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Robustly Stable Adaptive Control of a Tandem ofMaster–Slave Robotic Manipulators With ForceReflection by Using a Multiestimation Scheme

1

2

3

Asier Ibeas and Manuel de la Sen4

Abstract—The problem of controlling a tandem of robotic ma-5nipulators composing a teleoperation system with force reflection6is addressed in this paper. The final objective of this paper is7twofold: 1) to design a robust control law capable of ensuring8closed-loop stability for robots with uncertainties and 2) to use the9so-obtained control law to improve the tracking of each robot to10its corresponding reference model in comparison with previously11existing controllers when the slave is interacting with the obstacle.12In this way, a multiestimation-based adaptive controller is pro-13posed. Thus, the master robot is able to follow more accurately14the constrained motion defined by the slave when interacting with15an obstacle than when a single-estimation-based controller is used,16improving the transparency property of the teleoperation scheme.17The closed-loop stability is guaranteed if a minimum residence18time, which might be updated online when unknown, between19different controller parameterizations is respected. Furthermore,20the analysis of the teleoperation and stability capabilities of the21overall scheme is carried out. Finally, some simulation examples22showing the working of the multiestimation scheme complete23this paper.24

Index Terms—Adaptive control, force reflection, multiple25model, stability, switching, teleoperation.26

I. INTRODUCTION27

T ELEOPERATION systems have been widely studied dur-28

ing the last decades because they can extend the human29

reach and manipulation capabilities to remote places and/or30

hazardous locations. Teleoperation strategies can be found in31

a broad variety of applications such as in space and undersea32

explorations, nuclear operations, or medical applications [1].33

Recently, as the development of the virtual reality is in34

progress, related application areas have been extended to the35

entertainment and training fields [2]. A teleoperation system36

consists of a human operator (which classifies this kind of37

systems as those named as “man in the loop systems” [3], [4]),38

a master manipulator, a remotely located slave manipulator,39

and an environment/obstacle situated in the slave manipulator’s40

Manuscript received May 10, 2005; revised November 8, 2005 and January18, 2006. This work was supported in part by the MEC and the UPV-EHU underProjects DPI 2003-00164 and 9/UPV 00I06.I06-15263/2003, respectively. Thework of A. Ibeas was supported by MECD under FPU Grant P2002-2727. Thispaper was recommended by Associate Editor M. S. De Queiroz.

The authors are with the Departamento de Ingeniería de Sistemas y Au-tomática, University of the Basque Country, Bilbao 48080, Spain (e-mail:iebibhea@ehu.es; msen@we.lc.ehu.es).AQ1

Digital Object Identifier 10.1109/TSMCB.2006.874693

location. The human operator applies his/her intentional force 41

to the master manipulator to make the slave to perform a series 42

of commands. If the slave manipulator exactly reproduces the 43

master’s motions and the master manipulator accurately reflects 44

to the master the measured slave force, then the operator should 45

experience the same sensation as the slave manipulator does. 46

In other words, the teleoperation system would be completely 47

“transparent” [5], [6]. In this way, the final objective of the 48

teleoperated system is to achieve a good transparency condition 49

while remaining stable. Hence, the control of a tandem of 50

master–slave manipulators with force reflection constitutes a 51

challenge for control theorist because it requires not only to 52

face the problem of controlling robot manipulators but also 53

to achieve an adequate position and force tracking proper- 54

ties. Thus, “adaptive controllers” are an interesting alterna- 55

tive for the design of high-performance robotic manipulator 56

controllers for teleoperation systems due to their capabilities 57

for adapting to changing operation conditions. Much effort 58

has been done in this direction (see, for instance, [7]–[11]). 59

However, adaptive controllers may lead to a poor closed-loop 60

response in terms of reference tracking deviations, [12]–[17], 61

especially when the slave is interacting with an environment. 62

This situation leads to a poor transparency in the teleopera- 63

tion system. In this paper, a multiestimation-based adaptive 64

controller architecture is proposed to improve the position 65

and force tracking properties of teleoperation systems with 66

force reflection. 67

Thus, the idea of multiestimation is linked in this paper 68

with that of teleoperation for nonlinear uncertain dynamical 69

systems, which have not been before addressed in a unified 70

way in the literature. In particular, the multiestimation scheme 71

improves the behavior of the adaptive impedance-based control 72

law when the slave robot interacts with the environment. In 73

this way, the master robot is able to track more accurately 74

the motion and force of the slave robot when interacting with 75

an environment while improving the transparency condition 76

in comparison with existing single-model-based adaptive con- 77

trollers. Thus, the multiestimation scheme is a powerful tool 78

to improve the contact process performance for interactions 79

ranging between smooth impacts, associated with relatively 80

soft environments, and nonsmooth impacts, associated with 81

environments with a large stiffness, with the stability be- 82

ing guaranteed. Hence, a framework where teleoperated ro- 83

bot controllers capacities are enhanced in comparison with 84

existing adaptive controllers described in the literature [2], 85

1083-4419/$20.00 © 2006 IEEE

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2 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 36, NO. 6, DECEMBER 2006

Fig. 1. Teleoperation system architecture.

[8], [10], [18]–[22] is proposed by taking advantage of the86

new design possibilities added by the switching and multiple-87

model-based techniques [13]–[15], [23], [24]. In the general88

problem setting, a number of estimation algorithms running89

in parallel are used, each one being initialized by different90

estimated parameter values. At each time instant, a higher91

level switching structure selects in real time the element from92

the convex hull of the whole set of estimated parameter vec-93

tors, which is then used to parameterize an adaptive interval94

controller. The values of the coefficients of such a convex95

linear combination are selected at each time instant based96

on the values of a set of performance indexes evaluating the97

identification performance of each estimator. In this way, the98

so selected estimated vector parameterizes an impedance-type99

adaptive control law. Furthermore, the proposed control law is100

“modular” in the sense that the stability properties of the101

scheme are maintained irrespective of the estimation algorithms102

used if they possess some basic asymptotic estimation proper-103

ties. This modularity is obtained by adding a novel correcting104

term to the standard feedback linearization impedance control105

law used in the literature [8], [10], [20], [21]. The stability of106

the teleoperation system is proved, and explicit conditions on107

the influence of the human operator in the stability properties108

are discussed.109

The paper is organized as follows. First, the problem descrip-110

tion and modeling issues are described in detail in Section II.111

Section III deals with the multiestimation scheme architecture.112

The design of the control law is addressed in Section IV.113

Section V discusses the stability properties of the overall114

scheme. In Section VI, the numerical examples of the teleoper-115

ation system are carried out. Finally, conclusions end the paper.116

II. PROBLEM DESCRIPTION AND MODELING ISSUES 117

A. Teleoperation System 118

The force–force teleoperation control scheme displayed in 119

Fig. 1 has been chosen to implement the teleoperation ar- 120

chitecture [5]. According to Fig. 1, the force made by the 121

operator fop is transmitted to the slave robot, and the force it 122

measures in its end effector is transmitted to the master one. 123

Thus, it is necessary to dispose a force sensor at the slave’s end 124

effector to transmit the interaction force with the environment 125

to the master robot. However, a model for the environment 126

and interacting force is considered in this paper for computer 127

simulation purposes. For each robot, a multiestimation-based 128

adaptive controller is synthesized. Each controller is parameter- 129

ized in an independent way according to its associate estimator 130

from the corresponding multiestimation scheme, and thus, they 131

can be tuned separately. In this paper, both robots will be of 132

revolute joints because the stability proof is devoted to this kind 133

of mechanical systems. 134

B. Plant Description 135

The scheme is composed of two robots, namely, master 136

and slave. The subscript = m, s will be used to denote the 137

master or slave manipulators, respectively. Thus, a general 138

robotic manipulator with n rigid degrees of freedom is char- 139

acterized by a set of n generalized coordinates collected in 140

the vector 141

qT = [q1 q2 · · · qn] .

