2012_A Frequency-Domain Approach for Flexible-Joint Robot Modeling and Identification

8/17/2019 2012_A Frequency-Domain Approach for Flexible-Joint Robot Modeling and Identification

1/6

A Frequency-Domain Approach for

Flexible-Joint Robot Modeling and

IdentificationMaria Makarov ∗,∗∗ Mathieu Grossard ∗

Pedro Rodŕıguez-Ayerbe ∗∗ Didier Dumur ∗∗

∗ CEA, LIST, Interactive Robotics Laboratory, Fontenay aux Roses,F-92265, France (e-mail: [email protected])

∗∗ SUPELEC Systems Sciences (E3S), Control Department,Gif sur Yvette Cedex F-91192, France

Abstract: This paper proposes a control-oriented modeling and identification framework forflexible-joint robot arms using motor-side measurements only. From the perspective of model-based control strategies including an inner feedback linearization loop, the proposed method

allows an explicit treatment of the vibrational behavior induced by the flexibilities. A theoreticalmodel of the partially decoupled system is derived and a frequency-domain identificationprocedure allowing an estimation of the flexible parameters is detailed. The obtained descriptionof the system is experimentally validated on the CEA lightweight robot arm ASSIST.

Keywords: robotic manipulators, flexible arms, feedback linearization, control-oriented models.

1. INTRODUCTION

Over the last two decades modeling and control of flexiblerobots have attracted a special attention of the roboticcommunity (Dwivedy and Eberhard, 2006; De Luca andBook, 2008). These studies are all the more motivatedtoday by emerging applications in service, medical, spaceor industrial fields. Innovative mechanical designs providethe desired features for these applications, such as safetyin case of shared human-robot workspace, leading toan expansive development of lightweight robots (KUKA;DLR; ABB; Barrett Technology; Sugano Laboratory).These mechanisms are often intrinsically flexible due totheir slender structure and/or transmissions and can besubject to resonant modes. In this context, advancedcontrol techniques taking into account the flexibilities arerequired to reach a high control bandwidth for precisehigh-speed operation.

The present study focuses on flexible-joint robots, wherethe transmissions between the motors and the rigid linksare assumed to concentrate the essential part of the elastic-ities (possibly due to harmonic drives, transmission beltsor cable driven mechanisms) and are modeled as springs.When compared with the robot dynamics under standardrigid body assumptions, these flexibilities introduce sup-plementary degrees of freedom between the motor and the joint angles. To cope with this issue, a large number of solutions propose additional sensors to measure the elasticdeformations between the motors and the joints. Theseadditional measurements allow powerful and theoreticallywell founded control strategies such as flexible feedbacklinearization (De Luca and Book, 2008) or full state feed-

back (Petit and Albu-Schaffer, 2011; Albu-Schaffer andHirzinger, 2000). However, these relatively complex so-lutions can not always be implemented on robots in a

standard industrial configuration, i.e. equipped only withmotor position sensors and controlled in real-time at a highsampling rate. Possible control strategies in this case aremotor feedback which may be completed with feedforwardterms based on the desired joint reference trajectory andthe flexible model (De Luca, 2000).

Similarly to the above cited control strategies, the model-ing and identification approaches for flexible-joint robotsheavily depend on the available measurements and theintended use of the model. A control-oriented descriptionmust provide an adequate level of details while remainingexploitable for control design. The simplest model is thesingle joint model (the inertial couplings between the jointsbeing neglected), suitable for single-input single-output(SISO) control strategies. Such a physically parametrizedlinear model has been identified on an industrial robotby Östring et al. (2003). When a higher level of preci-sion is required, the coupled vibration effects have to betaken into account and a multivariable model has to be

considered. In most approaches the rigid body dynamicsare assumed to be known, from CAD estimates or exper-imental identification, reducing the identification problemto the stiffness parameters estimation. Following this ap-proach, Albu-Schaffer and Hirzinger (2001) use additional joint torque sensors to identify the elasticity and dampingseparately for each joint on testbed before the assembly of the robot. Oaki and Adachi (2009) employ additional linkaccelerometers in a gray-box modeling approach. Phamet al. (2001) propose an identification procedure basedon bandpass filtering which uses only motor-side measure-ments, identifying one joint at a time. Hovland et al. (2000)describe a frequency-based identification method under

linearizing assumptions for an industrial robot with twocoupled flexible joints. Nonlinear gray-box identificationand multivariable nonparametric methods for frequency

16th IFAC Symposium on System IdentificationThe International Federation of Automatic ControlBrussels, Belgium. July 11-13, 2012

978-3-902823-06-9/12/$20.00 © 2012 IFAC 583 10.3182/20120711-3-BE-2027.00127


2/6

Fig. 1. Rigid feedback linearization strategy for a robotmanipulator.

response function (FRF) estimation have been developedby Wernholt (2007) for industrial robots.

