A System Identification Technique for Haptic Devices

Samantha Tam, Eric Kubica, and David Wang

Abstract— The number of unknown parameters in a highdegree-of-freedom (DOF) robot makes traditional identificationtechniques difficult to implement. A non-conventional systemidentification method is proposed to model the Freedom 6S,a force feedback hand controller with 6 DOF. Compared toindustrial robots, haptic devices are back-drivable, and theyare characterized by relatively large forces and low friction.Our method is used to identify parameters such as the mass,centroid, inertia and friction of the manipulator. The numberof unknown parameters in the dynamic equations describingthe system can be reduced by physically locking down selectedjoints. This reduced parameter set can then be identified bya method of nonlinear optimization, using measured data as abasis for cost-function evaluation. By an appropriate sequenceof joint analysis, all the parameters can then be identified.This paper discusses the systematic locking order of jointswith the objective of reducing the number of parametersrequiring identification at each stage. The method is verified bycomparison between simulated and measured dynamic data.

I. INTRODUCTION

Robots with a high number of degrees-of-freedom (DOF)coupled with nonlinear friction characteristics present a greatchallenge for system identification. In some cases, the hapticdevices are simple enough that a basic transfer function canbe used based on linear approximation [1]. The most generalapproach for modeling haptic devices is done by designerswho have knowledge of the physical parameters [2], [3] andthese data are used to form the general dynamic equationsof the system. Our modeling approach addresses complexsystems that cannot be modeled by linear approximation andit is able to model the dynamics with unknown physicalparameters.

In this paper, the dynamic modeling of a 6 DOF hapticdevice, the Freedom 6S1, is studied. It has three direct-drivemotors controlling the translational motion of the base links.The roll-pitch-yaw motion is provided by the wrist joints andthe torque transmissions are done by tendons and pulleysactuated by three additional motors. It is difficult to disas-semble the device and no precise physical parameters (e.g.mass, inertia, centroid location) are available. In addition tothese unknowns, the friction effect in each joint has to be

S. Tam is with the Department of Systems Design Engineering,University of Waterloo, 200 University Ave. W., Waterloo, Canadas2tam@engmail.uwaterloo.ca

E. Kubica is with the Department of Systems Design Engineer-ing, University of Waterloo, 200 University Ave. W., Waterloo, Canadaekubica@kingcong.uwaterloo.ca

D. Wang is with the Department of Electrical and Computer Engineer-ing, University of Waterloo, 200 University Ave. W., Waterloo, Canadadwang@kingcong.uwaterloo.ca

1Manufactured by MPB Technologies Inc., Montreal, Quebec. www.mpb-technologies.ca

accounted for. As a result, the number of unknown systemparameters is further increased.

With many unknowns, it is advantageous to organize andto reduce the number of parameters to be identified at a time.It may be possible to reduce the number of parameters bydecoupling the wrist joints from the base joints. However,this approach still presents many unknowns to be solved ateach step. This paper proposes a non-conventional systemidentification technique to isolate the unknown parametersone joint at a time. Mathematically, only the dynamic equa-tions of the joints of interest are required for system identifi-cation. Physically, joint isolation can be done by locking thenon-relevant joints in place. Through this arrangement, onlythe unknown parameters of the dynamics of the one joint areaffecting the system response. By determining the parametersthat affect each locked joint’s configuration, a sequence ofwhich joints to lock can be established. Hence, the numberof parameters required for identification at any one time isreduced dramatically.

II. ASSUMPTIONS

Some assumptions are made to facilitate the use of thissystem identification approach and are discussed below.

1) Dynamic Model

To isolate the dynamic equation of any one joint, ananalytical dynamic model describing the full DOF of thedevice is required. This dynamic model contains all the ge-ometric unknowns of the system (e.g. mass, inertia, centroidlocation).

2) Coupling Effects

It is assumed that the coupling effects between joints arenegligible when only one joint is unlocked. This is enforcedphysically by restricting any movements from non-relevantjoints. Hence, it is valid to consider only one dynamicequation relevant to that joint.

3) Physical Lock

The joints of the manipulator are physically locked inplace by immobilizing the non-relevant joints. The size ofFreedom 6S and its configuration allow for tying down oflinks to inhibit joints movement.