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IBEAS AND DE LA SEN: ADAPTIVE CONTROL OF ROBOTIC MANIPULATORS WITH FORCE REFLECTION 3

Thus, the general dynamic equations of both the master and

AQ2

142

slave robotic manipulators are written in the joint space as143

M(q)q + C(q, q)q + F (q)+ G(q) + JT

(q)f ext = τ (1)

where M(q) ∈ Rn×n symmetric and positive definite is144

the so-called “inertia matrix,” and C(q, q)q ∈ Rn×1 is a145

vector containing the “Coriolis” and centripetal terms.146

Also, F (q) ∈ Rn×1 models the friction phenomena,147

G(q) ∈ Rn×1 is the gravity vector, and J ∈ R

n×r is the148

“Jacobian” of the manipulator, where r is the number of149

degrees of freedom of the manipulator in the task space. The150

external forces (and torques) are f ext ∈ Rr×1 (r ≤ 6), and151

τ ∈ Rn×1 is the joint torque vector. Note that it is possible152

to consider a different number of degree of freedoms in the153

master and in the slave manipulators. This allows the existence154

of teleoperation systems with redundancies in any master–slave155

robot. This possibility enhances the applications of telerrobotic156

systems allowing the avoidance of certain obstacles during the157

robot motion or adding robustness against actuator failures to158

the system (which is useful, for example, in space applications;159

see, for instance, [25], [26]). The general dynamic model (1)160

possesses a certain number of well-known structural properties,161

which can be found, for instance, in [10] and [20]. However, the162

following one, then used in the design of the adaptive control163

law, is formulated for convenience below (subscripts = m, s164

denoting each robot are omitted for the sake of simplicity).165

Property 1: The inertia matrix M and the vectors Cq and G166

depend linearly on a p′ × 1 vector P ′ of parameters containing167

the masses and inertias of the system [10], [20], so that (1)168

becomes169

M(q)q + C(q, q)q + F (q) + G(q) + JT (q)f ext

= W ′(q, q, q)P ′ + F (q) + JT (q)f ext = τ

where W ′(q, q, q) ∈ Rn×p′

is the “regressor” matrix. 170

C. Nominal Dynamical Model of the Robot171

Nevertheless, it is difficult to obtain an accurate model of a172

real robotic manipulator. This fact implies that the nominal dy-173

namic model of the robot, used for control law design purposes,174

is always inherently approximated, and thus, there always exists175

an “unmodeled dynamic component” in the closed-loop system176

due to the parametrical and same-order structural uncertainties.177

To illustrate the unmodeled dynamics effect, a nominal model178

for the real robot is considered in this section. In this way,179

the real robots are simulated by using (1), whereas the control180

law is generated by using the nominal model described below.181

Moreover, because we are designing a multiestimation-based182

controller, we will consider a set of models of the plant operat-183

ing in parallel, each described by the equation184

M(i)0 (q)q + C

(i)0 (q, q)q

+ F(i)0 (q) + G

(i)0 (q) + JT

(q)f ext

= W(q, q, q)P(i) + JT

(q)f ext = τ (2)

with i = 1, 2, . . . , Ne, Ne ≥ 1 being the number of different 185

plant models disposed for each robot, whereas the matrices 186

M(i)0 , C(i)

0 , F(i)0 , G

(i)0 stand for the nominal model descrip- 187

tion, W(q, q, q) ∈ Rn×p , and P

(i) ∈ R

p×1, where p is 188

the number of the unknown robotic parameters (which may be, 189

for example, masses or friction coefficients), is the nominal 190

parameter vector. It is supposed that the robot parameters for 191

the nominal model are unknown, and therefore, they will be 192

estimated leading to a set of estimated nominal parameter 193

vectors P(i)

. Note that the larger number of estimators we 194

dispose, the more the closed-loop response may be improved. 195

Thus, the practical number of estimators needed is determined 196

by the available outlay on hardware and the characteristics of 197

this. Also, in [27], a recent approach to the design of families of 198

estimation models is presented for multimodel controller design 199

from a theoretical point of view. However, a small number of 200

estimators can still achieve an important response improvement 201

as simulation examples in Section VI point out. 202

Furthermore, the dynamic equations of the robotic manip- 203

ulator (1) can be rewritten in terms of each reduced nominal 204

model (2) as 205[M

(i)0 (q) + Λ(i)

1 (q)]q + C

(i)0 (q, q)q + F

(i)0 (q)

+ G(i)0 (q) + JT

(q)f ext + Λ(i)2 (q, q)

= W · P (i) + JT

f ext︸ ︷︷ ︸nominal

+ Λ(i)1 (q)q + Λ(i)

2 (q, q)︸ ︷︷ ︸unmodelled

= τ (3)

where the error signals Λ(i)1 ,Λ(i)

2 of the real dynamics com- 206

pared with the nominal one are defined by 207

Λ(i)1 (q) =M(q)−M

(i)0 (q) (4.1)

Λ(i)2 (q, q) =C(q, q)q + F (q) + G(q)

− C(i)0 (q, q)q − F

(i)0 (q) − G

(i)0 (q).

(4.2)

Moreover, the following assumption concerning the dynam- 208

ics of the friction terms is made. 209

Assumption 1: There exist real constantsCF1, CF2, C(i)F01, 210

C(i)F02 ≥ 0, = m, s, such that the friction terms for both real 211

and nominal models satisfy 212

‖F (q)‖ ≤CF1‖q‖ + CF2‖q‖2∥∥∥F (i)0 (q)

∥∥∥ ≤C(i)F01‖q‖ + C

(i)F02‖q‖2

i = 1, 2, . . . , Ne.

213

Assumption 1 is feasible for most friction models used in 214

the literature, which include viscous, Coulomb, and Stribeck 215

effects [28], [29]. The following Property 2, then used in the 216

design of the estimation algorithm, can be directly stated using 217

standard robotic properties and Assumption 1. 218

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Property 2: There exist real constants χ(i)10, χ

(i)0 , χ

(i)1 ,219

χ(i)2 ≥ 0 for i = 1, 2, . . . , Ne such that Λ(i)

1 (q),Λ(1)2 (q, q)220

satisfy from (4.1) and (4.2), i.e.,221 ∥∥∥Λ(i)1 (q)

∥∥∥≤ Λ(i)1 =χ

(i)10∥∥∥Λ(i)

2 (q, q)∥∥∥≤ Λ(i)

2 (q) =χ(i)0 +χ(i)

1 ‖q‖+χ(i)2 ‖q‖2.

222

Remark 1: Note that a sole set of constants could be used for223

each robot defining, i.e.,224

(χ10, χ0, χ1, χ2) = max1≤i≤Ne

(χ

(i)10 , χ

(i)0 , χ

(i)1 , χ

(i)2

).

Moreover, the following assumption concerned with the pertur-225

bation signals is also made. 226

Assumption 2: A set of constants χ10, χ0, χ1, χ2 ≥ 0,227

= m, s, satisfying Property 2 is known. 228

Thus, according to Property 2 and Assumption 2, the unmod-229

eled dynamics is not supposed to be stable but only growing230

nonfaster that quadratically with ‖q‖ contrarily to the twin hy-231

pothesis in [16] and [30]. The knowledge of the upperbounds of232

the parametrical uncertainty signals (Assumption 2) is required233

to design a relative adaptation dead zone for the estimation al-234

gorithm to guarantee adequate estimation properties. However,235

this knowledge can be overcome by using large values for the236

constants involved or, even, including them in the estimation237

process as made in [31].238

D. Simulation Model for the Environment239

In this section, a model for the interaction force between the240

slave’s end effector and the obstacle is presented for simulation241

purposes. Thus, the interaction force between the robot and the242

obstacle will be decomposed as the superposition of two elastic243

forces. This is a feasible case to point out the stability properties244

because an elastic environment is one of the most critical cases245

dealt with for stabilization purposes [32], [33]. On the other246

hand, nonelastic interactions imply the dissipation of energy247

that contributes to stabilize the system.248

The environment modeling issue is summarized as follows.249

Assumption 3: The model for the normal force is given by250

f = fnn; n = ∇g/‖∇g‖; fn = Ken|∆x⊥| (5)

where Ken > 0 is the “normal stiffness” of the environment251

or interacting obstacle, |∆x⊥| is the normal penetration of the252

slave in the obstacle, and n is a unitary vector perpendicular to253

the object and outward-oriented [34]. Also, the tangential force254

is given by the “Salisbury” model [35], whose basic behavior is255

described in Fig. 2, where Ket > 0 is the “tangential stiffness”256

and t is a vector satisfying ‖t‖ = 1, t · n = 0, t · xs < 0 where257

x is the vector containing the position of the end effector of the258

manipulator, and xad is a vector called “adherence point.” 259

The above model includes two kinds of tangential interaction260

terms, namely, the “static,” µs, and the “dynamic,” µd, one [35]261

satisfying µs > µd > 0.262

Fig. 2. Simulation model for the tangential force.

