jamisola dynamics parameters 09

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DynamicsParametersdentificationof Rigid-BodyManipulatorshroughNaturaloscillationsRodrigo S. Jamisola,Jr.t, Elme] p. Dadiost, and Marcelo H. Ang, Jr.*

Abstract-This work will present an experimental procedurein identifying the moments of inertia, mass,and center of massof a rigid-body manipulator through natural oscillations.Basedon the responseof the manipulator, the values of the dynamicsparameters to be identified are adjusted in such a way thatthe desired natural oscillation is achieved. The experimentsto be performed on a manipulator under torque control arecomposedof two sets: mass and center of mass identification.and momentsof inertia identification. The first setof experimentswill use an openJoopcontrol,such hat the natural oscillation sminimized. The secondsetof experimentswill use a proportionalcontrol, such that the square of the natural oscillation is equalto the proportional gain. Thus each set of experiments has awell.defined objective function such that it can be treated asan optimization computation. The correct dynamics parametersare idenffied when the desired objective function is achieved.The proposed dynamics identification method is analyzed anda theorem is presentedto support the claims presented n thiswork, togetherwith simulationresul8.

Index Terms-moment of inertia,mass,centerof mass, denti-fication experiment, dynamics model, natural oscillation, torquecontrolI. INTRODUCTION

A full dynamics control of a robot manipulator can onlybe achieved with accurate information on the manipulator'sdynamics parameters.A very good performance in full dy-namics simultaneous orce and motion control shown in [l]was achievedbecause he manipulatordynamicswere properlymodeled and identified. The accuracyof dynamics modelinglies heavily on accuratevalues of the manipulator'sdynamicsparameters,namely, the momentsof inertia, mass,and centerof mass. Although some robot manufacturers do provideinformation on mass and center of mass parameters 2l, themoments of inertia are not provided. One possiblehesitationin providing the inertia values by manufacturers s the factthat at best, they only provide inertia values based on thetype of material used and CAD drawing of the physicalsystem. However, values derived through this method canbe inadequatebecauseof other factors like gear ratio andmotor inertia, which could offset the true valuesof the inertiaparameters.But even if all the dynamics parametersare provided by amanufacturer, rom the point of view of full dynamicscontrol,

, tDept. of Electronics and Communications Engineering and+Dept. of Manufacturing Engineering and Management, O" LaSalle University - Manila, 2401 Taft Ave, 1004 Manila, philippines{ rod r i go . j am i so l a , e tme r . dad i os }od l su . edu . ph .*Dept. of Mechanical Engineering, National University of Singapore, 21Lower Kent Ridge Road, Singapore 119077 mpeangh@nus . edu. sg.This work is supportedby the University ResearchCouncil of De La SalleUniversity - Manila.

it would be good to verify the accurateness f the given valuesespeciallywhen the given manipulatorhas achieved housandsof hours of operation. Thus, an experimental procedure toidentify the mass, center of mass, and inertia values can bemore appropriate.This work will proposean experimental procedure n iden-tifying mass, center of mass, and inertia parameters hroughnatural oscillation. The experiment sets the manipulator toachieve a linear second-ordersystem response and its os-cillation is measured. A well-defined value of the desired

natural oscillation transforms the identification experimentinto an optimization computation.The period of oscillation isminimized in the mass and center of mass dentification withthe manipulatorunder open-loop torque control. While in theinertia identification, the square of the period of oscillationis equal to the proportional gain, with the manipulator underproportional torque control.There are severalstudies n inertia parameter dentificationdesignedto accuratelymodel and control a physical system.Among such studies nclude a moment of inertia identificationfor mechatronic systemswith limited strokes [3] in utilizingperiodic position reference nput identification ofinertia basedon the time averageof the product of torque reference nputand motor position. This study showed he moment of inertiaerror is within *25Vo. On tracking a desired trajectory andnoting the error response, inertia parametersare identifiedusing adaptive feedback control [4] where angular velocitytracking are observed and inertia parameters are changed,while a globally convergent adaptive tracking of angularvelocity is shown in [5]. In other studies, east squareserrorin the response is used in inertial and friction parametersidentification for excavatorarms [6]. It is also used o identifyinertias of loaded and unloaded rigid bodies for test facilities[7], and in another experimental set up [8].Inertias that are dependent on the same joint variables,and is also referred to as the minimal linear combinationsofthe inertia parameters 9], can be combined to form lumpedinertias.This inertia model has been shown even n an earlierwork on a completemathematicalmodel of a manipulator [10].In most cases umped inertias, instead of individual inertias,are identified. This is because umped inertias are easier toidentify.