The present work addresses the identification problemfrom a control perspective. A modeling strategy is pro-posed for flexible-joint robots using only motor-side mea-surements. In particular, the presented method aims atproviding a physically parametrized model suitable both

for further control design and an effective identification. Tothis end, a theoretical description of a partially decoupledflexible system containing an inner feedback-linearizationloop based on a rigid model (Fig. 1) is proposed. The modelstructure being thus fixed, its multivariable frequency-domain identification offers valuable insights on the resid-ual flexible dynamics to be taken into account in the designof the outer-loop controller.

Section 2 recalls the standard dynamic modeling of rigidand flexible-joint robots. In Section 3, the model of theresidual system resulting from the inner model-based loopis derived. Section 4 details the identification methodol-ogy used for frequency-domain validation of this model

and its parameter estimation. In Section 5 the introducedmodeling strategy is applied to a lightweight robot armdeveloped at the CEA LIST in the context of the human-robot interaction and safe manipulation. The proposedtheoretical model is compared with the experimental mul-tivariable FRF allowing an estimation of the unknownflexible parameters.

Following notations are used throughout this paper. Ta-ble 1 introduces the physical variables and parameters. I ndenotes the identity matrix of dimension n.

Table 1. Notations

Name Signification or expression Units (SI)

θm motor angular position radτ m motor torque NmR reduction matrix -θ motor angular position after reduction

θ = R−1θm

rad

q joint angular position radτ motor torque after reduction τ = R τ m NmM L rigid-body inertia matrix kg m

2

J m diagonal matrix of rotor inertias kg m2

J matrix of rotor inertias after reductionJ = R2J m

kg m2

M rigid-body inertia matrix in rigid modelM = M L + J

kg m2

H Coriolis, centrifugal and gravity terms NmK diagonal joint stiffness matrix Nm rad−1

F v, vm joint and motor viscous friction Nm s rad−1

2. ROBOT DYNAMIC MODELS

This section recalls the classical dynamical models for an-link rigid manipulator and its flexible-joint counterpart.

2.1 Rigid dynamic model

Modeling The inverse dynamic model of a n-link rigidmanipulator can be obtained from the Lagrange formalismand is given by:

M (q )q̈ + H (q, q̇ ) + τ f = τ (1)

with τ ∈ Rn the motor torque vector after the reductionstage, q , q̇ and q̈ ∈ Rn the joint positions, velocitiesand accelerations vectors, M (q ) ∈ Rn×n the robot inertiamatrix, H (q, q̇ ) ∈ Rn the vector of Coriolis, centrifugal andgravitational torques, and τ f ∈ R

n the friction torque.

A rigid transmission is assumed between the motor anglesθ after the reduction stage and the joint angles q , so thatθm and q are connected by a purely algebraic relation:

q = θ = R−1

θm (2)

Feedback linearization The nonlinear and coupled dy-namic model (1) can be linearized and decoupled by feed-back (Khalil and Dombre, 2004) according to the schemesummarized in Fig. 1. The linearizing control torque is:

τ = M̂ (q )u + Ĥ (q, q̇ ) (3)

where the estimates M̂ (q ) and Ĥ (q, q̇ ) are updated ateach sampling time for the current position q and velocityq̇ . The new control vector is denoted u. In case of aperfectly known model, applying the control torque (3)to the system (1) of relative degree 2 leads to:

M̂ (q ) = M (q )

Ĥ (q, q̇ ) = H (q, q̇ ) ⇒ q̈ = u (4)

The linearized system (4) therefore consists of n indepen-dent double integrators, and linear SISO controllers canthus be applied to control each of them in an outer loop.

2.2 Flexible-joint dynamic model

In case of flexible-joint robots, relation (2) no longer holds.The elastic behavior in the transmission between motorsand links is represented by a torsional spring (Fig. 2).