E. Model for the Human Operator 263

Inasmuch as telemanipulation systems are considered as man 264

in the loop systems [3], [4], a model for the operator performing 265

the task must also be considered in the closed-loop defini- 266

tion (shown in the telemanipulation scheme, Fig. 1, the block 267

representing the operator). Several models for the operator 268

behavior in the control loop have been proposed in the literature 269

[5], [6], [36]–[39]. A typical model for the operator is given by 270

a second-order linear matrix differential equation formalized as 271

follows. 272

Assumption 4: The model for the operator is described by 273

Mopx+Bopx+Kopx = fman (6)

where the real positive definite matrices Mop, Bop,Kop define 274

the operator impedance, and fman is a force applied to the 275

extreme of the operator hand. 276

Remark 2: Different authors point out different values for 277

the inertia, damping, and elastic coefficients for the operator 278

model (see [5], [6], [36], [37], [39]–[41]). The values of the 279

operator impedance determine the allowable values for the 280

target matrices, which guarantee closed-loop stability according 281

to the stability theorem below. Thus, the use of different models 282

leads to different stability requirements. However, to design a 283

system that remains insensitive regardless of the operator, an 284

algorithm to online adjust the value of the target matrices is 285

then proposed in Section IV. 286

Nevertheless, one always has to be cautious about the ob- 287

tained results when a human operator is involved in the closed- 288

loop as Flemmer et al. have pointed out in [42] because the 289

second-order linear model is an approximation of the complex, 290

probably nonlinear, behavior of the user. To perform the simula- 291

tion examples with a simulated operator connected in the loop, 292

the Kazerooni and Moore data [37], (Mop = 4.54 kg, Bop = 293

6.83 N · s/m, Kop = 12.5 N/m) have been used to implement 294

the user. Hence, simulations describe a case study when a 295

certain human operator is manipulating the device despite the 296

fact that the system remain bounded for any operator due to the 297

implementation of the online updating algorithm for the target 298

matrices introduced then in Section IV. 299

F. Closed-Loop Robot Reference Model 300

The control algorithm is based on an impedance-type control. 301

Thus, under an external force f ext, we want the robot to behave 302

as the mass-damped reference system described by 303

Mt xd(t) +Bt xd(t) +Kt xd(t) = υf ext(t) (7)

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IBEAS AND DE LA SEN: ADAPTIVE CONTROL OF ROBOTIC MANIPULATORS WITH FORCE REFLECTION 5

where xd is a vector containing the end effector’s task space304

desired position as well as a parameterization for the orienta-305

tions for each robot, = m, s, and υ is a scaling factor that306

allows to resize the force transmitted between both manipula-307

tors. This property would be useful when both robots are of308

different kinds, sizes, or powers. In that case, the scaling of the309

corresponding workspaces allows to manage a robot of large310

size with another of reduced one and vice versa. The targeted311

matrices Mt = MTt > 0, Bt = BT

t ≥ 0, Kt = KTt ≥ 0 are312

chosen diagonal to decouple the robot dynamics in the task313

space. It is recommended in the literature to chose reference314

models for both robots with the same dynamic behavior, ex-315

cluding the value of the gain υ [43]. Thus, to then establish316

the stability theorem under this recommended situation, the317

following assumption concerning the reference model of the318

master and slave manipulator is made.319

Assumption 5: It is assumed that both the master and slave320

robot possess the same dynamic behavior, i.e.,321

Mtm = Mts = Mt; Btm = Bts = Bt; Ktm = Kts = Kt.

322

Furthermore, it is remarked in several papers ([43], [44], and323

references therein) that it is convenient to change the impedance324

reference model when the system contacts an obstacle. Thus,325

once the contact has been detected, according to a prescribed326

threshold for the measured force at the slave’s end effector,327

the reference impedance is modified along the directions of328

contact to maintain stability, whereas the same reference model329

is maintained for the rest of degrees of freedom [41], [45].330

The modification of the reference model should be gradual331

between both sets of parameters to reduce the effect of the332

switching transient. There is a diversity of opinion about what333

constitutes the best parameterization for a master–slave ref-334

erence model [46]–[48]. In general, the target impedance in335

free motion should be selected to minimize the fatigue of the336

operator handling the master. Thus, small values for the inertia337

and dumping matrices are the most appropriate option. On338

the other hand, the reference model in the contact directions339

should be selected to block the master and slave manipulators340

motion in the directions of contact. Hence, high values for341

the damping and matrix inertia are recommended. In this way,342

exhaustive simulation work has revealed that an increment of343

the value of the dumping matrix a factor of a thousand in the344

directions where a contact has been detected provides a good345

behavior serving this factor as a rule of thumb. Furthermore,346

Raju et al. [47] and Bejczy and Handylykken [46] report347

that there seem to be different best combinations of force-348

feedback gain υm for different tasks. A set of guidelines for349

selecting an adequate parameterization for the reference model350

(7) for different teleoperation tasks can be found in [5], [40],351

[45]–[47], and [61]–[64].352

III. PARALLEL MULTIESTIMATION SCHEME353

ARCHITECTURE AND UPDATING SUPERVISION354

RULES OF THE ACTIVE ESTIMATOR355

The architecture of the parallel multiestimation scheme is356

depicted in Fig. 3.357

Fig. 3. Multiestimation scheme architecture.

There exists Ne estimation algorithms running in parallel. 358

Each one is different from another in what is concerned with 359

the estimates parameter vector initialization and/or the kind 360

of estimation algorithm. Hence, a set of estimated parameter 361

vectors P(i)

k,, i = 1, 2, . . . , Ne is available at each sampling 362

instant k, whereas a continuous estimated parameter vector can 363

be defined by P(i)

(t) = P(i)

k,, t ∈ [kTs, (k + 1)Ts). At each 364

time interval, the controller is parameterized by a vector P k, 365

belonging to the convex hull of the estimated parameter vectors 366

of the parallel multiestimation scheme, i.e., 367

P k, ∈ Co

P(1)

k,, P(2)

k,, . . . , P(Ne)

k,

:=

P k, : P k,=

Ne∑i=1

α(i)k, P

(i)

k,,

Ne∑i=1

α(i)k, ≤1, α

(i)k, ≥0

.

(8)

Such a parameter vector should guarantee the closed-loop 368

stability and asymptotic reference tracking properties while 369

improving the transient contact closed-loop response of the 370

overall system compared with traditional single-model-based 371

adaptive control schemes. All the estimation algorithms are 372

chosen to be of the same type for both robots, initialized by 373

different estimated parameter vectors. 374

A. Estimation Algorithm 375

For the implementation of the parameter estimation, a dis- 376

crete algorithm will be used, although the problem is continu- 377

ous in its initial formulation. Denote the sampling period as Ts. 378

At each sampling time tk = kTs, the joint torque, joint posi- 379

tions, velocities, and accelerations are supposed to be available. 380

In particular, the measurable joint torques are the control input. 381

Furthermore, acceleration data should be obtained because real 382

values for the signals have to be included in the estimation 383

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algorithm. Due to the fact that acceleration measurements are384

not usually directly available from sensors in standard robot385

manipulators, a causal linear strictly stable filter (1/F(D)) in386

the time derivative D = d/dt (D0 = 1, Di+1 = DiD) will be387

used to avoid the problem of obtaining the higher order deriva-388

tives of the joint velocities. In Section V-C, an example of how389

the acceleration measurement is avoided during estimation is390

pointed out. In this way, only joint velocities and positions have391

to be measured. This assumption is not quite restrictive because392

they are usually available for measurement in typical robots. In393

this way, the filtered identification errors for each (master and394

slave) robot are defined as E(i)τf = (1/F(D))E(i)

τ , with E(i)τ395

defined as396

E(i)τ =τ −τ

(i) =WP

(i)

+Λ(i)2 +Λ(i)

1 q=WP(i)

+η(i)

(9)

where η(i) defines the unmodeled dynamic contribution as-397

sociated with the ith model, and τ(i) is the estimated joint398

torque vector given via (1) by399

M(i)0 (q)q + C

(i)0 (q, q)q

+ F(i)

0 (q) + G(i)

0 (q) + JT (q)f ext = τ

(i)0 (10)

and τ is the control torque for the robot defined by the control400

law described completely in the sequel. Also, the filtered signal401

η(i)f = (1/F(D))η(i)

represents an upperbound of the contri-402

bution of the parametrical uncertainties η(i) according to (4.1)403

and (4.2) and Assumption 2, i.e.,404 ∥∥∥η(i)

∥∥∥ =∥∥∥Λ(i)

2 + Λ(i)1 q

∥∥∥ ≤ Λ(i)2 + Λ(i)

1 ‖q‖

=χ(i)0 + χ

(i)1 ‖q‖+χ

(i)2 ‖q‖2+χ(i)

10 ‖q‖ = η(i) .

