An instrumental variable approach for rigid industrialrobots identification

Alexandre Janot a,n, Pierre Olivier Vandanjon b, Maxime Gautier c

a ONERA the French Aerospace Lab, Centre Midi-Pyrénées, 2 Avenue Edouard Belin, BP 74025, 31055 Toulouse Cedex 4, Franceb French Institute of Science and Technology for Transport, Development and Networks, LUNAM University, Ifsttar, IM, EASE, F-44341 Bouguenais, Francec University of Nantes and IRCCyN, 1, rue de la Noë, BP 92 101, 44321 Nantes Cedex 03, France

a r t i c l e i n f o

Article history:Received 26 April 2012Accepted 21 December 2013Available online 22 January 2014

Keywords:RoboticsDynamics identificationLeast squares techniquesInstrumental variable technique

a b s t r a c t

This paper deals with the important topic of rigid industrial robots identification. The usual identificationmethod is based on the use of the inverse dynamic model and the least-squares technique. In order toobtain good results, a well-tuned derivative bandpass filtering of joint positions is needed to calculate thejoint velocities and accelerations. However, we can doubt whether the bandpass filter is well-tuned ornot. Another approach is the instrumental variable (IV) method which is robust to data filtering andwhich is statistically optimal. In this paper, an IV approach relevant for identification of rigid industrialrobots is introduced. The set of instruments is the inverse dynamic model built from simulated datawhich are calculated from the simulation of the direct dynamic model. The simulation assumes the samereference trajectories and the same control structure for both the actual and the simulated robot andis based on the previous IV estimates. Furthermore, to obtain a rapid convergence, the gains of thesimulated controller are updated according to IV estimates. Thus, the proposed approach validatesthe inverse and direct dynamic models simultaneously and is not sensitive to initial conditions. Theexperimental results obtained with a 2 degrees of freedom (DOF) planar prototype and with a 6 DOFindustrial robot show the effectiveness of our approach: it is possible to identify 60 parameters in3 iterations and in 11 s.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The usual robots identification method is based on the use of theinverse dynamic model (IDM) and the least-squares technique (LS).This method, called as Inverse Dynamic Identification Model withLeast Squares technique (IDIM-LS) has been validated on severalindustrial robots, see Khalil and Dombre (2002), Swevers, Verdonck,and De Schutter (2007), Hollerbach, Khalil, and Gautier (2008), andGautier, Janot, and Vandanjon (2012) and the references giventherein. In order to obtain good results, a well-tuned derivativebandpass filtering of joint positions is needed to calculate the jointvelocities and accelerations. However, even with the guidelinesgiven in Gautier et al. (2012) to tune the bandpass filter, the userscan doubt whether the IDIM-LS estimates are unbiased or not.

This leads us to try other methods: the Total Least Square (TLS)(Gautier, Vandanjon, & Presse, 1994;Gautier & Briot, 2012; Xi, 1995),the Extended Kalman Filter (EKF) (Gautier & Poignet, 2001; Kostic,de Jager, Steinbuch, & Hensen, 2004, the Set Membership Uncer-tainty (Ramdani & Poignet, 2005), an algorithm based on LinearMatrix Inequality (LMI) tools (Calafiore & Indri, 2000a), a maximum

likelihood (ML) approach (Olsen & Petersen, 2001; Olsen, Swevers,& Verdonck, 2002. Though these approaches are interesting, theydo not really improve the IDIM-LS method coupled with anappropriate bandpass filtering, their robustness to data qualityand/or data filtering is not addressed and some of them are quitetime consuming.

Another approach is the Instrumental Variable technique (IV)which was introduced by Reiersøl in 1941 (Reiersøl, 1941). Thismethod deals with the problem of noisy observation matrix and canbe statistically optimal. In Söderstrom and Stoica (1989), Garnier andWang (2008), Young (2011), and Gilson, Garnier, Young, and Van denHof (2011) and the references given therein, different IV methods arestudied for linear systems. However, these works are theoretical-oriented and they are validated on low-dimensional systems. Thismay explain why there are few real-world applications of IV method(Garnier, Gilson, Young, & Huselstein, 2007; Liu, Yao, & Gao, 2009 andespecially in robotics (Puthenpura & Shina, 1986; Vandanjon, Janot,Gautier, & Khatounian, 2007; Xi, 1995; Yoshida, Ikeda, & Mayeda, 1992.This shows that a gapmust be bridged between the theory and controlengineering practices.

In this paper, an IV approach which is relevant for the identifica-tion of rigid industrial robots is proposed. The set of instruments isthe inverse dynamic model built from simulated data which arecalculated from the simulation of the direct dynamic model. Thesimulation assumes the same reference trajectories and the same

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/conengprac

Control Engineering Practice

0967-0661/$ - see front matter & 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.conengprac.2013.12.009

n Corresponding author. Tel.: þ33 5 62 25 27 73.E-mail addresses: Alexandre.Janot@onera.fr (A. Janot),

pierre-olivier.vandanjon@ifsttar.fr (P. Olivier Vandanjon),Maxime.Gautier@irccyn.ec-nantes.fr (M. Gautier).

Control Engineering Practice 25 (2014) 85–101

control structure for both the actual and the simulated robot and isbased on the previous IV estimates. In addition, in order to obtain avalid set of instruments, the gains of the simulated controller areupdated according to the IV estimates. This algorithm called asIDIM-IV method validates the inverse and direct dynamic modelssimultaneously, improves the noise immunity of estimates withrespect to corrupted data in the observation matrix and has a rapidconvergence.

A condensed version of this work has been presented in Janot,Vandanjon, and Gautier (2009, 2012). However, the theoreticalproofs are missing, the tuning of control-loops is missing, the gainsupdate of the simulated controller is not well introduced and thestatistical efficiency of the IDIM-IV method is not addressed. Inthis paper, the detailed proofs to enlighten the theoretical under-standing of the IDIM-IV method and additional experimentalresults are given. At last, the statistical efficiency of the IDIM-IVmethod is experimentally addressed.

The rest of the paper is organized as follows: Section 2 reviewsthe usual identification technique of robots dynamic parameters.Section 3 presents the IDIM-IV identification method. SCARAprototype robot and experimental results are presented in Section4 while TX40 robot and experimental results are presented inSection 5. Finally, Section 6 is the conclusion.

2. IDIM-LS method: inverse dynamic identification model andleast squares method

2.1. Inverse dynamic model of robots

The inverse dynamic model (IDM) of n-moving-links robotscalculates the (n�1) vector of joint torques τidm as a function ofgeneralized coordinates and their derivatives (Khalil & Dombre,2002). It is given by

τidm ¼MðqÞ €qþNðq; _qÞ; ð1Þ

where q, _q and €q are respectively the (n�1) vectors of generalizedjoint positions, velocities and accelerations, M(q) is the (n�n)inertia matrix, Nðq; _qÞ is the (n�1) vector of centrifugal, Coriolis,gravitational and friction torques.

The modified Denavit and Hartenberg (DHM) notation allowsobtaining an IDM linear in relation to a set of base parameters β(Khalil & Dombre, 2002)

τidm ¼ IDMðq; _q; €qÞβ; ð2Þ

where IDMðq; _q; €qÞ is the (n� b) matrix of basis functions of bodydynamics and β is the (b�1) vector of base parameters.

The base parameters are the minimum number of dynamicparameters from which the IDM can be calculated. They areobtained from standard dynamic parameters by eliminating thosewhich have no effect on the IDM and by regrouping some of themwith linear relations (Gautier & Khalil, 1990; Mayeda, Yoshida, &Osuka, 1990. The standard parameters of a link j are XXj, XYj, XZj,YYj, YZj and ZZj the six components of the inertia matrix of link j atthe origin of frame j, MXj, MYj and MZj the components of the firstmoment of link j, Mj the mass of link j, Iaj the total inertia momentfor rotor and gears of actuator j, Fvj and Fcj the viscous andCoulomb friction parameters of joint j.

Because of uncertainties (measurement noise, model mismatch…),the ðn� 1Þ vector of actual joint torques τ differs from τidm by an errore i.e.

τ¼ τidmþe¼ IDMðq; _q; €qÞβþe: ð3Þ

Eq. (3) represents the Inverse Dynamic Identification Model (IDIM).The offline identification of base parameters β is considered, given the

measured or estimated offline data for τ and ðq; _q; €qÞ, collected whilethe robot is tracking some planned trajectories.

2.2. Data acquisition

Usually, data available from controllers of robots are themeasurements of q and the measurements of vτ the ðn� 1Þ vectorof control signals.

Usually, robots are position-controlled and the proportional-derivative (PD), the proportional-integral-derivative (PID), thecomputed torque (flatness control) and the passive controls arethe laws which are often applied (Khalil & Dombre, 2002). Whenidentifying the base parameters, the PD control is preferred to theothers because it is easy to tune and an excellent tracking is notnecessary (Gautier et al., 2012).

At last, the motors are current-controlled with a proportional-integral (PI) control. The current closed-loop has a bandwidthgreater than 500 Hz. Then, within the frequency range of bodydynamics (less than 10 Hz), the transfer function of the currentclosed-loop is modeled as a static gain (Gautier et al., 2012).

The joint torques are calculated from the control signals withthe following relation

τ¼Gτvτ ; ð4Þwhere Gτ is the ðn� nÞ diagonal matrix of drive gains. The diagonalcomponents of Gτ have a priori values given by manufacturers.These values can be checked with special tests, see e.g. Gautier andBriot (2012).

2.3. Data filtering

In (3), q is estimated with q obtained by filtering the measure-ments of q through a low-pass Butterworth filter in both theforward and the reverse directions using filtfilt Matlab function.This filter has a flat amplitude characteristic without phase shift inthe range [0 ωf q], ωf q being the filter cutoff frequency. We chooseωf qZ5ωdyn, ωdyn being the maximum bandwidth of the jointposition-loop (Gautier et al., 2012). ð _q; €qÞ are calculated offlinewithout phase shift using a central differentiation algorithm oflow-pass filtered positions q. In doing so, the distortion is avoidedwhile the coefficients of IDMðq; _q; €qÞ are calculated (Gautier et al.,2012).