Fig. 2. Schematic representation of a flexible joint.

The state of the system is therefore composed of both themotors and links coordinates : x = (θ θ̇ q q̇ )T ∈ R4n. Underthe assumption that the angular velocity of the rotors isdue only to their own spinning (De Luca and Book, 2008),the flexible-joint manipulator is modeled by the reduceddynamic model as follows :

M L(q )q̈ + H (q, q̇ ) + τ fa + K (q − θ) = 0 (5)

J θ̈ + τ fm + K (θ − q ) = τ (6)

with K the joint stiffness matrix, M L the rigid body inertiamatrix, J the diagonal rotors inertia matrix and τ fa andτ fm respectively the joint and motor friction torques.

16th IFAC Symposium on System IdentificationBrussels, Belgium. July 11-13, 2012

584


3/6

3. CONTROL-ORIENTED MODELING APPROACH

In this section the flexible model of a partially linearizedrobot arm using only motor-side measurements is firstderived, then a physically parametrized transfer matrixform of this system is proposed and illustrated on a 2degrees-of-freedom (dof) system.

3.1 Flexible model of a partially linearized robot arm

In the reduced measurements case when only the motor-side signals are available, a partial feedback linearizationbased on the rigid model (1) may be attempted. Thisstrategy is described by the system (Σ) in Figure 1 withinput u and output θ, and composed of the robot describedby (5) and (6) with the inner loop (3). Since the latterdoes not take into account the flexibilities, (Σ) does notconsist on n independent double integrators as expectedin the ideal linearization case. It is instead still nonlinear,coupled and affected by resonant flexible modes. In order

to evaluate these effects, equations (5) and (6) under thefeedback law (3) are expressed in the motor variables. Thefriction terms τ fa and τ fm are assumed to represent theviscous friction contribution with coefficients F v and F vm:

τ fa = F v q̇, τ fm = F vm θ̇ (7)

From (6) we obtain :

q = K −1J θ̈ + K −1F vm θ̇ + θ −K −1τ (8)

Differentiating (8) twice and replacing τ from (3), (5) canbe reformulated as :

θ(4) + A3θ(3) + A2θ̈ + A1 θ̇

= B2ü + B1 u̇ + B0u + dH (9)

with the matrix coefficients Ai and Bj ∈ Rn×n dependingon q and q̇ :

A3 = J −1F vm + LJ (10)

A2 = J −1K (I + M −1L J ) + F vF vm

A1 = J −1KM −1L (F v + F vm)

B2 = J −1 M̂ θ

B1 = L M̂ θ + 2J −1 ˙̂M θ

B0 = J −1KM −1L M̂ θ + L

˙̂M θ + J

−1 ¨̂M θ

L = J −1KM −1L F vK −1

dH = ˆH θ −H q + F vK

−1 ˙̂

H θ + M LK

−1 ¨̂

H θNote that the estimates of M (q ) and H (q, q̇ ) = H q areapproximated by their evaluation in the motor variable θas denoted by M̂ θ and Ĥ θ. This approximation is justifiedin a local study by the relatively slow variations of arm in-ertia with the robot configuration. The difference betweenH q and Ĥ θ is seen as a disturbance. The validity of thesimplifying assumptions successively applied throughoutthis section is confirmed by simulations of the flexiblerobot dynamics.

3.2 Transfer matrix form

Following a frequency-oriented approach, the obtainedsystem is then rewritten under a more easily interpretabletransfer matrix form. In a first-order approximation, the

Fig. 3. Local representation of the system (Σ).

derivatives of M̂ θ in the expression of B0 are neglected.This assumption is justified around a given configurationq = q 0 due to small variations of the robot inertia matrix.With this assumption, (9) can be locally written as an LTIsystem in the Laplace domain:

A(s)Θ(s) = B(s)U (s) + dH (11)

⇒ Θ(s) = A(s)−1B(s)U (s) + A(s)−1dH (12)

⇒ Θ(s) = A(s)−1B(s)U (s) + d̃H (13)

with

A(s) = s4I n + s3A3 + s2A2 + sA1 (14)

B(s) = s2B2 + sB1 + B0 (15)

The term dH is seen as a filtered output perturbation d̃H and is not intended to be identified but instead treatedwith the help of robust control techniques in a future outer-loop control strategy.