(11)

Let ak = ‖WkfΓkΓk+1ΓkWTkf‖ and δ

(i)k = ϑ

(i)k +405

‖WkfΓ(i)k W

Tkf‖ with Wkf = (1/F(D))Wk, so that the406

P(i)k estimate vector is given at each sampling time by407

(12.1)–(12.3), shown at the bottom of the page.408

Initial values for P(i)

0, and Γ(i)0 = Γ(i)T

0 > 0 are arbitrary 409

and bounded, for some real design constants µ(i) > 1, where 410

ξ(i)k, ∈ [ε(i)

, 1], ∀k ≥ 0, is the forgetting factor of the esti- 411

mation algorithm with ε(i) > 0, s(i)

k, ∈ [0, 1] and i = 1, 2, 412

. . . , Ne. The free-design scalar parameter ϑ(i)k ∈ (0,∞) acts 413

as a gain modifying the adaptation rate. The inclusion of a 414

relative adaptation dead zone in the estimation algorithm acts 415

freezing the adaptation process when the identification error 416

lies below a prescribed bound related to the unmodeled dy- 417

namics upperbound amplitude allowing to guarantee adequate 418

estimation properties [49]. Each matrix Γ(i)k must be reset to its 419

original positive definite value Γ(i)0 when its minimum eigen- 420

value lies down a prescribed positive threshold to avoid the con- 421

vergence of Γ(i)k to a positive semidefinite matrix. Moreover, 422

it is possible to use different estimation algorithms for each 423

estimation model. For instance, the use of Landau–Lozano-type 424

estimation algorithms [49], which possess a larger number of 425

free-design parameters, is recommended to get more control 426

by the designer on the converge rate and properties of the 427

estimates. However, in this paper, we have used the same 428

estimation algorithm for all the estimators to point out the 429

relevance of the switching process to improve the closed-loop 430

response. The estimation algorithm possesses the following 431

properties. 432

Lemma 1: The estimation algorithm defined by 433

(12.1)–(12.3) has the following properties: 434

(i) ‖P (i)

k ‖ <∞, ∀k ≥ 0, provided that ‖P (i)

0 ‖ <∞; 435

(ii) limN→∞∑N

k=1(s(i)k ‖E(i)

τkf‖2)/(ϑ(i)k + ‖WkfΓ

(i)k × 436

WTkf‖) <∞; 437

(iii) limk→∞(s(i)k ‖E(i)

τkf‖2)/(ϑ(i)k + ‖WkfΓ

(i)k W

Tkf‖) = 438

0; 439

(iv) limN→∞∑N

k=1(s(i)2k ‖E(i)

τkf‖2)/(ϑ(i)k + ‖Wkf × 440

Γ(i)k W

Tkf‖) <∞; 441

(v) limN→∞∑N

k=η ‖P(i)k − P

(i)k−β,‖2 <∞, for any finite 442

β > 0; 443

(vi) limk→∞ ‖P (i)k − P

(i)k−β,‖ = 0, for any finite β > 0. 444

The proof can be carried out following [49]. Basically, 445

Lemma 1 states that the estimation algorithm for each estimated 446

P(i)

k+1, = P(i)

k, +s

(i)k,Γ

(i)k,W

TkfE

(i)τkf

ϑ(i)k, +

∥∥∥WkfΓ(i)k,W

Tkf

∥∥∥ ; P(i) (t) = P

(i)k, , ∀t ∈ [kTs, (k + 1)Ts) (12.1)

Γ(i)k+1, =

1

ξ(i)k,

(Γ(i)k, − s

(i)k,

Γ(i)k,

WTkf

Wkf Γ(i)k,

ϑ(i)k,

+∥∥Wkf Γ

(i)k,

WTkf

∥∥)

s(i)k, = 0

Γ(i)k, s

(i)k, = 0

Γ(i) (t) = Γ(i)

k, , ∀t ∈ [kTs, (k + 1)Ts) (12.2)

s(i)k, =

0 if∥∥∥E(i)

τkf

∥∥∥ ≤ µ(i)k,

√(ξ

(i)k, δ

(i)k, + a

(i)k

)/ξ

(i)k, δ

(i)k, η

(i)kf

1 − µ(i)

√(ξ

(i)k, δ

(i)k, + a

(i)k,

)/ξ

(i)k, δ

(i)k,

η(i)kf∥∥E(i)τkf

∥∥ otherwise.

(12.3)

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parameter vector possesses standard estimation properties [49],447

ensuring the boundedness of the estimated parameter values for448

all time and their convergence to a finite limit.449

Remark 3: The above properties also hold for an estimated450

parameter vector belonging to the convex hull of the estimated451

parameter vectors, P k, ∈ CoP (1)

k,, P(2)

k,, . . . , P(Ne)

k, , with452

constant coefficients as can be concluded from the proof of the453

single estimation algorithm properties [49]. 454

B. Identification Performance Index455

The aim of the subsequent performance index with forgetting456

factor λ ∈ (0, 1] is to quantify the goodness of each estimation457

algorithm at each time instant and to govern the switching rule458

in between the various estimators within the multiestimation459

scheme, i.e.,460

J(i)w (k) =

k∑ζ=k−NC

λk−ζ E(i)TτζfQζE

(i)τζf (13)

where Nc > 0 is the sliding constant horizon size of the super-461

visory index (13), = m, s, and tς = ςTs and Qζ = QTζ ≥ 0462

are weighting matrices that allow the design of different ad hoc463

weights for each component of the identification error through464

time within each sliding-time optimization horizon window.465

Note that the identification performance index (13) is a measure466

of the long-term accuracy of each identification algorithm. The467

switching rule for the adaptive controller reparameterization468

is obtained from the performance index (13) by one of the469

following proposed methods.470

C. Selection of the Parameter Vector Used to Parameterize471

the Adaptive Controller472

In this section, three different switching policies between473

elements of (8) are proposed.474

Switching Rule 1 (Convex Combination of Estimates): The475

α(i)k, coefficients from the convex linear combination (8) are476

modified according to the flowchart for Switching Rule 1. Each477

robot is running the above algorithm updating the weighting478

functions at discrete times, which translates into continuous-479

time weighting functions α(i) (t) = α

(i)k,, t ∈ [kTs, (k + 1)Ts).480

The supervisory rule works as follows: if the elapsed time481

from the last previous α updating is smaller than the “residence482

time” NDTs (with the residence number of samples being483

ND), then the parameterization of the adaptive controller484

is kept unchanged, and hence, the same values for the α485

coefficients as before are used. Otherwise, a new controller486

reparameterization is given according to the algorithm above487

based on the identification performance index (13). The resi-488

dence time is subject to a minimum lowerbound constraint to489

guarantee closed-loop stability, which is a typical requirement490

in switching systems [13], [15], [16]. Note that the adaptive491

controller reparameterization only takes place at certain time492

instants, whereas all the estimators in the parallel multiestima-493

tion scheme are running at all time.494

Switching Rule 2 (Selection of a Single Estimate Vector):495

The switching rule chooses online one single estimator from the496

parallel multiestimation scheme that leads to the best closed- 497

loop behavior according to the minimization of (13). This is a 498

particular case of the convex combination of estimates when 499

one of the coefficients is unity and the remaining ones are 500

zero. In this way, at each sampling time tk = kTs, there exists 501

an estimator ck such that α(i)ck,

= 1, whereas the remaining 502

coefficients are zero. Notice that this is a particular case of 503

Switching Rule 1. Thus, Switching Rule 2 is formally expressed 504

as the flowchart for Switching Rule 2. 505

This algorithm runs for both robots simultaneously and in an 506

independent way. Likewise, a minimum residence time between 507

consecutive controller reparameterizations has to be respected 508

to guarantee closed-loop stability. However, the said minimum 509

residence time may be unknown in advance despite its existence 510

being ensured in [59]. A possible way to select the residence 511

time is to tentatively initialize it with a small value to correct 512

any potential erroneous choice of the initial values of the 513

estimates as fast as possible. 514

However, the use of a very small value for the residence 515

time might lead to instability [15], [16]. Hence, an algorithm 516

to online tune the value of the residence time is proposed in 517

Section V to be calculated by the system online through its 518

operation if the selected initial value does not fit the stability 519

requirements. 520

Furthermore, the subsequent Switching Rule 3 supplies an 521

online valid residence time without requiring a priori knowl- 522

edge on the parameterizations of the master–slave tandem. 523

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Switching Rule 3 (Residence Time Autotuning): Define524

H(t) = [ eT eT ]M[ eT eT ]T , where M = MT > 0 is a525

prefixed design matrix. The switching rule is described by the526

flowchart for Switching Rule 3.527

The above algorithm is running for = m, s (i.e., both528

master and slave). It states that onceH has decreased a value of529

γ > 0 from the last controller reparameterization (i.e., it satis-530

fied H(t+lastSwitching + TD) ≤ H(t−lastSwitching) − γ with531

TD > 0), then the system is allowed to change the controller532

parameterization within the time interval [TD, t] in which533

H(t+lastSwitching+ t) ≤ H(t−lastSwitching

)− γ/2.534

IV. BASIC ROBOT CONTROL ALGORITHM535

The following control law for the master and the slave robots536

is proposed:537

τf =(1/F(D))τ

τ =1

1 − ρ

[M0J

−1

(xd + kv(xd − x)