The IDIM given by (3) is sampled at a measurement frequencyf m while the robot is tracking reference trajectories ðqr ; _qr ; €qrÞ. Weobtain an over-determined linear system of nm equations and bunknowns given by

YfmðτÞ ¼Wfmðq; _q; €qÞβþρfm; ð5Þwhere YfmðτÞ is the ðnm � 1Þ sampled vector of τ, Wfmðq; _q; €qÞ isthe ðnm � bÞ sampled matrix of IDMðq; _q; €qÞ, ρfm is the ðnm � 1Þsampled vector of e.

τ is perturbed by high-frequency disturbances that are rejectedby the closed-loop control. These torque ripples are eliminated byusing a parallel decimation procedure which low-pass filters inparallel Yfm and each column of Wfm and resamples them at alower rate, keeping one sample over nd. This parallel decimation iscarried out with decimate Matlab function. In (6), the low-passfilter cutoff frequency ωf p ¼ 2πn0:8f m=ð2ndÞ is tuned to keep Y andWðq; _q; €qÞ within the same frequency range of dynamics. Accord-ing to Gautier et al. (2012) a good choice is ωf pZ2ωdyn.

After data acquisition and the parallel decimation of (5), weobtain the following over-determined linear system

YðτÞ ¼Wðq; _q; €qÞβþρ; ð6Þwhere YðτÞ is the ðr � 1Þ measurements vector built from actualtorques τ, Wðq; _q; €qÞ is the ðr � bÞ observation matrix built from

A. Janot et al. / Control Engineering Practice 25 (2014) 85–10186

ðq; _q; €qÞ, ρ is the ðr � 1Þ vector of error terms, r¼ nnne withne ¼ nm=nd is the number of rows in (6).

In Y and W, the equations of each joint j are regroupedtogether. Thus, Y and W are partitioned so that

YðτÞ ¼Y1

⋮Yn

264

375;Wðq; _q; €qÞ ¼

W1

⋮Wn

264

375; ð7Þ

where

Yj ¼τjð1Þ⋮

τjðneÞ

264

375;

Wj ¼IDMjðqð1Þ; _qð1Þ; €qð1ÞÞ

⋮IDMjðqðneÞ; _qðneÞ; €qðneÞÞ

2664

3775; IDMjðqðdÞ; _qðdÞ; €qðdÞÞ

is the jth row of the ðn� bÞ matrix of the basis functionsIDMðqðdÞ; _qðdÞ; €qðdÞÞ given by (2).

Yj and Wj represent the ne equations of a subsystem j.Using the base parameters and tracking “exciting” reference

trajectories (Calafiore, Indri, & Bona, 2000b; Gautier & Khalil, 1992;Swevers, Ganseman, Tukel, DeSchutter, & VanBrussel, 1997, a well-conditioned matrix W is obtained. A good conditioning number ofW means that the base parameters are well excited.

2.4. IDIM-LS estimates and statistical analysis

Because robots are nonlinear Multiple Inputs Multiple Outputs(MIMO) systems, ρ is assumed to have zero mean, to be seriallyuncorrelated and to be heteroskedastic with a clustered form i.e. tohave a diagonal covariance matrix Ω partitioned so that Ω¼diagðs2

1Ine ⋯ s2j Ine ⋯ s2

nIne Þ where Ine is the ðne � neÞ iden-tity matrix.

s2j is the error variance calculated from Ordinary LS (OLS)

solution of the subsystem j (see Gautier, 1997) for the technicaldetails)

YjðτjÞ ¼WjðIDMjðq; _q; €qÞÞβþρj; ð8Þ

Thus, we use the Weighted LS (WLS) estimator to estimate β. WLSestimates of β are given by

βLS ¼ ðWTΩ�1WÞ�1WTΩ�1Y: ð9Þ

We use the subscripts LS instead of WLS because weighting opera-tions improve the efficiency of estimates without affecting theirconsistency (Davidson & Mackinnon, 1993; White, 1980). Weightingoperations normalize the deviation of error terms in (6). Indeed, with

ρ¼Ω�1=2ρ;

one obtains EðρρT Þ ¼Ω�1=2EðρρT ÞΩ�1=2 ¼Ω�1=2ΩΩ�1=2 ¼ Irwhere Ir is the ðr � rÞ identity matrix and EðdÞ is the expectationoperator. Furthermore, if the heteroskedasticity is well taken intoaccount, one has

s2ρ;LS ¼ jjΩ�1=2Y�Ω�1=2WβLSjj2=ðr�bÞ � 1:

The covariance matrix of WLS estimates is given by

ΣLS ¼ ðWTΩ�1WÞ�1: ð10Þ

s2βLSðiÞ

¼ΣLSði; iÞ is the ith diagonal coefficient of ΣLS . The relativestandard deviation %sβLSðÞ

of βLSðiÞ, the ith component of βLS , is then

given by

%sβLSðiÞ¼100nsβLSðiÞ

jβLSðiÞjfor jβLSðiÞja0: ð11Þ

The IDIM-LS identification method is illustrated in Fig. 1.

2.5. Limitations of the IDIM-LS method and brief theoreticalbackground of instrumental variable method

To obtain unbiased IDIM-LS estimates, the measurements of qand vτ must be accurate enough at high sampling rate and thebandpass filter described in the previous subpart must be well-tuned. At last, the direct dynamic model (DDM) given by (12) isvalidated a posteriori

MðqÞ €q¼ τidm�Nðq; _qÞ: ð12ÞAn alternative to the IDIM-LS method is the Instrumental Variable(IV) method which deals with noisy observation matrices andwhich can be statistically optimal (Young, 2011).

The LS estimates are unbiased if the following assumptionholds (Davidson & Mackinnon, 1993, chap. 7),

EðWTρÞ ¼ 0: ð13ÞIn this case, W is not correlated with ρ. A violation of assumption(13) leads to biased LS estimates (Davidson & Mackinnon, 1993).Since Wðq; _q; €qÞ is constructed with noisy and filtered data, we candoubt whether Wðq; _q; €qÞ is correlated with ρ or not.

The idea developed by Reiersøl in 1941 (Reiersøl, 1941) consistsin introducing an ðr � bÞ instrumental matrix denoted as Z suchthat (6) becomes

ZTY¼ ZTWβþZTρ:With the following assumptions,

EðZTWÞ exists; is finite and full rank b; ð14Þ

EðZTρÞ ¼ 0; ð15Þthe simple IV estimator provides unbiased estimates given by

βSIV ¼ ðZTWÞ�1ZTY:

It is important to note that the distribution of ρ does not matter forthis analysis. Only the critical conditions (15) and EðρÞ ¼ 0 must beverified (Davidson & Mackinnon, 1993). The normality assumptionmatters for model reduction because the test statistic is mainly basedon the chi square distribution (Davidson & Mackinnon, 1993; White,1980. However, in White (1980), the author shows how to calculate a

Fig. 1. Scheme of the IDIM-LS method.

A. Janot et al. / Control Engineering Practice 25 (2014) 85–101 87

covariance matrix which is robust to heteroskedasticity. The interestedreaders can refer to this work.

The main problem of IV technique is the choice of a valid ðr � bÞinstrumental matrix Zwhich fulfills the assumptions (14) and (15).For identification of physical parameters, the best choice consistsin constructing Z from simulated data only. These simulated dataare the outputs of an auxiliary model which is the noise-freemathematical model of the system. The simulation of the auxiliarymodel can be constructed on the previous IV estimates denoted as

βk�1

IV . Such iterative methods were widely studied and theoreti-cally validated on linear systems (see the references given inintroduction). However, these works are mostly theoretical-oriented and they were validated on low-dimensional linearsystems. For industrial robots identification, they cannot be usedas this. This shows that a gap must be bridged between the theoryand control engineering practices.

In the next section, we aim at bridging this gap by proposing anIV approach relevant for rigid industrial robots identification.

3. An instrumental variable method relevant for identificationof rigid industrial robots

3.1. Choice of a valid instrumental matrix

Because there are as many instruments as columns in W, i.e.z¼ b, the system is said “just identified” (Davidson & Mackinnon,1993). In this case, the true model is assumed to be

Y¼Wðqnf ; _qnf ; €qnf Þβþey ¼Wnfβþey;

where ey is the ðr � 1Þ vector of measurement noise, ðqnf ; _qnf ; €qnf Þare the noise-free vectors of joint positions, velocities and accel-erations respectively, Wðqnf ; _qnf ; €qnf Þ is the noise-free observat-ion matrix denoted as Wnf from now. We make the followingassumption.

EðWTnf eyÞ ¼ 0:

With W¼Wðq; _q; €qÞ, we can write

W¼Wnf þV;

where V is a ðr � bÞ matrix of error terms uncorrelated with Wnf

i.e. EðWTnfVÞ ¼ 0.

Since we have ρ¼ ey�Vβ,W is correlated with ρ and this leadsto biased LS estimates.

A ðr � bÞ valid instrumental matrix is

Z¼Wnf ¼Wðqnf ; _qnf ; €qnf Þ: ð16Þ

Indeed, with

W¼ ZþV¼Wnf þV;

one obtains

EðZTWÞ ¼ EðWTnfWnf ÞþEðWT

nfVÞ ¼ EðWTnfWnf Þ:

Finally, the two following relations hold

rankðEðZTWÞÞ ¼ rankðEðWTnfWnf ÞÞ ¼ b; ð17Þ

EðZTρÞ ¼ EðWTnfρÞ ¼ 0: ð18Þ

Hence, with Z¼Wnf , the assumptions (14) and (15) hold.Now, we must choose and simulate an auxiliary model to

construct an instrumental matrix Z which is as close as possible toZ given by (16). This is the topic of the following subsection.