Finally, the local model (13) can be expressed around q 0in the transfer matrix form :

Θ(s) = G(s)U (s) + d̃H (16)

with G(s) ∈ Cn×n. The bloc diagram of this system isshown in Fig. 3.

3.3 Discussion on a 2-dof example

The term Gij(s) of the matrix G in (16) represents theinfluence of input uj on output θi. For a 2-dof arm case(n = 2), Gij(s) has the following form :

Gij(s) = a0(s + a1)(s

2 + a2s + a3)(s2 + a4s + a5)

s(s + b1)(s + b2)(s2 + b3s + b4)(s2 + b5s + b6)

(17)

with ai and bj real scalar coefficients.

Without viscous motor and joint friction, the terms L, A3,A1 and B1 vanish so that

A0(s) = s4I n + s2A02 (18)

B0(s) = s2B02 + B00 (19)and (17) reduces to:

G0ij(s) = a0(s

2 + a3)(s2 + a5)

s2(s2 + b4)(s2 + b6)

(20)

with ai and b

j real scalar coefficients.

As the decoupled double integrator model expected in thecase of an ideal feedback linearization (4) is often usedfor the design of the outer-loop controller in Fig. 1, it isimportant to observe the differences that might appear ina case of a robot affected by joint flexibilities, in order totake them explicitly into account.

Figure 4 shows the typical profile of the FRF of the 2-dof system G in a given configuration q 0, assuming aperfectly known rigid part of the model, with and without


585


4/6

friction, compared with the ideal double integrator. Notethat the double integrator behavior initially expected onthe diagonal terms is now affected by anti-resonancesand resonances due to a finite stiffness matrix K . Thesystem including viscous friction behaves only as a simpleintegrator at low frequencies. Besides, the system is stillcoupled due to non-null extradiagonal terms.

Fig. 4. Frequency response of the flexible model with andwithout friction.

4. IDENTIFICATION METHODOLOGY

This section details the identification procedure employedto validate the previously derived model. The method forexperimental multivariable FRF estimation is described,including the choice of exciting input signals and theestimator used (see Wernholt (2007) for more details).

4.1 Input signals choice

Exciting input signals for FRF measurements must bechosen carefully. Periodic signals are preferred as theyavoid leakage issues in the Discrete Fourier Transform(DFT) computations and improve the signal-to-noise ratio(SNR) by averaging over periods. Sums of sinusoids alsocalled multisine signals appear as an attractive solution toachieve a certain spectrum at precisely specified frequen-cies in only one experiment, thus reducing the measure-ment time. Random phase multisines are defined as :

u(t) =

N f

k=1Akcos(ωkt + φk) (21)

with ωk ∈

2πlN pT s

, l = 0, 1..N p2 − 1

. N f is the number

of excited frequencies ωk, Ak the signal amplitudes, φkthe random phases uniformly distributed on [0, 2π], T s thesampling time, and N p (even) the length of the signal.Odd random phase multisines can be used to excite onlythe odd harmonics

ωoddk ∈

2π

N pT s(2l + 1), l = 1..

N p4 − 1

(22)

The choice of such signals has been suggested in (Schoukenset al., 2001) to obtain the best linear approximation of a dominantly linear system under nonlinear distortions.

Random phase multisines allow averaging the FRF overseveral realizations to reduce the nonlinear effects, andthus combine the benefits of random and periodic signals.

4.2 Multivariable frequency response estimation

Consider a linear system G with nu inputs u and ny out-puts y. Assuming periodic data affected by measurementnoise V , with U (ωk) and Y (ωk) the DFTs of the inputand output at frequency ωk, the following input-outputrelation holds:

Y (ωk) = G(ωk)U (ωk) + V (ωk). (23)

To estimate G(ωk) ∈ Cny×nu , data from ne ≥ nu different

experiments is collected:

Y (ωk) = G(ωk)U (ωk) + V (ωk) (24)

with U U U (ωk) ∈ Cnu×ne and Y Y Y (ωk) ∈ C

ny×ne .