+ kp(xd − x)− Jq

)+ V0 + F0 + G0 + JT

fext − ρτ

](14)

ρ(t) = 1 − C2

‖e(t)‖α + C2(15)

where 1/F(D) is a strictly stable linear filter, and538

M0, C0, F 0, G0 are the estimated nominal model dynamical539

matrices selected from the convex hull (7) of the parallel540

multiestimation scheme according to any Switching Rule (i.e.,541

1, 2, or 3) to parameterize the adaptive controller (14). J is 542

the manipulator Jacobian with kp = kTp > 0, kv = kTv > 0, 543

e = x − xd, α ≥ 2, and C2 > 0, and τ is the estimated 544

joint torque for the selected active estimator (9). In [50], and 545

references therein, some methods to avoid singularity proximity 546

during the robot operation and, thus, noninvertibility of the 547

Jacobian are discussed. Note that the control law (14) can be 548

equivalently rewritten in an implicit form as 549

τ =F(D)τf

= M0J−1

(xd + kv(xd − x) + kp(xd − x)

− Jq + ρJM−10 (τ − τ)

)+ C0q + F0 + G0 + JT

fext

which can be interpreted as an acceleration resolved control law 550

[8] with a correcting factor (ρJM−10 (τ − τ)) [59]. 551

This correcting factor allows to separate the estimation al- 552

gorithm properties from the closed-loop stability results in 553

comparison with classical techniques [8]. In these ones, the esti- 554

mation algorithm structure is derived from Lyapunov’s stability 555

theorem, so that the inconvenient contribution of positive terms 556

to the Lyapunov function time derivative is cancelled by the 557

updating algorithm structure. Fig. 4 displays the scheme for the 558

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IBEAS AND DE LA SEN: ADAPTIVE CONTROL OF ROBOTIC MANIPULATORS WITH FORCE REFLECTION 9

Fig. 4. Schematic representation of the basic robot control law.

proposed basic control law for each robot. As Fig. 4 shows,559

the feedback linearization loop is not able to cancel exactly the560

nonlinear robot dynamics due to the existence of unmodeled561

dynamics component. By applying the control law (14) and562

(15) to the robotic manipulator described by (1), we obtain the563

following closed-loop equation:564

1/F(D)(e + kve + kpe − (ρ − 1)JM−1

0

×(W(q, q, q)P + Λ2 + Λ1q

))= v(t) (16)

where P = P − P is the parametrical error vector, v(t)565

is the initial condition response of the filter 1/F(D) and566

vanishes exponentially because the filter is exponentially sta-567

ble because it is linear and strictly stable, and e = x − xd568

for = m, s. To ensure the nonsingularity of M0 during569

parameter adaptation, it may be necessary to restrict the al-570

lowable variation range of the unknown parameters by using571

projection over an appropriate set of parameter values when572

necessary [20].573

V. STABILITY PROPERTIES OF THE ADAPTIVE SCHEME574

A. Main Stability Result575

To establish the main stability result, the subsequent assump-576

tion is made.577

Assumption 6: The trajectory in the joint space q(t) satisfies578

det J(q(t)) = 0,∀t ≥ 0. 579

This assumption means that the Jacobian remains nonsingu-580

lar during the manipulator movement provided that for initial581

conditions det J(q(0)) = 0. The following theorem ensures the582

stability of the overall teleoperated system described generi-583

cally in Fig. 1.584Theorem 1 (Closed-Loop Stability): Consider a teleopera-585

tion system whose master and slave dynamics are described586

by (1) subject to Assumptions 1 and 2. Assume also that 587

the adaptive controller is given by (14) and (15), satisfying 588

the basic estimation algorithm Lemma 1(i)–(iii) properties and 589

Assumptions 3–6. Then, the following items hold: 590

(i) The single estimation scheme (14) and (15) for the tele- 591

operated system is robustly stable provided that either 592

Mt > υmsMop, Bt ≥ υmsBop

and Kt +R

[Ken 00 Ket

]≥ υmsKop (17.1)

or 593

Mt < υmsMop, Bt ≤ υmsBop

and Kt +R

[Ken 00 Ket

]≤ υmsKop (17.2)

hold, where Mop, Bop, Kop denotes the above defined 594

operator model, υms = υm/υs is the ratio of the ref- 595

erence model gains. and R denotes the rotation matrix 596

from the (t, n) axis to the base where the vector xs is 597

expressed. As a result, all the signals in the closed-loop 598

system are bounded for all t ≥ 0. 599

(ii) Assume that the target matrices satisfy (17). Then, the 600

current multiestimation-based closed-loop system is ro- 601

bustly stable under any Switching Rule (1, 2, or 3), 602

provided that a minimum positive residence time TD 603

between successive changes of the parameterization of 604

the adaptive controller is respected. 605

(iii) If there are no uncertainties in the system, [i.e., η = 0 in 606

(9)–(11)] with the designer’s knowledge, then the closed- 607

loop system is asymptotically stable, provided that 608

Theorem 1[(i) and (ii)] conditions hold. Thus, all the 609

tracking errors of the system tend to be zero for both the 610

single and the multiestimation schemes. 611

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The proof of this theorem is given in Appendix. Note that612

because the inverse of the Jacobian is used in (14), the stability613

is not global and has to be restricted to joint space trajectories614

avoiding Jacobian singularities (Assumption 6). This is a gen-615

eral drawback in robotic control strategies involving the inverse616

of the Jacobian. Nevertheless, there is extensive work in the lit-617

erature dealing with this problem while implementing strategies618

to avoid Jacobian singularities (see [50] and references therein).619

Thus, these results ensure the applicability of the proposed620

control law to a large variety of manipulators in a wide variety621

of working operations just incorporating any of the techniques622

proposed in the literature to the control algorithm. Also, the623

presence of an unmodeled dynamics component in the closed-624

loop prevents the system from possessing asymptotic stability.625

In the case when the designer knows that this unmodeled626

component is not present, the asymptotic stability can be stated627

according to Theorem 1(iii).628

Furthermore, Theorem 1 emphasizes the influence of the629

human operator behavior, which is crucial to guarantee the630

stability of the scheme. Also, the ratio υm/υs acts gaining631

the coupling between both robots. Note that the stability of632

the multiestimation scheme is subject to a minimum residence633

time between switchings to different controller parameteri-634

zations. This is a typical requirement in switching systems635

[13]–[16], [23], [51] being either adaptive or composed of636

fixed models. Moreover, note that the presented control law637

is modular in the sense that the closed-loop stability holds638

irrespectively of the estimation algorithm used while giving639

to a bounded estimated parameters through time (see [53] and640

references therein). In the subsequent section, a brief outline of641

the stability proof structure is given.642

B. Guidelines for the Proof of Theorem 1643

The proof of Theorem 1 is carried out through the following644

steps.645

1) First, asymptotic properties of the basic estimation algo-646

rithm are stated in Lemma 1. In particular, boundedness647

of the estimated parameter vector (for bounded initial648

conditions) and prediction error square summability are649

established. This property is needed to ensure the asymp-650

totic tracking properties of the schemes in the absence of651

uncertainties in the system [53].652

2) Then, the tracking errors for each robot e = x −653

xd, e, e, = m, s, are proved to be bounded when654

the single-model-based adaptive control scheme is used655

considering the fact that the estimated parameter vec-656

tors are bounded via Lemma 1. This result is stated657

in Lemma A1.1.658

3) Thus, to prove scheme’s signal boundedness, the refer-659

ence model for each robot is detailed with the inter-660

dependence between them clearly pointed out. Starting661

with these equations, boundedness of all signals is finally662

proved in the Theorem 1(i) proof using the boundedness663

of the tracking errors.664

4) Hence, the proof of Theorem 1(ii) is presented based665

on the single estimation scheme. With the basic results666

of stability of the coupled adaptive scheme, the proof is667

extended to the multiestimation scheme using the same 668

steps as those pointed out in [59], which highlights the 669

modularity and structure of the proof. 670

5) Finally, the (nominal) case when no uncertainties are 671

present is considered in the same way as in [59]. Asymp- 672

totic converge of the tracking errors to zero is then proved, 673

completing the Theorem 1 proof. In this way, Theorem 1 674

is fully proved. 675

Remark 4: The stability analysis can be extended to more 676

complex environments than that considered in Section II-D. In 677

this way, the contact model might be given by the more general 678

interaction force common in models like [52], [54], [55] 679

fcontact = kstxs + kvelxs + kacxs

with kst, kvel, kac being positive semidefinite matrices. In 680

this case, the stability of the system is also guaranteed (see 681

Remark 6 in Appendix). Furthermore, nonlinear models lin- 682

early upperbounded as ‖fcontact‖ ≤ kst‖xs‖+ kvel‖xs‖ with 683

kst, kvel ≥ 0 as proposed, for instance, in [56], also guarantee 684

the stability (see the extension of the stability proof for this kind 685

of environments in Appendix). 686

C. Tuning Process for the Residence Time and Target Matrices 687

Nevertheless, the residence time for Switching Rules 1 and 688

2 and target matrices satisfying conditions (17) are especially 689

difficult to set. Also, the residence time for Switching Rules 1 690

and 2 must be initialized while being larger than the minimum 691

residence time, which guarantees stability. Thus, a procedure to 692

overcome these difficulties is described above. 693

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IBEAS AND DE LA SEN: ADAPTIVE CONTROL OF ROBOTIC MANIPULATORS WITH FORCE REFLECTION 11

Fig. 5. Robot manipulator.