3.2. Choice and simulation of a valid auxiliary model

For robots, an appropriate auxiliary model is the direct dynamicmodel (DDM) given by (12). The DDM simulation assumes thesame reference trajectories and the same control structure forboth the actual and the simulated robot. Furthermore, the DDMsimulation is based on previous IV estimates. At step k, where k isthe kth IV estimates, the simulated joint accelerations are given by

MðqS; βk�1

IV Þ €qS ¼ τS�NðqS; _qS; βk�1

IV Þ: ð19ÞBy integrating (19), we obtain the ðn� 1Þ vectors of simulated jointvelocities _qS and positions qS. The ðn� 1Þ vector of simulatedtorque τS is given by τS ¼ Gτvτ;S, where vτ;S is the ðn� 1Þ vector ofsimulated control signals.

Like the measurements, the simulated data are sampled at ameasurement frequency f m. The ðnm � bÞ instrumental variablematrix is then

Zfm ¼WδfmðqS; _qS; €qS; βk�1

IV Þ;

where WδfmðqS; _qS; €qS; βk�1

IV Þ is the ðnm � bÞ sampled matrix of

IDMðqS; _qS; €qS; βk�1

IV Þ.Each column of Zfm is low-pass filtered and resampled at a

lower rate, keeping one sample over nd (parallel decimation).Finally, we have:

Z¼WδðqS; _qS; €qS; βk�1

IV Þ:Unfortunately, a simple DDM simulation is not enough to buildZ defined by (16). Indeed, the DDM simulation is based on the

previous IV estimates βk�1

IV . In Janot, Vandanjon, and Gautier

(2013), we have shown that we may obtain ZaZ¼Wnf if the

choice of initial values β0

IV is not relevant. In fact, a bad choice of

β0

IV may lead to algorithm divergence or invalid IV estimatesbecause (14) and (15) may be violated (Janot et al., 2013). To avoidthis, we propose an insensitive-to-initial conditions IV algorithm.This assumes that the following condition

ðqSðβk

IV Þ; _qSðβk

IV Þ; €qSðβk

IV ÞÞ � ðq; _q; €qÞ8 βk

IV ð20Þ

is satisfied at any iteration k, starting with k¼ 0.Furthermore, if assumption (20) holds, then (16) is fulfilled.This is possible by tacking the same control structure for both

the actual and the simulated robot with the same performancesgiven by the bandwidth, the stability margin or the closed-loop

poles. Because the parameters of the simulated robot βk

IV change ateach iteration k, the gains of the simulated controller must be

updated according to βk

IV .For example, let us consider a joint j PD control. We recall that

the PD control is usually used for robots identification because it iseasy to tune and excellent tracking is not required (Gautier et al.,2012).

The joint j IDM can be written as a decoupled double-integratorperturbed by a coupling torque i.e.

τj ¼ τidmj¼Mj;jðqÞ€qj�pj:

pj is considered as a perturbation given by

pj ¼ � ∑n

ia jMj;iðqÞ€qi�Njðq; _qÞ;

where Mj;iðqÞ is approximated by a constant inertia Jj, given by

Jj ¼ ZZjþ Iajþmaxq

ðMj;iðqÞ�ZZj� IajÞ: ð21Þ

A. Janot et al. / Control Engineering Practice 25 (2014) 85–10188

Jj is the maximum value of inertia with respect to q. This givesthe smallest stability margin of the position closed-loop while qvaries. It must be taken at least as ZZjþ Iaj which is calculated froma priori CAD values. The joint j DDM is approximated by a double-integrator system

€qj ¼ðτjþpjÞΜ j;jðqÞ

� ðτjþpjÞJj

:

Hence, it makes sense to use linear techniques to tune theperformances of a joint j closed-loop.

Now, let us consider a joint j PD control of the actual robot asillustrated in Fig. 2. The control input is given by

ντj ¼ akpjakvj ðqrj �qjÞ�akvj _qj; ð22Þ

where ντj is the current reference of motor j, akvj is the joint jfeedback gain of the velocity loop and akpj is the joint j feedbackgain of the position loop.

The joint j torque is given by

τj ¼ agτj vτj ; ð23Þ

where agτj is the actual value of gτj ,aJj is the actual value of Jj.

To tune the tracking performances of qrj , the transfer functionqj=qrj is calculated with pj ¼ 0

Hj ¼qjqrj

!¼ 1ðaJ j=agτj

akvjakpj Þs2þð1=akvj Þðsþ1Þ

¼ 1ð1=aω2

njÞs2þð2aζ j=

aωnj Þðsþ1Þ

where aωnj is the actual natural frequency which characterizes theclosed-loop bandwidth and aζj is the actual damping coefficientwhich characterizes the closed-loop stability margin.

We have

aωnj¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiakvj

akpj

agτjaJj

sand aζj ¼

12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiakvjakpj

agτjaJj

;

vuutand this gives

akpj ¼aωnj

2aζjand akvj ¼ 2aζj

aωnj

aJjagτj :

ð24Þ

The closed-loop performances are chosen with the desired twopoles of a second-order polynomial characterized by dωnj and

dζj .dωnj is the desired natural frequency which characterizes theclosed-loop bandwidth and dζj is the desired damping whichcharacterizes the closed-loop stability margin.

Because the actual values are unknown, the gains are calculatedwith (24) where the unknown actual values aωnj ,

aζj ,aJj , agτj are

replaced by their desired values dωnj ,dζj and by their a priori

values apJj,apgτj respectively. Then, it comes

akpj ¼dωnj

2dζjand akvj ¼ 2dζj

dωnj

apJjapgτj

: ð25Þ

Now let us consider a joint j PD control of the simulated robotas illustrated in Fig. 3.

The variables ðqS;j; _qS;j; €qS;j; τS;jÞ present in Fig. 3 are computed bynumerical integration of (12).

The control structure of the simulated robot is the same as theactual one illustrated in Fig. 2 where agτj is replaced by apgτj .

aJ j is

replaced by Jkj calculated with β

k

IV and skpj ,skvj are the gains of the

simulated controller.These gains are calculated in order to keep the same perfor-

mances for the simulated closed-loop and for the actual one. Then,it follows,

skpj ¼dωnj

2dζj¼ akpj and

skvj ¼akvj

Jkj

apJj¼ 2dζj

dωnj

Jkj

apgτj: ð26Þ

akpj does not depend on the parameters values. Hence, we keepskpj ¼ akpj . But,

skvj must be updated with Jkj at each iteration k.

It is important to note that only the gain skvj in the simulatedclosed-loop is modified during the iterative procedure. The gainsof the actual robot are not modified.

The relation (26) allows to keep ðqSðβk

IV Þ; _qSðβk

IV Þ; €qSðβk

IV ÞÞ �ðq; _q; ; €qÞ8 βk

IV at each iteration k. Finally, after simulating the DDMwith the gains which are updated by (26), after sampling of

simulated data and parallel decimation of Zfm, we obtain

Z¼WδðqS; _qS; €qS; βk�1

IV Þ �Wðqnf ; _qnf ; €qnf Þ ¼Wnf ¼ Z: ð27Þ

In the following sections, it is reasonable to make the followingapproximation Z� Z.

For algorithm initialization, we propose to take a regular inertia

matrix MðqS; β0

IV Þ for having a good initialization for numericalintegration of the DDM. It can be obtained with

β0

IV ¼ 0; except for; Ia0j ¼ 1; for j¼ 1; n: ð28Þ

The inertia of actuator j is taken into account in the IDM (1) asτrj ¼ Iaj €qj.

Then, with the regular initialization given by (28), the inertia

matrix becomes the identity matrix, i.e. MðqS; β0

IV Þ ¼ In, which isthe best regular matrix.

Another point is the choice of initial conditions of state vector,ðqSð0Þ; _qSð0ÞÞ, to integrate the DDM. Because the measured states areassumed to be noisy, we choose ðqSð0Þ; _qSð0ÞÞ ¼ ðqrð0Þ; _qrð0ÞÞ whichis close to ðqð0Þ; _qð0ÞÞ. Since the closed-loop transient response dueto different initial conditions differs between the actual andsimulated signals during a transient period of 5=dωn (settling timecorresponding to a second-order transfer function with a unitarydamping), the corresponding data samples are eliminated fromidentification data.

Compared with the other IV algorithms, the gains update per-formed at each iteration of the IDIM-IV method allows us obtaining

ðqSðβk

IV Þ; _qSðβk

IV Þ; €qSðβk

IV ÞÞ � ðq; _q; €qÞ8 βk

IV leading to Z�Wnf 8 βk

IV .

Thus, the relations (14) and (15) hold 8 βk

IV .

Fig. 2. Actual joint j PD control.Fig. 3. Simulated joint j PD control.

A. Janot et al. / Control Engineering Practice 25 (2014) 85–101 89

This algorithm which is not sensitive to initial values β0

IV is thefirst contribution of our approach.

3.3. Calculation of IDIM-IV estimates

YfmðτÞ and Wfmðq; _q; €qÞ are built according to (5) and Zfm isbuilt according to the procedure described in the previous section.

Yfm and each column of Wfm and Zfm are low-pass filtered andresampled at a lower rate, keeping one sample over nd. Then weobtain

ZTYðτÞ ¼ ZTWðq; _q; €qÞβþZTρ;

where Z is our ðr � bÞ instrumental variable matrix; Y and W aredefined by (6).

In Y, W and Z, all equations of each joint j are regroupedtogether. Thus, like Y and W, Z is partitioned so that

ZðqS; _qS; €qSÞ ¼Z1

⋮Zn

264

375with Zj ¼

IDMjðqSð1Þ; _qSð1Þ; €qSð1ÞÞ⋮

IDMjðqSðneÞ; _qSðneÞ; €qSðneÞÞ

2664

3775;

IDMjðqSðdÞ; _qSðdÞ; €qSðdÞÞ is the jth row of the ðn� bÞ matrix of thebasis functions IDMðqSðdÞ; _qSðdÞ; €qSðdÞÞ given by (2).