A similar work that identified lumped inertias throughnaturaloscillation s shown n [11]. The shown ull dynamicsidentification method wasthe foundation n a successful mple-mentation of a mobile manipulator [1] performing an aircraftcanopy polishing at a controlled normal force of 10 N. Thebiggest challenge n implementing the dynamics dentification10

Philippine Computing Journal, Vol. 4, No. 2, December 2009, pp. 10-18.


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in [11] is its need for a simplified symbolic model in order toidentify the lumped inertias. Deriving the simplified symbolicmodel can become very computationally expensive when thenumber of degrees of freedom of the manipulator increases.In addition. the mass and'Centerof mass were not identifiedand were assumedas given. r

This work differs from [11] in the sense hat the identifi-cation procedure proposed in this work does not require thesimplified symbolic model because he individual inertias areidentified, and not the lumped inertias. The mass and centerof mass are individually identified as well. These two pointscomprise the major contributions of this work.

More recent identification procedures of manipulator armdynamics include the use of torque data flzl, floating-basemotion dynamics [3], iterative learning [14], neural networkaided identification [15], set membership uncertainty [16],and simultaneousdentificationand control [17]. Identificationproceduresthat are more robot specific include tl8l, tl9l,l2}l, I2ll. Some examples of identification proceduresforhumanoid robot dynamics used neural network [22], andfuzzy-stochastic unctor machine[23]. Recenthumanoidrobotmodeling and control include fuzzy neural network [24]'global dynamics [25], and ground interaction control [26].

In most identification procedures he identification experi-ment is dependenton minimizing the leastsquareserror basedthe robot response to a predefined path which excites thedynamics parameters.The controller used n the experimentalprocedures s the same confioller that were used in the actualrobot control. This approachcan be at a disadvantage ecausecharacterizinga path that isolates the dynamics contributionof each parameter can be hard to find. Thus in some cases,the accurateness f each ndividual physical parametervaluesare at times compromised.

The experimental procedure proposed in this work makeuse of an independentcontroller and is dependenton eachlink achieving natural oscillation. The estimated values ofthe excited physical parameter can be verified by noting thechange n the period of oscillation basedon its predeterminedvalue. The parameter estimates are adjusted such that thedesiredncrin'l ^r -^'-;;l .o'iiiariu, or a glven lmk is achieved.This isolateseach ndividual physicalparameterestimate o thesystemresponse.

This work will propose an experimental procedure, basedon natural oscillation, to identify the individual moment ofinertia, mass,and center of massparametersof rigid-body ma-nipulators.The objective t to make the rigid-body manipulatorachieve inear second-order esponsesuchthat a desiredvalueof its natural oscillation is known. When the desired naturaloscillation is reached, he correct parametersare found. Thiswork is aimed at providing an alternative in dynamics iden-tification methods hat is easily implementable, hat identifiesindividual dynamicsparameters, nd with comparablyaccurateresults. A theorem is presented o support the mathematicalprinciples behind the experimental procedure together withexperimentalresults.

Fig. l. Subfigure (a) shows an upright pendulum nrming about point A dto gravity and an initial displacement from equilibrium position. Subfigu(b) shows the free body diagram.

II. OVERVIEWA. Linear Second-Order SYstems

The sum of the kinetic energy K and potential energy Pof an upright pendulum, as shown in Fig' l, consisting ofslenderbar with mass tn and length I is l27l

and s constant,where1 is the inertia of the slenderbar,g is thgravitational constant,and 0 is the angulardisplacement romthe vertical axis. Thking the time derivative and assuming smaangulardisplacements,he aboveequationcan be expresseda

**!s:o (dtz 2l -with constant frequency of oscillation o)2 : mglf2l . Thsystm n (2) is a linear second-order ystemsimilar to a masspring oscillator shown in Fig. 2 which is definedby

(1 "+k r :o (clt. mwhere the frequct,ugo' pgcillation a2 : kI m. The relationshin (2) shows the frequ3' ey of oscillation being dependentothe physical characteristicof the pendulum. Thus a differependulum with different physical characteristics will givedifferent value of the frequency of oscillation. When no torqugiven at point A, the only moment acting on the pendulum that due to gravity. When an initial displacernent is given, will causethe system to respond with angular acceleration anachievenatural oscillation.