The algorithm works as follows: Starting with initial values694

for the matrices and the residence time, if the norm of the695

master’s state vector exceeds a prescribed threshold, then we696

consider that the system is unstable, and we proceed to modify697

the residence time and target matrices. The modification is698

applied sequentially. First, the residence time is increased by699

a small quantity. If the master’s state vector still lies above700

the threshold, then at the next step, we would increase the701

value of the target inertia matrix. In the case that the system702

was still unstable, at the next step, the target dumping would703

be the variable increased until the norm of the mater’s state704

reduced below the given threshold. Note that a positive time705

interval between modifications Tnorm TD is stated to allow706

the algorithm enough time to appreciate the effect of modifi-707

cations in the system behavior. Moreover, the algorithm stops708

in finite time because finite large values for the target matrices709

and the residence time stabilize the system from Theorem 1(i)710

and (ii). The modification values σ1, σ2, σ3, and σ4 are usually711

expressed as a small percentage (1%–5%) of the initial values712

of the variables.713

Remark 5: Inasmuch as Switching Rule 3 provides online714

an appropriate value for the residence time, this parameter can715

be removed from the above algorithm when using Switching716

Rule 3. 717

VI. ANALYSIS OF THE TELEOPERATION SYSTEM718

PERFORMANCE THROUGH SIMULATION719

A. Analysis of the Closed-Loop System Under the Parallel720

Multiestimation Scheme721

1) Real and Reduced-Order Nominal Robot Models: The722

following specific two-link robotic manipulator shown in Fig. 5723

is used in the computer simulations to simulate the real robot.724

In fact, this will be the model for both the master and the slave725

manipulators.726

Inasmuch as the same robot model is used for both manipula-727

tors, and in order not to make the notation too cumbersome, the728

subscript = m, s is omitted. Thus, the generalized coordinates 729

qT = [ q1 q2 ] are chosen to be the angles defined in Fig. 5. A 730

mathematical dynamical model of the robot can be obtained by 731

using the above assumptions. 732

Assumptions 7.1–7.4: 733

(7.1) The elements have masses m1,m2, and the center of 734

mass of each element is placed at a length of lc1, lc2 from 735

the previous revolute joint. 736

(7.2) Inertia tensors are diagonal for both elements. Further- 737

more, the inertia tensor is defined by a unique element 738

for both arms, I1 for the first element and I2 for the 739

second one. 740

(7.3) The robot base is fixed to the ground. 741

(7.4) There is no gravitational effect because the manipulator 742

is assumed to operate in a horizontal plane. 743

By using Assumption 5, the following dynamic matrices 744

[(18.1)–(18.3), shown at the bottom of the next page] are 745

obtained for the real robot where s1 = sin(q1), c1 = cos(q1), 746

and so on is the typical notation in robotics [8], [10], [20]. 747

Also, a detailed friction model suitable for industrial controller 748

design [57] is introduced in (18.3) for each revolute joint. This 749

model is just considered to simulate the real robot behavior in 750

the computer. The reduced order nominal model, denoted by 751

M0(q), and (C0(q, q)q), is obtained, making lc1 = l1, lc2 = l2, 752

and I1 = I2 = 0 in (18.1)–(18.3). 753

The above conditions allow to consider a simpler nominal 754

dynamical model for designing the control law. This reduced 755

nominal model is obtained at the expense of the presence an 756

unmodeled dynamic component in the closed loop. Also, note 757

that a simplified model for the friction terms has been stated in 758

(19), where νi, ki are the friction coefficients, i.e., 759

F0(q)T = [ ν1q1 + k1sign(q1) ν2q2 + k2sign(q2) ] . (19)

Thus, this nominal model has been obtained considering a set 760

of mechanical assumptions on the robot manipulator. Hence, 761

different assumptions may lead to different nominal models. 762

Moreover, (1) can be represented in a linear parameter form, 763

for estimation purposes, as 764

Wf · P +(JTf ext

)f

= τ f (20)

PT

= [m1 m2 ν1 ν2 k1 k2 ] (21)

Wf =1

F (D)W

=

l21Dq1

F (D)W12F (D)

q1F (D) 0 sign(q1)

F (D) 0

0 W22F (D) 0 q2

F (D) 0 sign(q2)F (D)

(22)W12 = l22(Dq1 +Dq2) + l1l2c2(2Dq1 +Dq2)

+ l21Dq1 − l1l2s2q22 − 2l1l2s2q1q2

W22 = l1l2c2Dq1 + l22(Dq1 +Dq2) + l1l2s2q21

where F (D) has degree larger than unity to make Dqi/F (D) 765

causal, whereas Wf is implemented in (10) for estimation 766

purposes. The signal v(t), associated to the initial condition 767

response of the strictly stable filter F (D), vanishes exponen- 768

tially, and it can be neglected in the estimation process at the 769

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expense of worsening of the estimated values. Notice that no770

acceleration measurement is needed in the estimation algorithm771

because the acceleration value is obtained in (22) through a772

causal filter via the operator D/F (D), which appears in the773

filtered version of (22) (Wf ). In this way, the estimation process774

can be performed just measuring joint positions and velocities.775

2) Numerical Example:776

Robotic tandem: It is composed of two robots that are777

of the same type as that described in Section VI. The para-778

meters are as follows: l1 = l2 = 1 m, lc1m = lc2m = 0.4 m,779

lc1s = lc2s = 0.5 m; m1m = 5 kg, m2m = 3.5 kg, m1s =780

10 kg, m2s = 17.5 kg; and I1m = 0.8 kg ·m2, I1s = 4 kg ·781

m2, I2m = 1.1 kg ·m2, I2s = 3.6 kg ·m2; whereas the rest of782

the parameters are [57] α01 = 35, α11 = 1.1, α21 = 0.9, β11 =783

50, β22 = 65, α01 = 38, α11 = 1, α21 = 0.95, β11 = 55, and784

β22 = 60.785

Multiestimation scheme: It is formed by four estimators786

running in parallel. These estimators are initialized by787

P(1)T

0m = [ 25 15 4 5 8 10 ]

P(2)T

0m = [ 23 11 2.5 3.5 18 16 ]

P(3)T

0m = [ 12 6.5 −0.5 2 27 25 ]

P(4)T

0m = [ 7 4 −2 1 35 32 ]

P(1)T

0s = [ 1 700 3.9 0.001 0.001 1.7 ]

P(2)T

02 = [ 3 535 2.5 0.05 0.01 1.22 ]

P(3)T

02 = [ 6 235 1.5 0.2 0.05 0.72 ]

P(4)T

02 = [ 9.9 130 0.5 0.5 0.1 0.2 ].

The conventional adaptive scheme composed of a single es-788

timation algorithm used for comparison purposes is initialized789

by P(1)

0 . The parameters of the estimation algorithm are ξ(i)k =790

0.95, ∀k ≥ 0, Γ(i)0 = 104I6×6, i = 1, . . . , 4, and Qζ = I2×2;791

whereas the Γ(i)k matrix is reset when its minimum eigenvalue792

lies down 0.001, µ(i) = 1.2, χ(i)

01 = 0.5, χ(i)0 = 0.5, χ(i)

1 =793

1.5, χ(i)2 = 1.5, and ϑ(i)

k = ϑ(i) = 1.794

Control law parameters: The control matrices are795

kp = 750I2×2, kv = 150I2×2, whereas ND = 4 samples,796

Ts = 0.001 s, and α = 2 in (15). The filter is F (s) = 106/(s+797

103)2. It has been chosen with a large bandwidth to make the798

estimation well-behaved when interacting with the obstacle,799

and some signals are fast varying.800

Fig. 6. Task space motion for both robots. Normal stiffness: Ken = 105 N/m.