The partitions of Yand W are given in (7).Yj, Wj and Zj represent the ne equations of a subsystem j.Because ρ is assumed to be heteroskedastic (see Section 2), the

IV estimates are given by

βk

IV ¼ ðZTΩ�1WÞ�1ZTΩ�1Y: ð29Þ

This solution is called as Weighted IV estimates (WIV).Like with the LS techniques, these weighting operations nor-

malize the variance of error terms.The covariance matrix of IV estimates is given by

ΣIV ¼ ðZTΩ�1WÞ�1ZTΩ�1ZðWTΩ�1ZÞ�1: ð30Þ

s2

βk

IV ðiÞ¼ΣIV ði; iÞ is the ith diagonal coefficient of ΣIV . Relative

standard deviation %sβk

IV ið Þis given by

%sβk

IV ðiÞ¼100ns

βk

IV ðiÞ

βk

IV ðiÞjfor jβk

IV ðiÞ a0;j ð31Þ

where βk

IV ðiÞ is an IDIM-IV estimation of βðiÞ at step k.This process is iterated until its convergence i.e.

jjρkjj�jjρk�1jjjjρk�1jj

rtol1 and maxi ¼ 1; :::; b

jβk

IV ðiÞ� βk�1

IV ðiÞjjβk�1

IV ðiÞjrtol2; ð32Þ

where jjρkjj is the 2-norm of ρ at step k.The parameters tol1 and tol2 are values ideally chosen to get a

good compromise between rapid convergence and accuracy. A goodcompromise consists in choosing tol1 and tol2 between 2.5% and 5.0%.If tol1 and tol2 are too small (less than 1%), then we obtain accurateidentified values but the algorithm converges slowly. On the otherhand, if tol1 and tol2 are too large (greater than 10%), then thealgorithm converges quickly but the identified values are inaccurate.

Compared with the IDIM-LS method and the other methodscited in the introduction, the IDIM-IV method uses the inverseand the direct dynamic models. Thus, these models are validatedsimultaneously. This is the second contribution of our approach.

3.4. Scheme of the IDIM-IV method

The IDIM-IV method is illustrated in Fig. 4 and is summarizedas follows:

Compute the inverse and direct dynamic models with theNewton–Euler equations (Khalil & Dombre, 2002);

Compute W and Y according to ;(5) and (7)Step 0: initialize IDIM-IV with the regular initialization given by(28);

While jjρk jj� jjρk� 1jjjjρk� 1 jj Ztol1 and max

i ¼ 1;:::;b

jβk

IV ðiÞ� βk� 1IV ðiÞj

jβk� 1

IV ðiÞjZtol2

� �do:

Step k: Simulate the DDM by updating gains of thesimulated controller with (26);

Compute Z� Z¼Wnf as described in Section 3.2;Compute IV solution with (29);

End of while.

The direct and inverse dynamic models can be calculatedwith SYMOROþ software developed by the IRCCyN Robotics team.SYMOROþ software can calculate the kinematic and dynamic modelsfrom the robot geometric parameters (Khalil & Creusot, 1997). Inaddition, the number of operations (additions and multiplications) isoptimized in order to have a reduced calculation-time. It comes thatthe IDIM-IV method is a “fully-automated” identification method. Thisis the third contribution of our approach.

Fig. 4. Scheme of the IDIM-IV method. Fig. 5. SCARA prototype robot.

A. Janot et al. / Control Engineering Practice 25 (2014) 85–10190

3.5. Some comments about the IDIM-IV algorithm

ρ is assumed to have zero mean and to be serially uncorrelated.We do not try to identify the parameters of a stable filter that

colors ρ as done with Box–Jenkins model (see e.g. Gilson et al.,2011). In practice, it is very difficult (if not impossible) to choose ana priori structure for such filters. According to the Monte CarloSimulations (MCS) performed in Gilson et al. (2011), the IV methodgives unbiased values even though ρ is colored. We have per-formed MCS on SCARA robot (see Section 4). The results obtainedwith the IDIM-IV method agree with those exposed in Gilson et al.(2011). Hence, the IDIM-IV method is robust to statistical assump-tions made on ρ.

The parallel decimation performed to calculate Y, W and Z canbe related with the prefilters used to perform “optimal IVidentification” (Garnier & Wang, 2008; Gilson et al., 2011). Thedecimate filter cutoff frequency must be compatible with theclosed-loop bandwidth. This leads us to choose ωf p42ωdyn

(Gautier et al., 2012). It comes that the decimate filter and the“optimal prefilter” have the same properties within the frequencyrange [0 ωdyn]. Finally, the use of decimate filter affects theefficiency of IDIM-IV estimates only (unbiasedness is not affected,White, 1980). This point was validated with MCS and will beemphasized during experimental validations.

When using the IDIM-IV method, it is assumed that thecontroller is known. In Gilson et al. (2011), the authors havedeveloped different IV approaches when the controllers areunknown to users. Unfortunately, such algorithms may be ineffi-cient while identifying industrial robots. Indeed, their controllerscontain nonlinear components such as switches, offset compensa-tions, saturation and digital filters with time delay compensationmaking the identification of controllers very difficult even thoughthe structure is known to the users. As an example, we tried toidentify the parameters of the TX40 robot controller (see Section 5)without success, although the controller structure was known tous. Finally, it is much easier to perform gains update given by (26).

0 2 4 6 8 10 12-20

-15

-10

-5

0

5

10

15

20 Joint 1

Mot

or to

rque

(Nm

)

time (s)

Measurement: YfmEstimation: YeError = Yfm - Ye

0 2 4 6 8 10 12-1.5

-1

-0.5

0

0.5

1 Joint 2

Mot

or to

rque

(Nm

)

time(s)

Measurement: YfmEstimation: YeError = Yfm - Ye

Fig. 7. SCARA robot, direct comparisons with IDIM-IV estimates, well-tuned data filtering.

0 500 1000 1500 2000 2500-5

-4

-3

-2

-1

0

1

2

3

4

5Residue of estimation

Nm

Samples

Fig. 8. SCARA robot, IDIM- IV error, histogram of IDIM-IV error and estimated Gaussian.

Fig. 6. DHM frames of SCARA robot.

Table 1IDIM-LS and IDIM-IV estimates of SCARA robot, well-tuned data filtering.

Parameters βLS %sβLS β3IV

%sβIVNominal values

ZZ1 3.450 0.13 3.450 0.15 3.45Fv1 0.013 128.0 0.010 150.0 XFc1 0.782 0.38 0.780 0.38 XZZ2 0.063 0.49 0.063 0.51 0.06LMX2 0.241 0.50 0.240 0.55 0.25LMY2 �0.007 10.7 �0.0064 12.82 0.00Fv2 0.021 0.95 0.021 1.04 XFc2 0.130 0.28 0.132 0.3 X

jjY�WβLSjj=jjYjj ¼ 3% jjY� ZβIV jj=jjYjj ¼ 3%

A. Janot et al. / Control Engineering Practice 25 (2014) 85–101 91

4. SCARA prototype

4.1. Presentation and modeling

In this section, the IDIM-LS and IDIM-IV methods are carriedout on a 2 DOF planar prototype robot (see Fig. 5). This direct-drive

prototype is suitable for the study of the IDIM-IV method becauseit emphasizes nonlinear coupling torques. Indeed, for industrialrobots with gear ratio greater than 50, these nonlinear effects aredivided by 2500. Moreover, the model of this robot depend on8 parameters only. This facilitates the study of identificationmethods with respect to several conditions. This robot and its

0 2 4 6 8 10 12-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04 Joint position tracking errors: joint 1

rad

time (s)

q-qrqs-qrqs-q

0 2 4 6 8 10 12-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

Joint position tracking errors: joint 2

rad

time (s)

q-qrqs-qrqs-q

Fig. 9. Tracking errors at iteration k¼0: blue, error between actual position and reference; red: error between simulated position and reference; black: error betweensimulated and actual positions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

Table 3Errors (%) relative to the reference trajectories.

Iteration k Joint j¼1 Joint j¼2

0 1 2 3 0 1 2 3

100njjqsj �qrj jj=jjqrj jj 1.6 1.4 1.2 1.2 1.56 1.5 1.4 1.4

100njj_qSj � _qrj jj=jj_qrj jj 8 7.2 6.1 6.1 12.5 11.3 9.2 9.2

100njj€qSj � €qrj jj=jj€qrj jj 24 23 20 20 31 30 26 26

0 2 4 6 8 10 12-4

-3

-2

-1

0

1

2

3

4Referecence, simulated, measured joint position 1 and tracking error

rad

time (s)

MeasurementSimulationErrorReference

0 2 4 6 8 10 12-2

-1.5

-1

-0.5

0

0.5

1

1.5Referecence, simulated, measured joint position 2 and tracking error

rad

time (s)

MeasurementSimulationErrorReference

Fig. 10. Joint position at iteration k¼0; blue: measured position; red: simulated position; green: reference; black: tracking error between reference and simulated position.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

Table 2Errors (%) relative to the actual filtered trajectories.

Iteration k Joint j¼1 Joint j¼2

0 1 2 3 0 1 2 3

100njjqSj � qjjj=jjqjjj 0.5 0.49 0.47 0.47 0.5 0.49 0.49 0.49

100njj_qSj � _qjjj=jj _qjjj 2.2 2.1 2 2 4.5 3 2.7 2.7

100njj€qSj � €qjjj=jj €qjjj 6.5 6 5.4 5.4 9.5 9.2 9 9

A. Janot et al. / Control Engineering Practice 25 (2014) 85–10192

nominal parameters are well-known (see Table 1). Thus, we cancheck the physical meaning of estimates.