Next we consideran invertedpendulum as shown n Fig. This time, a torque t1 is given at point A as shown in thfree-body diagram of Subfigure (b). With the torque sent the system that is equal to the moment due to gravity, thinverted pendulum will now "float" against gravity. Whean initial push is given, the inverted pendulum will achieangular acceleration such that the effective torque sent to thsystem s

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(b)a)

| / d0 \2 IK + P : ; , \ ; ) - l m s t c o s o (


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. . 70 - l =ko | :0 .I (7)where fko: ct2. Therefore, the inertia is correctly modeledwhen the frequency of oscillation az : kp.D. Multi-Link Rigid-Body Dynamics : : !

The generalcaseof the multi-linked rigid body dynalrtiis ofa manipulator will be presented.The torqueto be sent to eachjoint of the robot is computed by taking the kinetic eneigy'Kand potentihl energyP and solving the Lagt'ange'equdtibnofT : K * P [ 2 8 1 ; ' : ' r r

Fig.2. A linear second-orderystem: pring-massscillator.

r (a) , (b)

!ig. f. Subfigure (a) shqws an inverte{ pendulum, Subfigure (b) shows thefree body diagram such that the gravitational compensaiion is sent to thesystemall the time which results into a systep that is floating against gravity.

. . l :76 : I0 + lmglsin9, (4 )that is, the effective torque seiit ar poim A is the sum ofthe given inittal'push'that resulted into an acceleration andthe torque sent to balance off gravity. An oscillation occurs'because he total enexgy s in the processof being convertedfrom a purely potential energy to a ptrrely kinetic energy, andvice versa.B. Mass and Center of Mass ldentification

A torque control in (4) will equate the torque sent to thephysical systemto the torque sent to control it such that,. . I I .ra: I6 * izglsin0 : iu+ ;ngisine (5)L L

where i denotes the approximation of the actual physicalsystem. Setting an open-loop control, u : 0, and assumingsmall angular displacements hat is around *15 degrees, 5)becomes

.. ml -fiti ^0+ -:1f g0:o (6)which is a linear second-order ystem where co2 (mt -nri )g 2I. The minimum crrcorrespond to ml : rhi.

| :C, Moment of Inertia ldentificationA proportional gain is set to (5) such.that the iontrolequation s expressed s u: -k70 where kp:is the propor-tional gain. Assuming that the,gravitationalterm was corrctlymodeled, 5) can be expressed s a linear second-ordervstem

' , ' (8)such he oint torgug vector s expressed,asi , ,i:, , , , r

, 1 . 1 . ,r : A (q )0+c (q , Q)+g (q ) . , , : , , , . , ( g )ThesymbolA(q) is the oinrspacenertiamatrix, (q,q) is theCoriolisandcentrifugdlorces ector, (q) is thegiavitationaltefms vector, and ri,Q,q are the joint spaib acieleraiion,velocity,artddisplacement.he'controlequatibni givenr'aslzel

u : {d , k,(Q - qd) - kp(q- qd ) (10)where ko is the proportional gain, ku is the'derivatiVe gaih,and the isubscriptd denotes':desired alue!. iir,aaaititin,toassumingsmall displacements' uring ideirtification prticediiib,velocity for each link is assumed'small such'thal Cofiolisand centrifugal forces are less dorninanticomparbd to thegtavitatioiral forces, and the inertia matfix.'Theseassumptionswould lead to a linear-secondorder system that is equivalentto the single degree-of-freedom DOF) case.In U U, it is shown hat the umped nertia terms found in theinertia matrix A(q) are the same erms found in Coriolis andcentrifugal forces vector c(q,8). Once all the lumped inertiaterms are identified in A(q), all the inertia terms ar alreadyfound.III. THE PROPOSEDDYNAMICS IDENTIFICATIONMETHOD