In case of a nonlinear system, a linear approximation of the input-output relation is sought. Different estimatorscan be used to compute the FRF estimate Ĝ from (24).In (Wernholt, 2007) the harmonic mean estimator hasbeen reported as providing better estimates in case of lowSNR, for instance at low frequencies or at resonances. The

harmonic mean estimator is defined as :

Ĝ(ωk) =

1

N e

N em=1

Ĝ[m]

−1−1

(25)

with

Ĝ[m] = Y [m](ωk)U

[m](ωk)−1

(26)

⇒ Ĝ[m]−1 = U [m](ωk)Y

[m](ωk)−1

(27)

assuming ny = nu and ne = N e×nu experiments in orderto partition the system (24) into N e integer number of blocs of size nu × nu to average over.

5. A CASE STUDY - THE ASSIST ROBOT ARM

In this section the previously described modeling andidentification procedure is applied to the CEA robot armASSIST. The obtained experimental results are comparedwith the theoretically derived model in Section 3.

5.1 System description

For the flexible model validation purposes, the 7-dof AS-SIST robot arm is considered without loss of generality asa two-joint manipulator, the other five rotational dof beingfixed. This system therefore corresponds to the theoreticalmodel proposed in Section 3.3. The two joints of interest

are the shoulder j1 and the elbow j2 (Fig. 5) so thatthe robot motion is restricted to the vertical plane. The joint actuators are based on a screw-and-cable mechanismto ensure a high mechanical backdrivability, essential forsafe human-robot interaction (Jarrasse et al., 2008). DCmotors driven by PWM servo amplifiers in torque modeare employed. The motor shafts are equipped with incre-mental position encoders. The robot arm is controlled bya real-time dedicated controller running VxWorks, with asampling time T s = 3ms.

Detailed results on the rigid system modeling and controlcan be found in (Makarov et al., 2011). The control schemebased on the rigid feedback linearization (Fig. 1) is applied.

This strategy provided satisfactory results in the trajec-tory tracking experiments at low speed. However, cable-based transmissions introduce joint flexibilities which can


586


5/6

Fig. 5. CAD view of the ASSIST arm with 2 actuated joints j1, j2 considered in this paper, and 5 other dof.

Fig. 6. Arm configurations used for the experimentalfrequency response estimation.

not be neglected for efficient high-speed operation anddisturbance rejection, and require a flexible identification.

5.2 Experimental identification and model validation

Experimental protocol The local model (16) dependingon the robot configuration q 0, the identification experi-ments have been performed in 7 different configurationsshown in Fig. 6. They correspond to three angular posi-tions for each of the 2 joints, q mini = −π/2 rad, q

medi = 0

rad and q maxi = π/2 rad. Odd random phase multisinesare used as input signals for the closed-loop identification,with the odd frequencies selected over the range 0.5-12Hzfrom the grid (22). Each joint is excited separately, with5 realizations of the input signal, resulting in ne = 5 × 2experiments. The used signal length is N p = 212 points.The multivariable frequency response is estimated usingthe harmonic mean estimator (25).

Experimental results Figures 7 to 10 show the obtainedfrequency responses of the 2-dof ASSIST arm in the testedconfigurations. They represent the transfers G11, G22,G21 and G12 according to the notation introduced inSection 3.3. All configurations display an antiresonancefollowed by a resonance varying between 4.5Hz and 8.5Hz.The resonances are grouped around 8Hz with deviation of 1Hz for all configurations except for P1a. In bended con-figurations (P2a, P2b, P3a, P3b) additional low-frequency

resonances appear on G22. The observed resonant behaviorconfirms the necessity of an active vibration dampingcontrol strategy to achieve high control bandwidth.

Fig. 7. Experimental frequency responses of the transferG11 from u1 to θ1.



Comparison with the theoretical model In Fig. 11 theflexible model (16) is superimposed with the experimentalfrequency response in two configurations (P1b and P2b).The rigid body parameters are assumed known fromprevious identification experiments (see Makarov et al.(2011)). The unknown parameters K , F v and F vm in theflexible model (16) are adjusted in the frequency domainto match the resonance and the law frequency behavior. Inthe present case, the identified joint stiffness parametersare K 1 = 1150 Nm/rad for j1 and K 2 = 220 Nm/rad for j2.

The difference in the resonance frequency between thesetwo experiments is entirely explained by the variation of the rigid body inertia matrix with the configuration q 0.


587


6/6


Fig. 11. Experimental and theoretical MIMO frequency re-sponse of the approximately decoupled flexible systemin configurations P1b and P2b.