The geometry description of the robot uses the modifiedDenavit and Hartenberg (DHM) notations (Khalil & Dombre,2002) illustrated in Fig. 6.

The robot is direct driven by 2 DC permanent magnet motorssupplied by PWM amplifiers.

The IDM depends on 8 base parameters:

β¼ ½ ZZ1R Fv1 Fc1 ZZ2R LMX2 LMY2 Fv2 Fc2 �T ;

with ZZ1R ¼ ZZ1þ Ia1þM2L2, ZZ2R ¼ ZZ2þ Ia2 and L¼0.5 m is the

first link length.In the case of SCARA robot, the parameters LMX2 and LMY2 are

identified instead of MX2 and MY2 respectively.The 8 columns of IDMðq; _q; €qÞ are automatically calculated with

SYMOROþ software (Khalil & Creusot, 1997). Their expressions canbe found in Gautier et al. (2012).

The control is the PD structure given by (21) with J1 ¼ ZZ1RþZZ2Rþ2LMX2 and J2 ¼ ZZ2R. The actual gains are calculated with(25) taking a desired damping dζj ¼1 (for joint 1 and 2)

corresponding to no overshoot. The desired natural frequencydωnj is chosen according to the driving capacity without saturationof joint drive. In motion control community, the bandwidth ofposition-loop is limited by the electro-mechanical cutoff frequencyωEM given by, (Gautier et al., 2012),

ωEMj¼ JjK

2τj=RAj

for j¼ 1; 2

where Kτj is the electromagnetic motor torque constant and RAjis

the motor armature resistance.For this robot, we obtain a full bandwidth with dωf

n1¼ 1 ðrad=sÞ

and dω fn2 ¼ 10 ðrad=sÞ.

We tried different controls including PID control and feedfor-ward velocity/acceleration which give better tracking accuracy. Weobtained the same results. This shows that the IDIM-IV method isnot sensitive to the control structure, needs only a simple easy-to-tune control and does not need an excellent tracking to succeed.Consequently, we present the results obtained with the PD control.

Data measurement frequency is f m ¼ 200 ðHzÞ.Torque data are calculated with (4) while the positions are

obtained through incremental encoders (2000 and 5000 (lines/rev) for joint 1 and 2 respectively) with a 4-fold subdivision ofeach encoder line (8000 and 20,000 (pulses/rev) for joint 1 and2 respectively).

The DDM simulation is carried out with the same referencetrajectories and with the same PD structure as the actual SCARArobot. The reference trajectories ðqr ; _qr ; €qrÞ are fifth order poly-nomials. Since we have condðWðq; _q; €qÞÞ ¼ 25, the system (6) is wellconditioned. The parameters are well excited (Gautier & Khalil,1992).

The gains of the simulated controller are updated with (26)where dζ j ¼ 1, dωn1 ¼ 1 ðrad=sÞ and dωn2 ¼ 10 ðrad=sÞ. This givesskp1 ¼ 0:5 ðs�1Þ and skp2 ¼ 5 ðs�1Þ.

0 2 4 6 8 10 12-20

-15

-10

-5

0

5

10

15

20 Joint 1

Mot

or to

rque

(Nm

)

time (s)

Measurement: YfmEstimation: YeError = Yfm - Ye

0 2 4 6 8 10 12-1.5

-1

-0.5

0

0.5

1 Joint 2

Mot

or to

rque

(Nm

)

time(s)

Measurement: YfmEstimation: YeError = Yfm - Ye

Fig. 11. SCARA robot, direct comparisons with IDIM-IV estimates, no data filtering.

Table 5IDIM-LS and IDIM-IV estimates of SCARA robot, no data filtering.

Parameters βLS %sβLS β3IV

%sβIV

ZZ1 1.50 1.60 3.450 1.73Fv1 0.095 80. 0 0.013 384.0Fc1 0.55 23.3 0.80 4.38ZZ2 0.14 6.7 0.063 1.96LMX2 0.63 2.7 0.240 4.38LMY2 0.1 11.8 �0.0065 123.0Fv2 0.001 700.0 0.022 9.0Fc2 0.19 68.40 0.132 9.5

jjY�WβLS jj=jjYjj ¼ 80% jjY� ZβIV jj=jjYjj ¼ 6%

Table 4Convergence of IDIM-IV estimates of SCARA robot.

Parameters β0IV β

1IV β

2IV β

3IV

ZZ1 1 3.449 3.450 3.450Fv1 0 0.013 0.013 0.013Fc1 0 0.783 0.780 0.780ZZ2 1 0.063 0.063 0.063LMX2 0 0.242 0.240 0.240LMY2 0 �0.0060 �0.0064 �0.0064Fv2 0 0.020 0.021 0.021Fc2 0 0.133 0.132 0.132

Table 6

IDIM-IV estimates with simulated half full bandwidth, dωn ¼ dωfn=2.

Parameter β0IV β6IV%sβIV

ZZ1 1 3.451 0.15Fv1 0 0.010 150.0Fc1 0 0.778 0.38ZZ2 1 0.063 0.51LMX2 0 0.241 0.55LMY2 0 �0.0064 12.82Fv2 0 0.021 1.04Fc2 0 0.131 0.3

jjY� ZβIV jj=jjYjj ¼ 3%

A. Janot et al. / Control Engineering Practice 25 (2014) 85–101 93

Table 8Errors (%) relative to the reference trajectories.

Iteration k Joint j¼1 Joint j¼2

0 1 2 3 4 5 6 0 1 2 3 4 5 6

100njjqsj �qrj jj=jjqrj jj 2.1 2.5 1.7 2.1 1.8 1.6 1.6 2.1 1.8 1.5 1.8 1.8 1.6 1.6

100njj_qSj � _qrj jj=jj_qrj jj 10.5 8.0 10.7 8.1 10.6 8.0 8.0 10 11.5 10 7.5 10 8.0 8.0

100njj€qSj � €qrj jj=jj€qrj jj 41 45 41 38 35 33 33 41 45 41 37 35 33 33

Fig. 13. Link frames of TX40 Stäubli robot.

Table 7Errors (%) relative to the actual filtered trajectories.

Iteration k Joint j¼1 Joint j¼2

0 1 2 3 4 5 6 0 1 2 3 4 5 6

100njjqSj � qjjj=jjqjjj 0.75 0.90 0.6 0.7 0.6 0.54 0.54 0.8 0.7 0.65 0.7 0.7 0.67 0.67

100njj_qSj � _qjjj=jj _qjjj 4.0 3.0 4.0 3.0 4.0 3.0 3.0 4.0 4.6 4.0 3.0 4.0 2.8 2.8

100njj€qSj � €qjjj=jj €qjjj 14 17 14 12 11 11 11 14 16 15 12 11 11 11

0 500 1000 1500 2000 2500-5

-4

-3

-2

-1

0

1

2

3

4

5Residue of estimation

Nm

Samples

Fig. 12. SCARA robot, IDIM-IV error, histogram of IDIM-IV error and estimated Gaussian, no data filtering.

A. Janot et al. / Control Engineering Practice 25 (2014) 85–10194

The drive gains in (4) were identified with special tests whichare described in Gautier and Briot (2012). We obtain:apgτ1 ¼ 1:414 ðNm=VÞ; apgτ2 ¼ 0:845 ðNm=VÞ.

This leads to the following initial values for skvj :sk0

v1 ¼ 2=1:414¼ 1:4 ðVsÞ and sk0v2 ¼ 20=0:845¼ 23:67 ðVsÞ.

4.2. Comparison of IDIM-LS and IDIM-IV methods with a well-tunedbandpass filtering

The IDIM-LS and IDIM-IV methods are carried out with filteredpositions q, calculated with a 10 Hz forward and reverse Butter-worth filter and with velocities _q, accelerations €q calculated with acentral differentiation algorithm of q.

The cutoff frequency of Butterworth filter is tuned according tothe guidelines given in Gautier (1997) which are recalled inSection 2. The maximum bandwidth for the joint 2 is ωdyn ¼ 10

ðrad=sÞ and this leads us to choose ωf qZ5ωdyn; ωf qZ50 ðrad=sÞ ¼8 ðHzÞ. Then we choose a 10 Hz cutoff frequency.

The parallel decimation of Yfm, Wfm and Zfm is carried out witha lowpass Tchebyshef filter with a cutoff frequencyωf pZ2ωdyn; ωf pZ20 ðrad=sÞ ¼ 3:18 ðHzÞ. Then we choose a 4 Hzcutoff frequency. According to the relation ωf p ¼ 2πn0:8f m=ð2ndÞ,the sample rate f m is divided by a factor nd¼20.

The IDIM-LS and IDIM-IV estimates are given in Table 1. TheIDIM-IV method needs only 3 steps to converge. The IDIM-LSestimates stick to the IDIM-IV estimates and the estimates arecomparable with their nominal values. According to the Hausman0stheory (Hausman, 1978), the IDIM-LS estimates are unbiased. Hence,like the other identification methods cited in the introduction, theIDIM-IV method does not really improve the IDIM-LS methodassociated with a well-tuned bandpass filtering. This is because onehas Wðq; _q; €qÞ �Wnf leading to EðWðq; _q; €qÞTρÞ � EðWT

nfρÞ ¼ 0.Direct comparisons have been performed (see Fig. 7). The

estimated torques constructed with the IDIM-IV estimates fit themeasured torques (similar results are obtained with IDIM-LSestimates). Since we have jjY�WβLSjj=jjYjj ¼ 3% and jjY� ZβIV jj=jjYjj ¼ 3%, the identification results are of good quality.

The IDIM-IV error is plotted in Fig. 8. Since we havesρ;LS ¼ sρ;IV ¼ 1:025� 1:0, the variance of ρ is normalized and thisindicates that the heteroskedasticity is well taken into account. Inaddition, the histogram of IDIM-IV error plotted in Fig. 8 matches aGaussian distribution. Thus, all the statistical assumptions madeon ρ hold in practice.