Environment: The obstacle will be a vertical wall situated 801

at xobs = 1.2 m with an infinity height. Ktn = 100 N/m and 802

µs = 0.25, µd = 0.2. 803

Reference model: The reference model is Vdi(s)/ 804

Fexti(s) = 1/(s+ 5) for i = x, y in the case of free motion, 805

whereas in the constrained motion case, the reference model 806

changes to Vdi(s)/Fexti(s) = 1/(s+ 1000) in the direction in 807

which contact is detected. 808

Figs. 6–8 display the trajectory in the workspace described 809

by the master and the slave robots for different values for the 810

normal stiffness of the environment. From these figures, it can 811

be concluded that when a contact occurs, the multiestimation 812

scheme allows to improve the tracking performance of the 813

master robot tracking the slave’s robot trajectory as compared 814

with single-model-based adaptive schemes. Furthermore, the 815

proposed architecture reflects with more fidelity the sensation 816

the slave manipulator is experiencing, improving the trans- 817

parency of the teleoperation system. Simulation examples show 818

the usefulness of the scheme for environments ranging be- 819

tween a relatively low value and a hundred million. Figs. 9–11 820

display the active controller for the master robot under the 821

multiestimation scheme through time. This is the most relevant 822

controller because switching in the master robot controller is 823

responsible for the tracking improvement. Notice that the use 824

of a switching-based control law allows to improve the overall 825

closed-loop performance, due to the switching process within 826

the nominal multiestimation scheme when the slave bumps 827

into the wall. Fig. 8 shows the evolution of the estimated 828

M(q) =

[m1l

2c1 +m2

(2l1lc2c2 + l21 + l2c2

)+ I1 + I2 l2c2m2 + l1lc2m2c2 + I2

l2c2m2 + l1lc2m2c2 + I2 l2c2m2 + I2

](18.1)

(C(q, q)q)T = [−m2l1lc2s2q22 − 2m2l1lc2q1q2 m2l1lc2s2q

21 ] (18.2)

Fi(qi) =[α0,i + α1,ie

−β1,i|qi| + α2,i

(1− e−β2,i|qi|

)]sign(qi), i = 1, 2 (18.3)

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Fig. 7. Task space motion for both robots. Normal stiffness: Ken = 106 N/m.

Fig. 8. Evolution of the estimates for the slave.

parameters of the slave through time. The evolution of the829

estimates is altered when the slave contacts an obstacle, making830

the values of the parameters adapt to their new configuration.831

Moreover, it can be concluded from Fig. 12, for the evolution832

of the estimated parameters of the master, that their values also833

change although the master is not interacting with anything.834

The change in the master estimated parameters appears due835

to the coupling between both robots through the force–force836

scheme according to the Fig. 1 diagram. Also, Fig. 13 displays837

the force tracking property of the system. From Fig. 13, it838

can be concluded that the multiestimation scheme with the839

switching process is able to improve the force tracking response840

when the contact occurs, contributing to the improvevent of the841

transparency of the scheme. Moreover, Figs. 14 and 15 show842

the force reference tracking achieved by the multiestimation843

scheme. The scheme is seen to be able to track the desired force844

reference despite the uncertainty in the robot models. Note that845

because the slave manipulator is heavier that the master one, its846

Fig. 9. Switching map for the master robot.

Fig. 10. Switching map for the master manipulator.

response is slower due to its larger inertia. This fact explains 847

why the slave manipulator penetrates into the wall more than 848

the master in Figs. 6, 7, and 16. 849

VII. CONCLUSION 850

In this paper, a robust adaptive multiestimation-based scheme 851

has been designed for robotic manipulators with uncertainties, 852

composing a teleoperation system with force reflection. The 853

controller of each robot is composed of a bank of estimation 854

algorithms running in parallel and a switching logic that selects 855

an element from the convex hull of the estimated parameter 856

vectors to parameterize an adaptive controller. The multiesti- 857

mation architecture is capable of improving the master–slave 858

tracking properties when the latter interacts with an elas- 859

tic obstacle in comparison with classical single-model-based 860

adaptive controllers. Furthermore, the force tracking response 861

is also improved when using the multiestimation scheme. This 862

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Fig. 11. Switching map for the master manipulator.

Fig. 12. Evolution of the estimated parameters for the master.

implies that the contact process can be transmitted to the863

operator handling the master manipulator with higher fidelity,864

improving the transparency of the teleoperation scheme. More-865

over, the stability of the overall closed-loop scheme is proved866

as well. Finally, simulation examples showing the operation,867

properties, and usefulness of the proposed scheme are showed868

and discussed for different values of the environment stiffness.869

APPENDIX870

PROOF OF THE CLOSED-LOOP STABILITY871

Proof of Theorem 1: First, the following result then used872

to prove Theorem 1(i) is established.873

Lemma A1.1 (Boundedness of e): The tracking error e =874

x − xd for the single-estimation-based scheme is bounded875

for all time, subject to the condition of Theorem 1(i). (In876

the following, the fixed subscript is omitted for the sake of877

simplicity when no confusion is expected.)878

Fig. 13. Force tracking during the interaction. Normal stiffness: Ken =105 N/m.

Fig. 14. Comparison between the reference interaction force and the oneapplied by the slave. Normal stiffness: Ken = 106 N/m.

Proof of Lemma A1.1: Consider L(e, e) = 1/2[eT kpe + 879

‖e‖2] being positive definite because kp = kTp > 0. Its time 880

derivative is from (16), i.e., 881

L = − eT kv e+ (ρ− 1)eTJM−10

×[W (q, q, q)P + Λ2 + Λ1q

]+ w(t)

≤ − eT kv e+C ′

‖e‖α + C2‖e‖

×∥∥∥W (q, q, q)P + Λ2 + Λ1q

∥∥∥+ w(t) (23)

where w(t) = F (D)v(t) → 0 exponentially. Thus, it will be 882

zeroed in the sequel without loss of generality for stability 883

analysis purposes. C ′ = C2‖JM−10 ‖ is a finite constant 884

because the Jacobian is bounded from q being bounded for 885

revolute manipulators and the fact that the estimate vector is 886

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Fig. 15. Reference interaction force and the one applied by the slave.

Fig. 16. Task space motion for both robots. Normal stiffness: Ken =10, 000 N/m.

bounded from estimation algorithm property (i). Furthermore,887

there exist real constants µ1, µ2, µ3 ≥ 0 such that888

‖W (q, q, q)P + Λ2 + Λ1q‖ ≤ µ1 + µ2‖e‖ + µ3‖e‖2 from889

Assumption 2, (16), and the fact that the joint angles are890

bounded. Thus891

L ≤ −eT kv e+C ′

‖e‖α + C2

(µ1‖e‖ + µ2‖e‖2 + µ3‖e‖3

).

If α ≥ 2 in (15), then the positive term diverges at most at the892

rate of ‖e‖ as ‖e‖ diverges, whereas the negative one decreases893

accordingly to ‖e‖2. Then, L(e, e) is bounded at all time,894

and the closed-loop system exhibits global Lyapunov stability895

with ultimate boundedness because for large values of ‖e‖,896

L(e, e) < 0, and thus, L decreases, whereas for small values897

for ‖e‖, L(e, e) > 0, and L increases. Inasmuch as) L(e, e)898

is bounded for all time and for any bounded initial conditions899

e(0), e(0), it can be concluded that e and e are bounded for900

all time. Furthermore, the boundedness of L ensures that there 901

is no finite escape time in (16). Inasmuch as the above result 902

holds for both robots, one explicitly has that em, em,es, es are 903

bounded. Now, rewrite (16) as 904

e =(ρI+(1−ρ)JM−1

0 MJ−1)−1

×[−kv e−kpe+(ρ−1)JM−1

0

(W (q, q, qd)P +Λ2

)]. (24)

Note that the inverse of the matrix (ρI + (1 − ρ)JM−10 MJ−1) 905

in (24) always exists because it is the sum of two positive 906

definite matrices, the estimated matrix M0 is nonsingular for 907

all time, and the robot manipulator avoids singularities; there- 908

fore, the Jacobian is invertible from Assumption 4. Also, e is 909

bounded because the right-hand side of the above equation is 910

a continuous function, which only depends on e, e, which are 911

bounded, and the fact that e is always bounded in a rotary mo- 912

tion. Hence, one has that em, em, em,es, es, es are bounded, 913

and the proof is completed. 914

Proof of Theorem 1(i): The subsequent coupled 915

master–slave coupled equations follow from (5), i.e., 916

Mtmxdm+Btmxdm+Ktmxdm

=υm

(Mopxm+Bopxm+Kopxm−R

[Ken 00 Ket

]xs

)(25)

and the equivalent for the slave where it has been supposed that 917

xobs = 0 and xad = 0 without loss of generality for stability 918

analysis purposes. Moreover, R is the rotation matrix, which 919

transforms the (t, n) base to the reference of the position x. 920

Subtracting both equations, one gets 921

Mtmxdm +Btmxdm +Ktmxdm

=υmυs

(Mtsxds +Btsxds +Ktsxds)

= υms(Mtsxds +Btsxds +Ktsxds).