6. CONCLUSION

In this paper, a new modeling approach was proposedfor flexible-joint robots based on motor-side measurementsonly and an inner feedback linearization loop. A physicallyparametrized flexible model was identified and validatedusing experimental FRF measurements, providing valu-able insights for further control design of the outer-loopcontroller.

The observed resonances at relatively low frequencies mo-tivate active damping solutions to achieve a high control

bandwidth. Robust control strategies will be considered todeal with the flexible modes variation in loaded conditions.

REFERENCES

ABB (2011). FRIDA robot - an ABB conceptrobot for industrial dual-arm assembly applica-tions. URL http://www.abb.fr/cawp/abbzh254/8657f5e05ede6ac5c1257861002c8ed2.aspx.

Albu-Schaffer, A. and Hirzinger, G. (2000). State feedbackcontroller for flexible joint robots: a globally stableapproach implemented on DLR’s light-weight robots.In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2000), volume 2, 1087 –1093.

Albu-Schaffer, A. and Hirzinger, G. (2001). Parameteridentification and passivity based joint control for a7 dof torque controlled light weight robot. In IEEE

International Conference on Robotics and Automation (ICRA 2001), volume 3, 2852 – 2858 vol.3.

Barrett Technology (2011). WAM arm. URL http://www.barrett.com/robot/products-arm.htm.

De Luca, A. (2000). Feedforward/feedback laws for thecontrol of flexible robots. In IEEE International Con- ference on Robotics and Automation (ICRA 2000), vol-ume 1, 233–240.

De Luca, A. and Book, W. (2008). Robots with flexibleelements. In B. Siciliano and O. Khatib (eds.), Springer Handbook of Robotics , 287–319. Springer.

DLR (2011). Medical robotics - MIRO. URL http://www.dlr.de/rm/en/desktopdefault.aspx/tabid-3828/.

Dwivedy, S. and Eberhard, P. (2006). Dynamic analysisof flexible manipulators, a literature review. Mechanism and Machine Theory , 41(7), 749–777.

Hovland, G., Berglund, E., and Sørdalen, O. (2000). Iden-tification of joint elasticity of industrial robots. InExperimental Robotics VI , volume 250 of Lecture Notes in Control and Information Sciences , 455–464. Springer

Berlin / Heidelberg.Jarrasse, N., Robertson, J., Garrec, P., Paik, J., Pasqui,V., Perrot, Y., Roby-Brami, A., Wang, D., and Morel,G. (2008). Design and acceptability assessment of anew reversible orthosis. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2008), 1933–1939.

Khalil, W. and Dombre, E. (2004). Modeling, identification & control of robots . Butterworth-Heinemann.

KUKA (2011). Lightweight robot (LWR). URL http://www.kuka-robotics.com/en/products/addons/lwr.

Makarov, M., Grossard, M., Rodriguez-Ayerbe, P., andDumur, D. (2011). Generalized predictive control of an anthropomorphic robot arm for trajectory tracking.

In IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM 2011), 948 –953.

Oaki, J. and Adachi, S. (2009). Decoupling identificationfor serial two-link robot arm with elastic joints. In 15th IFAC Symposium on System Identification , volume 1,1417–1422.

Östring, M., Gunnarsson, S., and Norrlöf, M. (2003).Closed-loop identification of an industrial robot con-taining flexibilities. Control Engineering Practice , 11(3),291–300.

Petit, F. and Albu-Schaffer, A. (2011). State feedbackdamping control for a multi dof variable stiffness robotarm. In IEEE International Conference on Robotics and

Automation (ICRA 2011), 5561 –5567.Pham, M., Gautier, M., and Poignet, P. (2001). Identifi-cation of joint stiffness with bandpass filtering. In IEEE International Conference on Robotics and Automation (ICRA 2001), volume 3, 2867–2872.

Schoukens, J., Pintelon, R., Rolain, Y., and Dobrowiecki,T. (2001). Frequency response function measurementsin the presence of nonlinear distortions. Automatica ,37(6), 939–946.

Wernholt, E. (2007). Multivariable frequency-domain iden-tification of industrial robots . Ph.D. thesis, LinköpingUniversity, Sweden.


588

Documents

2012_A Frequency-Domain Approach for Flexible-Joint Robot Modeling and Identification