4) Other Relevant Effects

To have a complete model, other effects affecting themanipulator behaviour must be included for system identifi-cation. For the Freedom 6S, these effects include plant fric-tion, (amplifier) gain and cable drag. In particular, a frictionmodel is incorporated in the system model to account for thedominating nature of friction on the manipulator dynamics.

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

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0-7803-9354-6/05/$20.00 ©2005 IEEE 1240

The LuGre friction model [4] is chosen for Freedom 6Sbecause other non-dynamic friction models [5], [6] are notable to produce the nonlinearities observed from the device.The LuGre model embodies a combination of friction effectssuch as deflection of the contact surfaces, Stribeck effect,stiction, Coulomb and viscous frictions. It has been provento be capable of capturing friction characteristics observedin many applications [7], [8].

III. METHODOLOGY

This section outlines the steps required to identify theunknown parameters of the system.

A. Step 1: An Analytical Dynamic Model of the System

From previous work [9], a 6 DOF analytical dynamicmodel is developed for the Freedom 6S using the La-grangian approach. This is accomplished by determiningthe kinematics and subsequently the dynamics relationshipsusing symbolic programming in Maple2. A schematic of thelinkages and joints are shown in Fig. 1.

Fig. 1. Schematic of links and joints of Freedom 6S.

Symbolic simplifications are done at each intermediatecalculation to reduce the memory required to store the result.Some simplifications made include reducing the number ofunknowns in a 3x3 inertia matrix from 9 parameters to 6parameters by noting that an inertia matrix is symmetric. Thecomplete model presented in Equation (1) includes inertialforces (product of mass matrix, M, and joint acceleration,q), centrifugal and Coriolis forces (product of Christoffelmatrix, C, and joint velocities, q), and gravitation effect (φ).The input to the system is torque (τ ). The individual jointposition, q, is defined with a subscript starting from 0 (i.e.q={q0, q1, q2, q3, q4, q5}).

τ6×1 = M6×6 ∗ q6×1 + C6×6 ∗ q6×1 + φ6×1 (1)

By only allowing one joint movement at a time whilefixing all other joints, a 1 DOF dynamic equation can bedetermined by ignoring the dynamic coupling from other

2A mathematical application software with symbolic computation devel-oped by Maplesoft. www.maplesoft.com

fixed joints from Equation (1). From the analytical model,the resulting 1 DOF dynamic equation is a function ofmasses, inertia, the locked joints positions, and the positionand velocity of the free joint. For example, if only joint iis unlocked, q and q for all the other joints are zero. Therespective dynamic equation describing the joint would be:

τi = Mi,i ∗ qi + Ci,i ∗ qi + φi (2)

where τi is the input torque to joint i, Mi,i is the (i,i) elementof the mass matrix, qi is the joint acceleration of joint i, Ci,i

is the (i,i) element of the Christoffel matrix, qi is the jointvelocity of joint i, and φi is the gravitational effect of jointi.

Hence, six 1 DOF equations, each describing the dynam-ics of one joint, are formulated for different locked jointconfigurations.

B. Step 2: The Locking Sequence

Once the equations from Step 1 are developed, a “lockingsequence” can be established. On the Freedom 6S device, thejoints are constrained physically to ensure locking occurs.The positions of the locked joints need not to be at theirhome positions as long as no relative movement is allowedon that joint. If a torque input is given to the free jointand no joint displacements are recorded from the sensorsof the locked joints, the locked joints are considered well-restrained. This condition is enforced during each test runrecorded for system identification.