Table 11Convergence of IDIM-IV estimates, TX40 robot.

β0IV β

1IV β

2IV β

3IV β

0IV β

1IV β

2IV β

3IV

ZZ1R 1.0 1.24 1.25 1.25 Fc3 0.0 6.00 6.0 6.0Fv1 0.0 8.18 8.20 8.20 MX4 0.0 �0.01 �0.02 �0.02Fc1 0.0 6.54 6.54 6.54 Ia4 1.0 0.03 0.03 0.03XX2R 0.0 �0.47 �0.48 �0.48 Fv4 0.0 1.13 1.15 1.15XZ2R 0.0 �0.15 �0.16 �0.16 Fc4 0.0 2.26 2.27 2.27ZZ2R 1.0 1.08 1.09 1.09 MY5R 0.0 �0.025 �0.03 �0.03MX2R 0.0 2.20 2.21 2.21 Ia5 2.0 0.04 0.04 0.04Fv2 0.0 5.62 5.68 5.68 Fv5 0.0 1.90 1.90 1.90Fc2 0.0 7.75 7.77 7.77 Fc5 0.0 2.75 2.80 2.80XX3R 0.0 0.125 0.13 0.13 Ia6 1.0 0.009 0.01 0.01ZZ3R 0.0 0.12 0.12 0.12 Fv6 0.0 0.64 0.69 0.69MY3R 0.0 �0.60 �0.60 �0.60 Fc6 0.0 1.95 2.00 2.00Ia3 1.0 0.09 0.10 0.10 fvm6 0.0 0.61 0.63 0.63Fv3 0.0 2.00 2.03 2.03 fcm6 0.0 1.78 1.81 1.81

Table 9Geometric parameters of TX40 robot.

j sj αj dj θj rj

1 0 0 0 θ1 02 0 �π/2 0 θ2 03 0 0 d3¼0.225 m θ3 r3¼0.035 m4 0 π/2 0 θ4 r4¼0.225 m5 0 �π/2 0 θ5 06 0 π/2 0 θ6 0

Table 10IDIM-LS and IDIM-IV estimates of TX-40 robot, well-tuned data filtering.

βLS %sβLS βLS %sβLS β3IV

%sβIV β3IV

%sβIV

ZZ1R 1.25 1.1 Fc3 6.10 1.8 ZZ1R 1.25 1.3 Fc3 6.0 1.9Fv1 8.18 0.6 MX4 �0.02 16.0 Fv1 8.20 0.7 MX4 �0.02 20.0Fc1 6.57 2.2 Ia4 0.03 8.8 Fc1 6.55 2.6 Ia4 0.03 9.4XX2R �0.48 2.6 Fv4 1.14 1.4 XX2R �0.48 2.9 Fv4 1.15 1.5XZ2R �0.16 4.3 Fc4 2.30 2.5 XZ2R �0.16 4.8 Fc4 2.27 2.6ZZ2R 1.08 1.0 MY5R �0.03 13.0 ZZ2R 1.09 1.2 MY5R �0.03 14.0MX2R 2.20 2.5 Ia5 0.04 8.8 MX2R 2.21 2.9 Ia5 0.04 11.0Fv2 5.67 1.0 Fv5 1.88 1.8 Fv2 5.68 1.2 Fv5 1.90 2.0Fc2 7.76 1.8 Fc5 2.90 2.9 Fc2 7.77 2.1 Fc5 2.80 3.5XX3R 0.13 9.4 Ia6 0.01 9.4 XX3R 0.13 10.0 Ia6 0.01 10.9ZZ3R 0.12 7.6 Fv6 0.68 1.5 ZZ3R 0.12 8.8 Fv6 0.69 1.6MY3R �0.60 2.2 Fc6 2.10 2.5 MY3R �0.60 2.3 Fc6 2.00 2.8Ia3 0.09 8.8 fvm6 0.63 1.6 Ia3 0.10 9.2 fvm6 0.63 1.8Fv3 2.02 1.6 fcm6 1.80 3.7 Fv3 2.03 1.8 fcm6 1.81 4.2

jjY�WβLSjj=jjYjj ¼ 6% jjY� Zβ3IV jj=jjYjj ¼ 6%

A. Janot et al. / Control Engineering Practice 25 (2014) 85–101 95

The assumption (20) holds at each iteration k with a constantrelative error close to 0.5% for the positions, 5% for the velocities and10% for the accelerations (see Table 2 and Fig. 9). These results validatethe update procedure of the simulated PD control given by (26).

It can be seen in Table 3 and in Figs. 9 and 10, that thesimulated trajectories ðqS; _qS; €qSÞ are 3 to 5 times closer to theactual ones ðq; _q; €qÞ � ðq; _q; €qÞ than to the references ðqr ; _qr ; €qrÞ,with a relative error close to 1.5% for the positions, 15% for thevelocities and 30% for the accelerations. Computing Z with thereference trajectories ðqr ; _qr ; €qrÞ ðqr ; _qr ; €qrÞ leads to invalid esti-mates because the set of instruments is not valid (Table 4).

Then, the right assumption made in Section 3.2 is

ðqSðβk

IV Þ; _qSðβk

IV Þ; €qSðβk

IV ÞÞ � ðq; _q; ; €qÞ8 βk

IV (20), with a constant smallerror. This can be seen in Figs. 9 and 10, at iteration k¼0, with theregular initialization (wrong estimates).

4.3. Comparison of IDIM-LS and IDIM-IV methods withoutdata filtering

The IDIM-LS and IDIM-IV methods are carried out with themeasurements of q and with ð _q; €qÞ calculated by a central

0 1 2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60

80Direct comparison, joint 1

Nm

Time

measurementestimationerror

0 1 2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60Direct comparison, joint 2

Nm

Time

measurementestimationerror

0 1 2 3 4 5 6 7 8-30

-20

-10

0

10

20

30Direct comparison, joint 3

Nm

Time

measurementestimationerror

0 1 2 3 4 5 6 7 8-15

-10

-5

0

5

10Direct comparison, joint 4

Nm

Time

measurementestimationerror

0 1 2 3 4 5 6 7 8-20

-15

-10

-5

0

5

10

15

20

25Direct comparison, joint 5

Nm

Time

measurementestimationerror

0 1 2 3 4 5 6 7 8-10

-5

0

5

10

15Direct comparison, joint 6

Nm

Time

measurementestimationerror

Fig. 14. Direct comparisons of TX40 with IDIM-IV estimates; well-tuned data filtering.

A. Janot et al. / Control Engineering Practice 25 (2014) 85–10196

differentiation algorithm of q without low-pass filtering. Thereis no parallel decimation. The IDIM-IV method starts with theregular initialization.

The IDIM-LS and IDIM-IV estimates are given in Table 5. TheIDIM-LS estimates do not stick to the IDIM-IV estimates and theydo not match their nominal values given in Table 1. Since theIDIM-IV estimates given in Table 5 stick to those given in Table 1,the IDIM-LS estimates are biased. The IDIM-LS method failsbecause of the noise level in the observation matrix Wfmðq; _q; €qÞcoming from the differentiation of q without low-pass filtering. Infact, we have EðWTρÞa0.

The IDIM-IV method succeeds because the instrumental matrix

Zfm ¼WδfmðqS; _qS; €qS; βk�1

IV Þ is calculated with the simulated valuesðqS; _qS; €qSÞ which are very close to the actual ones ðq; _q; €qÞ because ofthe gains update. The set of instruments is thus always valid. Thisvalidation shows that the IDIM-IV method cancels the bias of IDIM-LS

method, coming from a noisy observation matrix Wfmðq; _q; €qÞ. Thisresult was expected because this is a property of IV methods(Hausamn, 1978).

However, we can notice that the IDIM-IV estimates have losttheir statistical efficiency compared with the IDIM-IV estimatescalculated with a parallel decimation. Indeed, the deviations givenin Table 5 are greater to those given in Table 1. This experimentalresult validates the theoretical approach described in the Sections3.2 and 3.5 and shows that the parallel decimation can be relatedwith the “optimal prefilters” used in Garnier and Wang (2008) andGilson et al. (2011).

Direct comparisons have been performed (see Fig. 11). Theestimated torques constructed with the IDIM-IV estimates fit themeasured torques. Since we have jjY� ZβIV jj=jjYjj ¼ 6%, the iden-tification results are of good quality. The IDIM-IV error is plottedin Fig. 12. Since we have sρ;IV ¼ 1:03� 1:0, the variance of ρ isnormalized and this shows that heteroskedasticity is well takeninto account. Furthermore, the IDIM-IV error distribution matchesa Gaussian distribution. This means that all the statistical assump-tions made on ρ hold in practice.

4.4. Robustness of the IDIM-IV method with respect to error in thesimulated closed-loop tuning, dωn

This section investigates the effect of an error between the actualvalue aωn and the simulated value dωn of the natural frequency.

The IDIM-IV method is performed taking half of the full valuesgiven in Section 4.1, dωn1 ¼ dω f

n1=2¼ 0:5 ðrad=sÞ and dωn2 ¼

dω fn2=2¼ 5 ðrad=sÞ, and the same procedure used to obtain the

results shown in Table 1, that is a 10 Hz lowpass Butterworth filterand a parallel decimation filter with a factor nd¼20.

The IDIM-IV estimates given in Table 6 converge in 6 steps.These estimates stick to those given in Table 1 obtained with a fullclosed-loop bandwidth.

The relative errors of positions, velocities and accelerations aregiven in Tables 7 and 8. It can be seen that assumption (20) holdsat each iteration k with constant relative errors larger but close tothe values obtained with the full bandwidth (Table 7). The relativeerrors are close to 0.5% for the positions, 3% for the velocities and10% for the accelerations.

