

A. Design

The previous control algorithm achieves regulation to con-stant force and position. If the motion of the end-effector alongthe surface is relevant and it is desired to vary the amount ofcontact force, a computationally more demanding scheme canbe devised which ensures force and position tracking. As in theregulation case, the force/position controller derives from a po-sition controller, where the component of thetime-varyingde-sired position trajectory along the direction normal to thesurface is scaled in order to achieve tracking of atime-varyingdesired force trajectory , i.e.,

(30)

where is a time-varying scaling factor.Consider the classical Slotine and Li [13] algorithm for posi-

tion control

(31)

where is chosen so that the matrix is skew-sym-metric and is a positive definite matrix; the vectorsandare given by

(32)

(33)

where and is a diagonal and positive definitematrix. Splitting (33) in the force and position task subspacesand using (30) gives

(34)

(35)

with obvious meaning of notation. By using passivity arguments[14] it is easy to show that and . Equation (35)implies that the position error can be obtainedas the output of the strictly proper, exponentially stable transfermatrix — denotes the Laplace variable—whoseinput is ; therefore [15] as . As forthe force task subspace, a scaling factor in (34) should besought so that gives as .

If the stiffness coefficient of the environmentwere known,it can be shown [16] that if is redefined as

(36)

with , then should be the solution of thedifferential equation

(37)

Substituting the solution to (37) into (34) gives

(38)

where is the force error; thus, from andit follows that and as

.When the stiffness coefficient is unknown, the solution to

(37) cannot be computed. By replacing with an estimatein (30), the control law (31) can be rewritten as

(39)

where

(40)

and . By defining and accounting for (37),the vector can be decomposed as

(41)

where is given in (38). By using passivity arguments, it canbe shown that and . This result stillensures convergence of the position error in the position tasksubspace ( ), but it does not guarantee convergence ofthe force error. The key point is to find an adaptive algorithm for

which achieves for the closed-loop system (9), (39),(40). Equation (37) suggests computing from the equation

(42)

VILLANI et al.: EXPERIMENTAL STUDY OF ADAPTIVE FORCE/POSITION CONTROL ALGORITHMS 783

where is an estimate of and

(43)

By virtue of (42), the vector in (40) can be computed as

(44)

The following theorem holds, whose proof is omitted forbrevity but can be found in [8].

Theorem 2: Consider system (9) under control law (39), withgiven in (44), and block diag . Further,

consider the adaptive law

(45)

with . Then , , all the signals in the closed-loop

system are bounded, and the tracking errors, , , tendasymptotically to zero. If, in addition, the matrix is diag-onal and there exist constants , and suchthat

(46)

then the equilibrium point isexponentially stable.

Remark 2: The proposed control law is given by the set of(39), (44) and (45); computation of gives

(47)

with

(48)

Hence, the original position control scheme achieves forcetracking capabilities by using nonlinear PD position feedbackand time-varying PID force feedback; as can be seen from (45),the force time derivative is required to implement the controllaw.

B. Experiments

In order to test the performance of the above force/positioncontrol algorithm, the four case studies described in Section IIIhave been executed.

As done for the force and position regulator, the choice ofthe control parameters in (39), (44) and (45) has been made byreferring to the locally linearized system about the desired equi-librium. Taking for a constant , and proceedingas above leads to the linearized dynamics

(49)

(50)

where , are taken as diagonal matrices. Again, a poleplacement technique has been adopted to find the initial guessfor the control parameters. Notice that the poles of the forceloop can be easily computed, which then facilitates the choice ofthe control parameters with respect to the third-order dynamicsdescribed by (28).

A fine tuning has been accomplished as a result of exten-sive MATLAB simulations and preliminary experimental testsfor the first type of environment (the sphere). The resulting pa-rameters are , diag , ,

diag , .The initial value of the variablehas been set to ,

corresponding to an estimated stiffness coefficient of 2000 N/m;notice that a large discrepancy has been intentionally introducedwith respect to the actual stiffness of about 5000 N/m in orderto test the effectiveness of the adaptive algorithm.