Fig. 2 has a list of unknown parameters to be determinedby the associated locked joint configuration. The cells inwhich the number “1” is listed means the correspondingparameters are to be determined from the respective configu-ration test. The nomenclature used to describe the parametersis as follows: “I” prefix refers to the inertia (e.g. I1-12 meansthe inertia of Link 1 and it is the inertia element (1,2) inthe 3x3 inertia matrix); “m” prefix refers to the mass ofthe link (e.g. m1 means the mass of Link 1); “x”, “y”, “z”prefixes refer to the centroid location of the link (e.g. x1means the x position of the centroid of Link 1). Starting withjoint 4, each configuration uses parameters determined bythe previous configuration in order to reduce the number ofunknown parameters for the configuration at the next step. Bylisting the parameters associated with each 1 DOF dynamicequation, it shows that an additional test of keeping joints0 and 1 free at the same time is necessary to determine allthe unknown parameters of the system. Thus, there are sevenconfigurations to be tested. Table I summarizes the lockingsequence of Freedom 6S. In this paper, only the results ofthe single joint tests are presented. The coupled joint test(Configuration 7 in Table I) will be included in future work.

C. Step 3: Additional Parameters

In addition to the Lagrangian dynamics, other effectsmust be incorporated to produce a more accurate model.These effects include gearing and amplifier gain, equilibriumpositions of the joints, cable drag, and friction. The LuGrefriction model is chosen to be used as part of the system

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Fig. 2. Parameters of links 1 to 7 listed by order of locking configuration.

identification of Freedom 6S. It treats the deflection ofcontact surfaces as an internal state to dynamically describethe behaviour of friction [4]. Other non-dynamic frictionmodels, such as simple kinetic and viscous friction models[5], [6] were implemented but they were not able to producethe nonlinearities observed from the device. Hence, theparameters associating with the LuGre friction model isincluded in the list of unknowns. Table II gives a summaryof the notation used to represent these parameters alongwith their descriptions. With these additional parameters, theoverall model describing the one non-constrained joint of themanipulator is shown in Fig. 3.

D. Step 4: Optimizing Locked Joints Parameters

After implementing the friction model and incorporatingall the parameters into the model shown in Fig. 3, a nonlinearoptimization is done in an effort to find parameters thatbest fit the measured data. Note that optimization is done

Fig. 3. Block diagram of locked joint test labeled with system parameters.

on the open-loop system rather than the closed-loop system,allowing friction effects to be more observable.

It was found that the amplifier gain of the system ishighly nonlinear. Chirp signals are first used to determine thefrequency ranges suitable for each joint. At high frequencies,the amplifier gain exhibits nonlinear behaviour. As a result,torque inputs within a certain frequency range are exploredto avoid the nonlinear gain regions. The ranges of frequencyused for experiments start from the lowest frequencies ofthe chirp signals to the highest frequencies at which theamplifier gains are still linear. For tests on each joint, fiveinput cosine torques with the same amplitude and withan uniformly increase in frequencies are used for jointexcitation. However, not all the measured data from theseinputs are used for optimization because of non-uniformsystem behaviour (see Section IV). Table III lists the set ofinputs used for each joint. At different frequencies, differenttypes of friction effects would dominate. We believe thesemeasured data provide a sufficient amount of information forthe optimization to find the suitable friction parameters.

The optimization utilizes a sequential quadratic program-ming method3. The inputs required are the initial guessesof the parameters with their respective lower and upperbounds. In order to make good initial guesses, a SolidWorks4

model of Freedom 6S is made by using the approximatedphysical measurement of the device. Values of inertia, massand centroid location from this SolidWorks model are used asinitial guesses for parameters listed in Fig. 2. In addition, an

3The command used in Matlab optimization toolbox is “fmincon”.4A solid modeling package. www.solidworks.com.

Configuration Joint0 Joint1 Joint2 Joint3 Joint4 Joint51 L L L L U L2 L L L L L U3 L L L U L L4 L L U L L L5 L U L L L L6 U L L L L L7 U U L L L L

TABLE I

THE LOCKING SEQUENCE OF JOINTS: L = LOCKED, U = UNLOCKED

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Notation DescriptionKp Plant gain ( gearing and amplifier gain )

epos Equilibrium positions of jointsKs Spring constant ( modeling cable drag )σ0 Stiffness of bristles ( friction parameter )σ1 Damping coefficient of bristles ( friction parameter )σ2 Damping coefficient of links ( friction parameter )vs Stribeck velocity ( friction parameter )Fs Stiction friction force ( friction parameter )Fc Coulomb fricton force ( friction parameter )