Finally, the IDIM-IV method is not very sensitive to an error in thesimulated closed-loop bandwidth provided that the control structureis known. However, the IDIM-IV method fails beyond 1/3 of the fullbandwidth i.e. with dωn rdωf

n=3. The distortion between the actualclosed-loop bandwidth and the simulated closed-loop bandwidth istoo large and the set of instruments is no longer valid.

5. Stäubli TX-40 robot

5.1. Presentation and modeling

Stäubli TX40 robot has a serial structure with six rotationaljoints. Its kinematics is defined by the DHM notation, Fig. 13(Khalil & Dombre, 2002).

The geometric parameters defining the TX40 frames are givenin Table 9: sj ¼ 0 means that joint j is rotational; αj and dj giverespectively the angle and the distance between zj�1 and zj alongxj�1; θj and rj give respectively the angle and the distancebetween xj�1 and xj along zj. Because all joints are rotational, θj

is the joint j position variable i.e. qj ¼ θj.TX40 robot is characterized by a coupling between joints 5 and 6.

This coupling effect adds two additional parameters: f vm6 the viscousfriction of motor 6 and f cm6 the dry friction of motor 6. All the detailsabout TX40 modeling are given in Gautier, Vandanjon, and Janot(2011). TX40 has 60 base dynamic parameters. The columns ofIDMðq; _q; €qÞ in (2) are obtained using the Newton- Euler recursivealgorithm. SYMOROþ software is used to automatically calculate thecustomized symbolic expressions of models (Khalil & Creusot, 1997).

The joint positions and the control signals are stored with ameasurement frequency f m ¼ 5 kHz. The DDM simulation is car-ried out with the same reference trajectories and with the samePD control structure as the actual TX40 robot. The gains of thesimulated controller are updated with (25). The IDIM-IV method isinitialized with all the base parameters at 0 except Iaj ¼ 1 for ja5

0 500 1000 1500 2000 2500-5

-4

-3

-2

-1

0

1

2

3

4

5IV error and +/-3*sigma

Nm

Samples

Fig. 15. IDIM-IV error and histogram of IDIM-IV error, well-tuned data filtering.

A. Janot et al. / Control Engineering Practice 25 (2014) 85–101 97

and Ia5 ¼ 2 because of the coupling effect. At last, we choosetol1 ¼ tol2 ¼ 2:5%.

A C MEX S-Function of SIMULINK on a 2011 laptop PC with INTELi7 CPU is used to run the DDM simulation. One iteration of IDIM-IV(DDM simulation and calculation of IV estimates) takes 3.5 s for a8 s trajectory. This result tends to show that the DDM simulation isnot a real burden.

Reference trajectories ðqr ; _qr ; €qrÞ are fifth order polynomials. Wehave condðWÞ ¼ 200. So, according to Gautier and Khalil (1992),the reference trajectories excite well the base parameters.

5.2. IDIM-LS and IDIM-IV methods with a well tunedbandpass filtering

The IDIM-LS and IDIM-IV methods are carried out with thefiltered position q, calculated with a 50 Hz Butterworth filter andwith velocities _q and accelerations €q, calculated with a centraldifferentiation algorithm of q. Butterworth filter is tuned accordingto the guidelines given in Gautier (1997) which are recalled inSection 2. The maximum bandwidth for joint 6 is ωdyn ¼ 60 ðrad=sÞleading to choose ωf q � 50 Hz. The parallel decimation is carried outwith a low-pass Tchebyshef filter with a cutoff frequencyωf p � 20 Hz. According to the relation ωf p ¼ 2πn0:8f m=ð2ndÞ, thesample rate f m is divided by nd ¼ 100.

The IDIM-LS and IDIM-IV estimates are given in Table 10. TheIDIM-IV method needs 3 steps to converge (see Table 11) and10.5 s. The IDIM-LS estimates stick to the IDIM-IV estimates.According to the Hausman0s theory (Hausman, 1978), the IDIM-LS estimates are unbiased. Direct comparisons have been per-formed, Fig. 14. The torques reconstructed with the IDIM-IVestimates fit the measured ones (similar results are obtained with

the IDIM-LS estimates). Since we have, jjY�WβLSjj=jjYjj ¼ 6% and

jjY� Zβ3

IV jj=jjYjj ¼ 6%, the identification results are of good quality.The error of IDIM-IV method and its histogram are plotted inFig. 15. Its histogram matches a Gaussian distribution and sinceone has sρ;IV ¼ 1:05� 1:0, the variance of ρ is normalized and thisshows that heteroskedasticity is well taken into account.

The error relative to filtered joint position calculated at eachstep k and for each joint j is given in Table 12. Since these relativeerrors are very small, less than 0.2%, relation (20) is alwayssatisfied. This result emphasizes the effectiveness of updatingthe gains of the simulated controller, (25). The set of instrumentsis always valid and this explains why the IDIM-IV method has arapid convergence.

From 60 base parameters, only 28 are essential to describeTX40 dynamics, the 32 other parameters can be canceled. Indeed,when removed from IDM, the norm of error ρ and the values ofthe 28 essential parameters do not vary significantly, less than 2%.

Finally, the results obtained with TX40 robot support thoseobtained with SCARA prototype robot.

5.3. IDIM-LS and IDIM-IV methods without bandpass filtering

The IDIM-LS and IDIM-IV methods are carried out with the

measurements of q and with ð _q; €qÞ calculated by a central differ-entiation algorithm of q without filtering. There is no paralleldecimation. The IDIM-IV method starts with the regular initialization.The IDIM-LS and IDIM-IV estimates are given in Table 13. Once again,the IDIM-IV method needs 3 steps to converge (see Table 11). Directcomparisons have been performed, Fig. 16. The torques reconstructedwith IDIM-IV estimates fit the measured ones. The identification

results are of good quality because we have jjY� Zβ3

IV jj=jjYjj ¼ 10%.The error of IDIM-IV method and its histogram are plotted in Fig. 17.Its histogram matches a Gaussian distribution and since one hassρ;IV ¼ 1:04� 1:0, the variance of ρ is normalized showing thatheteroskedasticity is well taken into account.

The IDIM-LS estimates do not stick to the IDIM-IV estimates.Since the IDIM-IV estimates given in Table 13 stick to those givenin Table 10, and according to the Hausman0s theory, the IDIM-LSestimates are biased. The IDIM-LS method fails because of thenoisy observation matrix Wðq; _q; €qÞ coming from the derivation ofq without filtering. In fact, we have EðWTρÞa0.

However, we can notice that the IDIM-IV estimates have losttheir efficiency compared with the IDIM-IV estimates calculatedwith a parallel decimation. Indeed, the deviations given Table 13are greater than those given in Table 10.

Table 13IDIM-LS and IDIM-IV estimates of TX40 robot, no data filtering.

βLS %sβLS βLS %sβLS β3IV

%sβIV β3IV

%sβIV

ZZ1R 0.06 5.5 Fc3 5.56 1.4 ZZ1R 1.25 2.6 Fc3 5.9 3.4Fv1 8.10 0.4 MX4 0.06 2.8 Fv1 8.25 1.7 MX4 �0.02 40.0Fc1 6.06 1.3 Ia4 0.01 11.5 Fc1 6.50 6.6 Ia4 0.03 13.0

XX2R �0.08 4.1 Fv4 1.20 1.9 XX2R �0.48 6.0 Fv4 1.16 1.9XZ2R �0.02 6.7 Fc4 2.30 3.5 XZ2R �0.16 10.0 Fc4 2.20 3.8ZZ2R 0.05 3.2 MY5R �0.02 8.1 ZZ2R 1.08 2.4 MY5R �0.03 21.7MX2R 4.20 0.7 Ia5 0.01 6.8 MX2R 2.20 5.8 Ia5 0.04 17.0Fv2 5.15 0.6 Fv5 1.84 1.9 Fv2 5.68 2.3 Fv5 1.95 2.6Fc2 8.26 0.9 Fc5 2.85 1.5 Fc2 7.73 4.1 Fc5 2.80 5.5

XX3R �0.01 20.0 Ia6 0.001 19.0 XX3R 0.13 20.0 Ia6 0.01 15.1ZZ3R �0.05 3.2 Fv6 0.68 2.2 ZZ3R 0.11 19.0 Fv6 0.69 2.2MY3R �0.30 1.8 Fc6 2.00 3.8 MY3R �0.60 4.2 Fc6 2.00 4.0

Ia3 0.05 2.2 fvm6 0.64 1.8 Ia3 0.10 15.0 fvm6 0.64 2.4Fv3 2.21 1.05 fcm6 1.74 3.62 Fv3 2.06 2.8 fcm6 1.79 5.8

jjY�WβLSjj=jjYjj ¼ 75% jjY� Zβ3IV jj=jjYjj ¼ 10%

Table 12Error relative to the filtered joint position.

‖qj �qS;j‖‖qj‖

k¼0 k¼1 k¼2 k¼3

Joint 1 (%) 0.080 0.078 0.078 0.078Joint 2 (%) 0.050 0.045 0.045 0.045Joint 3 (%) 0.050 0.048 0.048 0.048Joint 4 (%) 0.051 0.050 0.050 0.050Joint 5 (%) 0.100 0.097 0.097 0.097Joint 6 (%) 0.120 0.119 0.119 0.119

A. Janot et al. / Control Engineering Practice 25 (2014) 85–10198

It comes that the results obtained with TX40 robot supportthose obtained with SCARA prototype robot. Finally, they defini-tively validate the theoretical approach described in Section 3.

6. Conclusion

In this paper, an instrumental variable approach relevant foroffline identification of rigid industrial robots is introduced. Thismethod is called as IDIM-IV method.

The set of instruments is the inverse dynamic model built fromsimulated data calculated from the simulation of the directdynamic model of the robot. The simulation assumes the same

reference trajectories, the same control structure for both theactual and the simulated robot and is based on previous IDIM-IVestimates. In order to have a rapid convergence, the gains of thesimulated controller are updated according to IDIM-IV estimates.