In the experimental tests, the force time derivative neededin (45) is computed by using the first-order filter of transferfunction with s; this value has been tunedas a result of a tradeoff between time delay and noise reduction.The analog filter is then converted to a digital filter with theEuler rule, i.e.,

(51)

where is the sampling interval. Nonetheless, it should beremarked that a force sensor with high signal-to-noise ratio isavailable for the experiments.

The results for Case Studies 1 and 2 are reported in Fig. 8.The time history of the measured contact force shows that, aftera transient due to initial stiffness coefficient mismatching, goodforce tracking performance is achieved in both cases. Note alsothat the variable tends to a value close to the inverse of the bestavailable estimate of the stiffness coefficient, thus confirmingthe theoretical findings. Clearly, compared to the previous reg-ulator, the tracking performance is greatly improved. Small po-sition errors still occur at steady state for the latitude and lon-gitude, but this does not affect the force steady-state perfor-mance thanks to the integral action on the force error in (48).Further, for Case Study 2, good position tracking performanceis achieved, although oscillations are still present on both mea-sured force and end-effector position because of the rubbing onthe surface.

With reference to the second type of environment (the plane),the same control parameters as above have been chosen to testthe robustness of the scheme to contact with a stiffer surface. Be-cause of the same factor of 100 pointed out in Section IV-B, theparameters has been scaled to diag .All the other parameters are unchanged.

The results for Case Studies 3 and 4 are reported in Fig. 9. Thetime history of the contact force and the position errors showthat the tracking performance remains satisfactory. Interestinglyenough, a larger overshoot initially occurs on the force due tothe actual higher stiffness, andconverges to a value very closeto the best available estimate of the stiffness coefficient for bothcase studies.

Case Study 3 has been executed again to probe into the effectsof overestimating the stiffness coefficient rather than underesti-mating it as above. Hence, the initial value ofhas been setto 1/15 000 and 1/20000, respectively. In both cases, the resultsin Fig. 10 show that the overshoot disappears due to the actuallower stiffness, while convergence to the best available estimateof the stiffness coefficient continues to hold, remarkably.

Finally, Case Study 3 has been executed once more to testthe performance of the scheme when the end effector is at a

784 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 5, SEPTEMBER 2000

Fig. 8. Performance with force and position tracking controller (left: CaseStudy 1; right: Case Study 2; dashed: desired; solid: measured).

distance of 0.05 m from the plane. The control parameters havebeen chosen as above with . The results in Fig. 11show that the steady-state performance is good but the transientis affected by large errors and oscillations; also, in this case,does not converge to the estimated value.

VI. FORCE AND POSITION TRACKING CONTROLLERWITHOUT

FORCEDERIVATIVE

A. Design

A modification of the force/position tracking controller hasbeen recently proposed [17] which avoids the force time deriva-tive. This could be useful whenever force measurements are af-fected by significant noise, since the computation of the force

time derivative needed byin (47) could be troublesome.The new adaptive controller is defined as

(52)

Fig. 9. Performance with force and position tracking controller (left: CaseStudy 3; right: Case Study 4; dashed: desired; solid: measured).

Fig. 10. Performance with force and position tracking controller in Case Study3 with overestimated stiffness coefficient (left:�̂ = 1=15000; right: �̂ =

1=20000; dashed: desired; solid: measured).

VILLANI et al.: EXPERIMENTAL STUDY OF ADAPTIVE FORCE/POSITION CONTROL ALGORITHMS 785

(a) (b)

Fig. 11. Performance with force and position tracking controller in Case Study3 and initial noncontact (dashed: desired; solid: measured).

with in (44), , and

(53)

where the variable is computed according to (48).The following theorem holds, whose proof is omitted for

brevity but can be found in [17].Theorem 3: Consider system (9) under control law (52), withgiven in (44), given in (53), given in (48), and

block diag . Further, consider the update laws

(54)

(55)

with , , where is the first columnof the inertia matrix . Then , , all the signals inthe closed-loop system are bounded, and the tracking errors,

, , tend asymptotically to zero.Remark 3: The problem of the force time derivative is con-

cerned with computation of (45). In view of this, a modifiedupdate law (54) has been derived on the basis of (8) which gives

, where the estimate of the unknown stiffness coef-ficient has been updated according to (55).