TABLE II

ADDITIONAL MODEL PARAMETERS

Free Joint Amplitude (A) (N*m) Frequency (ω) (rad/s)J0 0.020 1.00

1.331.65

(1.975)(2.3)

J1 0.015 1.001.211.421.62

(1.83)J2 0.02 1.00

1.502.00

(2.50)(2.30)

J3 0.008 0.50.881.251.632.00

J4 0.005 0.51.131.752.38

(3.00)J5 0.008 0.5

1.001.502.002.50

TABLE III

INPUT TORQUE FOR LOCKED JOINT TESTS. THE INPUTS WITH

FREQUENCIES IN “()” ARE NOT USED FOR OPTIMIZATION DUE TO

NON-UNIFORM SYSTEM BEHAVIOUR.

inequality constraint is added to restrict the optimized valueof the stiction force to be larger than that of the Coulombfriction. The cost function used for evaluation is the sumsquared error between the measured data and the simulationdata. Time weighing factor is incorporated into the costfunction to minimize transient effect. The first half of eachdata set is weighed half as heavily than the remaining portionof the data set. The cost function used for optimization isshown in Equation (3).

J = Σni=1(λi × (yi − yi))2 (3)

where J is the cost function, λi is the time weighing factor,yi is the measured data, and yi is the simulated data.

The optimization is done for each locking sequence test

as indicated in Table I. Within each iteration, a quadraticprogramming subproblem is solved and the direction of thesearch is calculated at the end of each step. Convergenceis achieved when the search direction changes less than 2e-6 and the maximum constraint violation is less than 1e-65.Further information on this nonlinear search method can befound in [10].

E. Step 5: Parameter Verification

After identification of all the parameters, they can be usedin the full model (i.e. substituting values into the full DOFmodel) for verification. In this paper, each joint is verifiedindependently. In future work, we will verify the parametersin the closed-loop system of the full model.

IV. RESULTS

Using the method described in Section III, optimization isdone on data sets for each joint. For each test, a set of cosineinput torques is applied in sequence to the joint of interest(i.e. the unlock joint) as shown in Fig. 3. The resulting datasets are used to evaluate the cost function for optimizingthe parameters of each joint. Table III lists the input torquesgiven to the device for generating measured data used foroptimization. A fixed time step of 0.5ms and the ode4 Runge-Kutta solver are used for the simulation of joints 0 - 5.

The results obtained from simulation using the optimizedparameters and the measured data are presented in Figs.4 to 11. Not all five data sets for each joint are used foroptimization because some collected data has variations thataffected convergence. These are the “()” frequencies in tableIII. The results of using different frequency torque inputs arepresented for joint 0, showing a sample result set of differentinput frequencies applied to the same joint. For joints 1-5,only the results of torque input at the middle frequencies (J1:ω=1.42; J2: ω=1.50; J3: ω=0.88; J4: ω=1.75; J5: ω=1.00)are presented due to limited printing space. The simulationtime is chosen such that a relevant number of cycles can bedisplayed.

Fig. 4. Measured and simulated joint positions of Joint 0 with all otherjoints fixed.

5These are the default values used by Matlab.

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Fig. 5. Measured and simulated joint positions of Joint 0 with all otherjoints fixed.

Fig. 6. Measured and simulated joint positions of Joint 0 with all otherjoints fixed.

In general, the simulation closely matches the measureddata. The simulated joint positions of the base joints (Figs.4 to 8) show results that are in agreement with the measureddata. Since these joints are driven directly by motors, lessfriction effects are expected. The results for joints 1-2 haveconsistent amplitudes and frequencies with the measureddata. As for joint 0, a similar consistency is shown at thesteady state response.

Figs. 9 to 11 show the simulated joint positions of the wristjoints. The LuGre friction model is able to produce stictioneffects that agree with experiment. The optimized parametersproduce simulations that are able to match the frequency ofthe measured data completely, with acceptable variations inamplitudes. The measured data exhibits some randomness inamplitude (Fig. 9) that cannot be accounted for in the non-stochastic dynamic model. This may be caused the intrinsic

Fig. 7. Measured and simulated joint positions of Joint 1 with all otherjoints fixed.