From Assumption 5, the above equation becomes 922

Mt(xdm − υmsxds) +Bt(xdm − υmsxds)

+Kt(xdm − υmsxds) = 0 (26)

with Mt = MTt > 0, Bt = BT

t ≥ 0, and Kt = KTt ≥ 0, 923

which implies that (xdm − υmsxds), (xdm − υmsxds), and 924

(xdm − υmsxds) are bounded, so that xdm = υmsxds + ξ1, 925

xdm = υmsxds + ξ2, and xdm = υmsxds + ξ3 with ξ1, ξ2, ξ3 926

bounded. Also, because e = x − xd, e = x − xd, and 927

e = x − xd, = m, s, one obtains the following equation 928

from (25): 929

(Mt − υmsMop)xds + (Bt − υmsBop)xds

+(Kt − υmsKop +R

[Ken 00 Ket

])xds

= Mop(em + ξ3) +Bop(em + ξ2)

+Kop(em + ξ1)−R

[Ken 00 Ket

]es. (27)

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Inasmuch as the right-hand side of (27) is bounded from930

Lemma A1.1, (27) leads to bounded xds, xds, xds, provided that931

either932

Mt > υmsMop, Bt ≥ υmsBop, Kt +R

[Ken 00 Ket

]≥ υmsKop

or933

Mt < υmsMop, Bt ≤ υmsBop, Kt +R

[Ken 00 Ket

]≤ υmsKop

hold. Thus, xs, xs, xs are bounded as well because ‖xs‖ =934

‖es + xds‖ ≤ ‖es‖+ ‖xds‖ <∞ and so are the rest of vari-935

ables. Now, because xdm = xds + ξ1, xdm = xds + ξ2 and936

xdm = xds + ξ3, it can be concluded that xdm, xdm, xdm are937

bounded for all time, and hence, xm, xm, xm are bounded as938

well. Then, one has that q, x, q, x, q, x are bounded because939

the Jacobian is bounded for revolute manipulators and for940

both robots. Moreover, the estimated joint torque (16) and the941

control law (14) and (15) are bounded as well. Thus, all system942

variables remain bounded for all time, and item (i) is proved. 943

Remark 6 (Extension to Other Environments): According to944

Remark 4, the above stability proof can be extended to other945

linear environments. Thus, consider environments modeled by946

fcontact = kstxs + kvelxs + kacxs

with kst, kvel, kac being positive semidefinite matrices. This947

generalization allows to consider dissipative environments. The948

stability proof is identical until the end step in which (26) would949

read as950

(Mt − υmsMop + kac)xds + (Bt − υmsBop + kvel)xds

+ (Kt − υmsKop + kst)xds

= Mop(kacem + ξ3) +Bop(kvelem + ξ2)

+Kop(em + ξ1) − kstes

which is asymptotically stable provided that the following951

conditions (analogous to those stated in Theorem 1) hold:952

(Mt − υmsMop + kac) > 0

(Bt − υmsBop + kvel) ≥ 0

(Kt − υmsKop + kst) ≥ 0.

Note that the model for the environment is intended to be953

linear because the closed-loop target response has been selected954

to be linear. Furthermore, note that using the algorithm for955

adjusting the target matrices, the stability of the closed loop956

can be guaranteed due to the increase of the target matrix value.957

Consider a nonlinear environmental model subject to958

‖fcontact‖ ≤ kst‖xs‖ + kvel‖xs‖

(which might include a dumping term proportional to velocity)959

where kst, kvel ≥ 0 are now scalars. To extend the stability960

proof to this case, the above upperbounding can be rewritten 961

for a convenient arrangement of the constants kst, kvel as 962

‖fcontact‖ ≤ k1‖xext‖+@ (28)

with k1 ≥ 0, xText = [xds xds ] being an extended state space 963

vector, and @ ≥ 0 is the remaining bounded signal, which 964

appears due to the substitution of xdm = xds + ξ1 and so on 965

when defining xext. Then, (26) can be rewritten as 966

xext =[

0 I−M−1

ts Kts M−1ts Bts

]xext +

[0Ω

]with 967

Ω=

[ 0M−1

ts Mop(em+υmsxds+ξ3)+M−1ts Bop(em+υmsxds+ξ2)

+M−1ts Kop(em+υmsxds+ξ1)+M−1

ts fcontact

]

where M−1ts always exist because Mts is a positive definite 968

matrix. The formal solution of this equation is given by 969

xext = e

[0 I

−M−1ts Kts M−1

ts Bts

]t

xext(0)

+

t∫0

e

[0 I

−M−1ts Kts M−1

ts Bts

](t−τ)

Ω dτ

which can be upperbounded as 970

‖xext‖ = Ce−λt‖xext(0)‖

+

t∫0

Ce−λ(t−u) (σ1 + σ2 ‖xext‖) du (29)

because

∥∥∥∥exp[

0 I−M−1

ts Kts M−1ts Bts

]t

∥∥∥∥ ≤ Ce−λt for some 971

constants σ1, σ2 ≥ 0, C ≥ 1, and λ ≥ 0 because the reference 972

model is strictly stable [58]. Then, applying Gronwall lemma, 973

[38] to (29), it follows that xext is bounded, and thus, xds is 974

bounded as well. 975

Proof of Theorem 1(ii) and (iii): Having proved the sta- 976

bility of the single adaptive teleoperation scheme, the stability 977

of the multiestimation-based setup is carried out following the 978

steps of [59]. 979

ACKNOWLEDGMENT 980

The authors would like to thank the reviewers for their 981

helpful comments, which have helped to improve this paper. 982

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Asier Ibeas was born in Bilbao, Spain, on July11597, 1977. He received the M.Sc. degree in applied1160physics and the Ph.D. degree in automatic control1161from the University of the Basque Country, Bilbao,1162Spain, in 2000 and 2006, respectively.1163

He was recently appointed Associate Professor of1164electrical engineering at the Autonomous University1165of Barcelona, Barcelona, Spain. His research inter-1166ests include time-delayed systems, robust adaptive1167control, and applications of artificial intelligence to1168control systems design.1169

Manuel de la Sen was born in Arrigorriaga, Bizkaia, 1170Basque Country, Spain. He received the M.Sc. de- 1171gree (Honors) in applied physics (electronics and 1172automation) and the Ph.D. degree (High Honors) in 1173applied physics from the University of the Basque 1174Country, Bilbao, Spain, in 1975 and 1979, respec- 1175tively, and the Docteur-d´Etat-ès-Sciences Physiques 1176(specialité Automatique et Traitement du Signal) 1177(mention très honorable) from the University of 1178Grenoble, Grenoble, France, in 1987. 1179

He held several teaching and research positions in 1180the University of the Basque Country, where he is currently a Professor of 1181systems engineering and automatic control in the Department of Electricity 1182and Electronics and the Head of the Institute of Research and Development 1183of Processes IIDP. He was also a Visiting Professor in the University of 1184Grenoble, France; the University of Newcastle, New South Wales, Australia; 1185and the Australian National University, Canberra, Australia. He was the author 1186or coauthor of a number of papers in the fields of expert and adaptive systems, 1187mathematical systems theory, discrete and time-delay systems, and ordinary 1188differential equations, which are his research interests. 1189

Dr. de la Sen was a member of the Editorial Board of the Electrosoft 1190Journal (CML Mechanical and Computational Engineering Publications) and 1191is currently a ember of the Editorial Board of the journal Nonlinear Analysis, 1192Modelling and Control. He acts and has acted as a Reviewer for several 1193international journals and conferences of control theory and engineering and 1194applied mathematics. 1195

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AQ1 = The affiliation address does not match that in the author’s biography. Please check.AQ2 = Running head for approval.AQ3 = Please provide the dissertation title, publisher name and location in Ref. [41].AQ4 = Please provide the dissertation title, publisher name and location in Ref. [45].Notes: 1) The figures were renumbered. The original sequence was 1-2-3-4-5-9-12-14-7-10-13-16-8-11-

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