B. Experiments

Case Study 2 has been executed to test the effectiveness ofthe adaptive force/position controller without force derivative.Besides choosing the same , , and as above, thecontrol parameter in (55) has been set to . Theresults in Fig. 12 show that there is no significant degradationof the force and position tracking performance. Therefore, thiscontroller may represent a valid alternative to the original algo-rithm in case that force measurements would be too noisy.

VII. CONCLUSION

Two force/position control schemes for an industrial robot incontact with an elastically compliant curved surface have beenexperimentally tested in this work.

The force control objective has been obtained starting froma position controller, by properly scaling the original desired

Fig. 12. Performance with force and position tracking controller without forcederivative in Case Study 2 (dashed: desired; solid: measured).

end-effector position. Adaptive algorithms have been designedto compute the time-varying scaling factor, whose value de-pends on the unknown stiffness of the environment. For theforce and position regulation problem, a linear controller withgravity compensation has been used. On the other hand, for theforce and position tracking problem, a model-based controllerhas been used where the need of the force time derivative canbe avoided if desired.

The results in a number of case studies have permitted acritical comparison of the controllers when implemented on anindustrial setup under nonideal conditions. Clearly, the on-linecomputational burden is lower for the linear controller withgravity compensation, although tuning of the control param-eters is more straightforward for the model-based controller.Both controllers exhibit very good steady-state performance.As foreseen in theory, the tracking performance is better forthe model-based controller. Robustness tests to inaccurateestimate of the stiffness coefficient have revealed the superiorperformance of the model-based controller, and in particular aninitial overestimate of the stiffness coefficient produces betterresponses. Also, robustness tests to initial noncontact haverevealed the superior performance of the linear controller withgravity compensation.

The experimental study has confirmed that the controllerswork well in the case of contact with surfaces of unknownmedium stiffness (5000 to 10 000 N/m). For higher stiffness,the experimental results were less encouraging and have notbeen reported here. This fact should not surprise since theproposed schemes achieve control of the contact force throughthe local deformation of the surface. When such deformationbecomes negligible, instead, different control strategies shouldbe sought where the surface has to be modeled as a rigid(infinitely stiff) constraint.

ACKNOWLEDGMENT

The authors would like to thank the Associate Editor and theanonymous reviewers for helpful and constructive comments.

786 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 5, SEPTEMBER 2000

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Luigi Villani (S’94–M’96) was born in Avellino,Italy, in 1966. He received the Laurea degreein electronic engineering from the University ofNaples, Italy, in 1992. In 1996, he received theResearch Doctorate degree in electronic engineeringand computer science from the University of Naples.

Since then, he has been with the University ofNaples, where he is currently an Assistant Professorin the Department of Computer and SystemsEngineering. He was recently promoted to AssociateProfessor. From June to October 1995, he was a

Visiting Scholar at the Department of Automatic Control, National PolytechnicInstitute of Grenoble, France. His research interests include force/motioncontrol of manipulators and adaptive and nonlinear control of mechanicalsystems. He is coauthor of the booksSolutions Manual for Modeling andControl of Robot Manipulators(New York: McGraw-Hill, 1996; 2nd ed.,London, U.K.: Springer-Verlag, 2000) andRobot Force Control(Boston, MA:Kluwer, 1999).

Dr. Villani is a Member of the Conference Editorial Board of the IEEE Con-trol Systems Society.