Fig. 8. Measured and simulated joint positions of Joint 2 with all otherjoints fixed.

elasticity of the tendons used to transmit torques in the wristjoints, making their responses less predictable.

V. CONCLUSIONS AND FUTURE WORK

In this paper, a non-conventional system identificationtechnique is applied to the dynamic model of the Freedom6S. The resulting simulations match closely with the mea-sured data of the locked joints tests by incorporating theLuGre friction model. Discrepancies between the simulationand measured data can be attributed to two factors. First, theflexibility in amplitude observed in the tendon-driven jointsis not accounted for in the non-stochastic model of Freedom6S. It is speculated that the elasticity of the tendons greatlyaffects the wrist joint positions. This is acceptable as themodel is able to predict the average amplitude. Second, thecost function can be improved if a more suitable friction

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Fig. 9. Measured and simulated joint positions of Joint 3 with all otherjoints fixed.

Fig. 10. Measured and simulated joint positions of Joint 4 with all otherjoints fixed.

model is used to match the friction characteristic observed.The coupled joint tests for joints 0 and 1, and the verificationof parameters using the full 6 DOF model are part of theon-going work of the Freedom 6S modeling. These will beincluded in future publications.

The proposed system identification method is applicableto modeling haptic devices with a large number of unknownparameters. It requires an analytical dynamic model fromwhich a set of 1 DOF equations are extracted to describethe dynamic behaviour of a single joint. By only allowingone joint to move at a time, the number of system unknownsis reduced with the systematic locking sequence prescribed.With less parameters, optimization can be done on the modelin a time-efficient manner.

Fig. 11. Measured and simulated joint positions of Joint 5 with all otherjoints fixed.

VI. ACKNOWLEDGMENTS

The authors gratefully acknowledge the contribution ofHandshake VR6 and the Natural Sciences and EngineeringResearch Council of Canada.

REFERENCES

[1] N. Ando; K. Morioka; P. T. Szemes; H. Hashimoto, ”Development of6-DOF haptic interface for tele-micromanipulation systems,”IECON’01 IEEE 27th Annual Conf., Vol. 1, pp. 444-449, Denver, USA, Nov-Dec, 2001.

[2] N. Cauche; A. Delchambre; P. Rouiller; P. Helmer; C. Baur, C; R.Clavel, ”Rotational force-feedback wrist,” Proc. Int. IEEE Symposiumon Assembly and Task Planning., pp. 210-215, Besancon, France, July,2003.

[3] C.A. Avizzano; F. Bargagli; A. Frisoli; M. Bergamasco, ”The handforce feedback: analysis and control of a haptic device for the human-hand,” IEEE Int. Conf. on Systems, Man, and Cybernetics, Vol. 2, pp.989-994, Nashville, USA, 2000.

[4] C.C. de Wit; H. Olsson, K.J. Astrom, P. Lischinsky, ”A new modelfor control of systems with friction,” IEEE Trans. on Autom. Control.,Vol. 40, No. 3, pp. 419-425, March, 1995.

[5] G. Lai, ”Investigation Of A Haptic Device For Teleoperation,” MASc.Thesis, Department of Electrical and Computer Engineering, Univer-sity of Waterloo, 1999.

[6] C. Turner, ”Development of an Internet visual Telepresence System,”MASc. Thesis, Department of Electrical and Computer Engineering,University of Waterloo, 2000.

[7] D. Kostic; R. Hensen; B. de Jager; M. Steinbuch, ”Modelling andIdentification of an RRR-robot,” Proc. IEEE Conf. Decision andControl., pp. 1144-1149, Orlando, USA, Dec, 2001.

[8] Y. Zhu; P. R. Pagilla, ”Static and dynamic friction compensation intrajectory tracking control of robots,” Proc. IEEE Int. Conf. Roboticsand Automation., pp. 2644-2649, Washington, USA, May, 2002.

[9] S. Tam, ”Modeling and Friction Compensation of A Haptic Device,”MASc. Thesis, Department of System Design Engineering, Universityof Waterloo, under preparation.

[10] MATLAB and toolbox documentation, Optmization Toolbox, Con-strained Optimization, Matlab Release 13, The Math Works, Inc.,2003.

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