The difficulties for the choice of initial conditions are overcomewith a “regular initialization” of parameters. Initial state is givenby the initial values of reference trajectories.

An experimental validation is carried out on a 2 DOF prototypeand a 6 DOF industrial robot. The following points were checked:

� Like the other identification methods cited in the introduction,the IDIM-IV method does not really improve the IDIM-LSmethod associated with a well-tuned data filtering;

0 1 2 3 4 5 6 7 8-100

-80

-60

-40

-20

0

20

40

60

80Direct comparison, joint 1

Nm

Time

measurementestimationerror

0 1 2 3 4 5 6 7 8-100

-80

-60

-40

-20

0

20

40

60

80Direct comparison, joint 2

Nm

Time

measurementestimationerror

0 1 2 3 4 5 6 7 8-40

-30

-20

-10

0

10

20

30Direct comparison, joint 3

Nm

Time

measurementestimationerror

0 1 2 3 4 5 6 7 8-15

-10

-5

0

5

10

15Direct comparison, joint 4

Nm

Time

measurementestimationerror

0 1 2 3 4 5 6 7 8-25

-20

-15

-10

-5

0

5

10

15

20

25Direct comparison, joint 5

Nm

Time

measurementestimationerror

0 1 2 3 4 5 6 7 8-15

-10

-5

0

5

10

15Direct comparison, joint 6

Nm

Time

measurementestimationerror

Fig. 16. Direct comparisons of TX40 with IDIM-IV estimates; no data filtering.

A. Janot et al. / Control Engineering Practice 25 (2014) 85–101 99

� The IDIM-IV method is robust to the initialization of para-meters and state;

� The IDIM-IV method is robust to the closed-loop performancestuning errors between the simulated and actual closed-looprobot provided the same control structure;

� The IDIM-IV method has a rapid convergence.� Compared with the IDIM-LS method, the IDIM-IV technique is

particularly interesting because of the following reasons:� The calculation of joint velocities and accelerations is much

easier;� It cancels the bias in the IDIM-LS method due to errors in

bandpass filtering data, or no filtering at all;� It validates the direct and inverse dynamic models of robot

simultaneously. Up to now, the direct dynamic model wasvalidated a posteriori in simulation.

Future work concerns the validation of the IDIM-IV method ona flexible industrial robot.

References

Calafiore G., & Indri M. (2000a). Robust calibration and control of roboticmanipulators. In Proceedings of the American control conference (pp. 2003–2007). Chicago, Illinois June 2000.

Calafiore, G., Indri, M., & Bona, B. (2000b). Robot dynamic calibration: Optimalexcitation trajectories and experimental parameter estimation. Journal ofRobotic Systems, 18(2), 55–68.

Davidson, R., & MacKinnon, J. G. (1993). Estimation and inference in econometrics (p.1993) New York: Oxford University Press.

Garnier, H., Gilson, M., Young, P. C., & Huselstein, E. (2007). An optimal IV techniquefor identifying continuous-time transfer function model of multiple inputsystems. Control Engineering Practice, 15(4), 471–486.

Garnier, H., & Wang, L. (2008). Identification of continuous-time models from sampleddata (p. 2008) Springer-Verlag London Limited: Springer.

Gautier M., &Briot S. (2012). Global identification of drive gains parameters ofrobots using a known payload. In: Proceedings of the international conference onrobotics and automation (pp. 2812–2817). May 2012, Saint Paul, USA.

Gautier, M., & Khalil, W. (1990). Direct calculation of minimum set of inertialparameters of serial robots. IEEE Transactions on Robotics and Automation, 6(1990), 368–372.

Gautier, M., & Khalil, W. (1992). Exciting trajectories for the identification of theinertial parameters of robots. International Journal of Robotics Research, 11,362–375.

Gautier, M., & Poignet, P. (2001). Extended Kalman filtering and weighted least-squares dynamic identification of robot. Control Engineering Practice, 9(2001),1361–1372.

Gautier M. (1997). Dynamic Identification of Robots with Power Model,” InProceedings of the of IEEE international conference on robotics and automation(pp. 1922–1927). Albuquerque, USA.

Gautier, M., Janot, A., & Vandanjon, P. O. (2012). A new closed-loop output errormethod for parameter identification of robot dynamics. IEEE Transactions onControl System Technology, 21(4), 428 (433).

Gautier M., Vandanjon P., & Janot A. (2011). Dynamic identification of a 6 DOF robotwithout joint position data. In Proceedings of the IEEE international conference onrobotics and automation. Shangaï, China.

Gautier, M., Vandanjon P. O., & Presse C. (1994). Identification of inertial and drivegain parameters of robots. In Proceedings of the conference on decision andcontrol (pp. 3764–3769). Lake Buean Vista, USA.

Gilson, M., Garnier, H., Young, P. C., & Van den Hof, P. (2011). Optimal instrumentalvariable method for closed-loop identification. Control Theory & Applications,IET, 5(10), 1147–1154, http://dx.doi.org/10.1049/iet-cta.2009.0476.

Hausman, J. A. (1978). Specification tests in econometrics (1978). Econometrica, 46(6), 1251–1271.

Hollerbach, J., Khalil, W., & Gautier, M. (2008). Model identification (p. 2008)Springer-Verlag Berlin Heidelberg: Springer Handbook of Robotics, Springer.

Janot A., Vandanjon P. O., & Gautier M. (2009). Identification of robots dynamicswith the Instrumental Variable method, In Proceedings of the IEEE on interna-tional conference on robotics and automation (pp. 1762–1767). Kobe.

Janot A., Vandanjon P. O., & Gautier M. (2012). Identification of 6 DOF RigidIndustrial Robots with the Instrumental Variable Method. In Proceedings ofsymposium on system identification 2012. Brussels, Belgium, July 2012.

Janot, A., Vandanjon, P. O., & Gautier, M. (2013). Identification of physical parametersand instrumental variables validation with two-stage leastsquares estimator. IEEE Transactions on Control Systems Technology, 21(4),1386–1393.

Khalil, W., & Creusot, D. (1997). SYMOROþ: A system for the symbolic modelling ofrobots. Robotica, 15(1997), 153–161.

Khalil, W., & Dombre, E. (2002). Modeling identification and control of robots (p.2002) Taylor & Francis, Inc. Bristol, PA, USA: Taylor and Francis.

Kostic, D., de Jager, B., Steinbuch, M., & Hensen, R. (2004). Modeling andidentification for high-performance robot control: An RRR-Robotic arm casestudy. IEEE Transactions on Control System Technology, 12(6).

Liu, T., Yao, K., & Gao, F. (2009). Identification and autotuning of temperature-control system with application to injection molding. IEEE Transactions onControl System Technology, 17(6), 1282–1294.

Mayeda, H., Yoshida, K., & Osuka, K. (1990). Base parameters of manipulatordynamic models. IEEE Transactions on Robotics and Automation, 6(3),312–321.

Olsen, M. M., & Petersen, H. G. (2001). A newmethod for estimating parameters of adynamic robot model. IEEE Transactions on Robotics, 17(1), 95–100.

Olsen, M. M., Swevers, J., & Verdonck, W. (2002). Maximum likelihood identifica-tion of a dynamic robot model: Implementation issues. International Journal ofRobotics Research, 21(2), 89–96.

Puthenpura, S. C., & Sinha, N. K. (1986). Identification of continuous-time systemsusing instrumental variables with application to an industrial robot. IEEETransactions on Industrial Electronics, IE-33(3), 224–229.

Ramdani, N., & Poignet, P. (2005). Robust dynamic experimental identification ofrobots with set membership uncertainty. IEEE/ASME Transactions on Mecha-tronics, 10(2), 253–256.

Reiersøl, O. (1941). Confluence analysis by means of lag moments and othermethods of confluence analysis. Econometrica, 9(1), 1–23.

Söderström, T., & Stoica, P. (1989). System identification, 1989. Great Britain: PrenticeHall International Series in Systems and Control Engineering.

Swevers, J., Ganseman, C., Tukel, D., DeSchutter, J., & VanBrussel, H. (1997). Optimalrobot excitation and identification. IEEE Transactions on Robotics and Automa-tion, 13, 730–740.

Swevers, J., Verdonck, W., & De Schutter, J. (2007). Dynamic model identification forindustrial robots – Integrated experiment design and parameter estimation.IEEE Control Systems Magazine, 27(2007), 58–71.

Vandanjon P.-O., Janot A., Gautier M., & Khatounian F. (2007). Comparison of twoidentification techniques: Theory and application. In Proceedings of the 4th

0 2 4 6 8 10 12

x 104

-8

-6

-4

-2

0

2

4

6IV error and +/-3*sigma

Nm

Samples

Fig. 17. IDIM-IV error and histogram of IDIM-IV error; no data filtering.

A. Janot et al. / Control Engineering Practice 25 (2014) 85–101100

international conference on informatics in control, automation and robotics; SS onfractional order systems (ICINCO) (pp. 341–347). Angers.

White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and adirect test for heteroskedasticity. Econometrica, 48(4), 817–838.

Xi, F. (1995). Effect of non-geometric errors on manipulator inertial calibration.In Proceedings of the international conference on robotics and automationpp.(pp. 1808–1813). May 1995, Nagoya, Japan.

Yoshida K., Ikeda N., & Mayeda H. (1992). Experimental study of the identificationmethods for an industrial robot manipulator. In Proceedings of internationalconference on intelligent robots and systems, IROS 1992 (pp. 263–270). Raleigh,USA.

Young, P. C. (2011). Recursive estimation and time-series analysis: An introduction forthe student and practitioner (2nd Edition). Springer-Verlag Berlin Heidelberg:Springer Verlag.

A. Janot et al. / Control Engineering Practice 25 (2014) 85–101 101