Ciro Natale (S’ 94) was born in Caserta, Italy, in1969. He received the Laurea degree in electronic en-gineering from the University of Naples in 1995. Hereceived the Research Doctorate degree in electronicengineering and computer science from the Univer-sity of Naples in 2000.

From November 1998 to May 1999, he was a Vis-iting Scholar at the Institute of Robotics and SystemDynamics, DLR (German Aerospace Center). He iscurrently a Research Associate in the Department ofInformation Engineering at the Second University of

Naples. His research interests include force/motion control of robot manipula-tors, nonlinear control, passivity-based adaptive control, visual servoing of in-dustrial robots, active damping of structural vibrations.

Dr. Natale received the 1996 UCIMU award for the section “Robots: Designand Applications” for his Laurea thesis.

Bruno Siciliano (M’91–SM’94–F’00) was born inNaples, Italy, in 1959. He received the Laurea degreeand the Research Doctorate degree in electronic en-gineering from the University of Naples in 1982 and1987, respectively.

Since then, he has been with the University ofNaples, where he is currently an Associate Professorof robotics in the Department of Computer andSystems Engineering. He was recently promoted toProfessor. In 1986, he was a Visiting Scholar at theSchool Mechanical Engineering, Georgia Institute

of Technology, Atlanta. His research interests include inverse kinematics,redundant manipulator control, modeling and control of flexible arms,force/motion control and cooperative robots. He is coauthor of the books:Modeling and Control of Robot Manipulatorswith Solutions Manual(NewYork: McGraw-Hill, 1996; 2nd ed., London, U.K.: Springer-Verlag, 2000),Theory of Robot Control(London, U.K.: Springer-Verlag, 1996),Robot ForceControl (Boston, MA: Kluwer, 1999), and is coeditor of the bookControlProblems in Robotics and Automation(London, U.K.: Springer-Verlag, 1998).He was an Associate Editor of the ASME JOURNAL OF DYNAMIC SYSTEMS,MEASUREMENT, AND CONTROL from 1994 to 1998. He is on the EditorialBoards of ROBOTICA and the JSME INTERNATIONAL JOURNAL.

Dr. Siciliano served as an Associate Editor of the IEEE TRANSACTIONS ON

ROBOTICS AND AUTOMATION from 1991 to 1994He is a Member of ASME.He has held representative positions within the IEEE Robotics and AutomationSociety: Administrative Committee Member from 1996 to 1999, Vice-Presidentfor Publications in 1999, and Vice-President for Technical Activities since 2000.From 1996 to 1999 he was Chair of the Technical Committee on Manufacturingand Automation Robotic Control of the IEEE Control Systems Society.

Carlos Canudas de Witwas born in Villahermosa,Tabasco, Mexico, in 1958. He received the B.Sc.degree in electronics and communications from theTechnological Institute of Monterrey, Mexico, in1980. He received the M.Sc. and Ph.D. degrees inautomatic control from the Polytechnic of Grenoble,France, in 1984 and 1987, respectively.

Since 1987, he has been working at the same de-partment as “Directeur de recherche” at the CNRS,where he teaches and conducts research in the area ofadaptive and robot control. He is also responsible for

a team on control of electromechanical systems and robotics. He teaches under-graduate and graduated courses in robot control and stability of nonlinear sys-tems. He was a Visiting Researcher in 1985 at the Lund Institute of Technology,Sweden. His research topics include adaptive control, identification, robot con-trol, nonlinear observers, control of systems with friction, ac and CD drives,automotive control, and nonholonomic systems. He has written the bookAdap-tive Control of Partially Known Systems: Theory and Applications(Amsterdam,The Netherlands: Elsevier, 1988). He also edited two books,Advanced RobotControl(New York: Springer-Verlag, 1991) andTheory of Robot Control, in theSpringer Control and Communication Series.

Dr. de Wit was Associate Editor of the IEEE TRANSACTIONS ONAUTOMATIC

CONTROL from January 1992 to December 1997 and has been Associate Editorof Automaticasince March 1999.