In this section we will show a mathematical proof ofthe proposed method to support the claims in this work inidentifying mass, center of mass, and moment of inertia. Inaddition, corresp-onding lgorithms will be sfown.A. Mass and Centerof Mass Identification.,

Given a physical systernsuch hat each ink is influencedbygravitational force, mass and centerof massean be'identifiedby letting the systemachievenatural oscillationrthrough'theforce of gravity. It is ,assurnedhat the links move at smallangular displacementsaround *15 degrees away'from zerogravity.axis. Fricdon contribution is disregarded n this workand r.vill be.addressedn the future.

d lar l a r- l - l _ - - ? .at l0q1l 0q,- " ' .

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Theorem1; A multi-link rigid manipulatorwith revolutejoints is under orquecontroland s influencedby gravitationalforce, Its minimum frequency of oscillation o is achievedwhen the estimatedphysical parametervalues n the mathe-maticalmodelareclosest o thevalues n thephysical ystem.Proof: Case . SingleDegree-of-Freedom.Eqrpting thetorque betweenthe physical system and the mathematicalmodel esults n

t : t 6+s(0 ) : i u+E(O) , ( l l )where he controlequation : 6a k"(6 - Ot - kp@- ei.Settingz =0 results nto open-loop ontrol such hat

6*s(o) :3(o)r .

the natural requencyof oscillation and ko is the proportiongain.Proof: Case . Single Degree-of-Freedom.Equating htorque betweenthe physical sys0emand the mathematimodel. r : 16 8(e) iu+E(0), (1The proportional ontrol s set to u: -kog. and.he rest thecontrol erms reset o zero,&u 0 and0a 0d: 0a:0Becausehe gravity hasno influenceor is correctlymodelethe equationbecomes

. . i0 I =k o00 .I (1linear second-orderystem 30] su12) This is an undamPedthat,Because f relativelysmallangular isplacements,12)canbeconsideredas an undamped inear second-order ystem[30]where

a2 f(s@) g(0)). (13)Because is constant, o s minimumwheng(0) : g(0).Case II. Multiple Degree-of-Freedom.The torques sentto the physicalsystemand the mathematicalmodel are equalandcan be expresseds,

r :A (0 )6*c (0 ,d ) s (e ) A(e)o+e(0 ,e )g (0 ) . 14)With relativelysmall oint velocities,Coriolisand centrifugalforcesare ess dominant and are ignored.With small angulardisplacements,ravitationalerms can be consideredinear.Setting he controlequation :0 results nto0+r(e)-1(g(e)E(g))1. (ls)With the small angular displacementA(0) can be consideredconstant. As the correct parameter values are approached,the system becomes decoupled such that each link can beindependently considered as second-order undamped linearsystem [30] where

a2: f (s (o ) -g (0 ) ) . (16)The minimum ar is achievedwhen C(0): E(0). I

B. Moment of Inertia ldentificationGiven a physical system with its gravitational model cor-

rectly compensated (or is independent of the gravitationaleffect), the inertia parameters can be identified by lettingthe system achieve its natural frequency of oscillation. Theinertia'values will be adjusted until the desired frequencyresponse is achieved. The following corollary follows thetheorem presentedabove.

Corollary 1; A rigid manipulator underproportional torquecontrol is not influencedby gravity. The correct values of linkinertia parametersare identified when crl' : kp where al is

" i -0r- : ikp. (1When :1, then p: a2.

Case II. Multiple Degree-of-Frcedom.The torquesseto both the physical systemandthe mathematicalmodel aequalandexpresseds,c A(0)0c(9,6) g(e) A(e)u+e(0,e)E(0). 2

Setting he proportional ontrol o u: -kr_O an{ the restthecontrol erms reset o zero, u :0 and0a - 0d: 0a:0With slow velocities he Coriolis and centrifugal erms are edominant and can be ignored. And becausegravity has influence r is correctlvmodeled.d+.n e)-14(g)kpo 0. (2

As the system approacheshe correct parametervaluesbecomesdecoupledsuch hat each ink can be independeconsideredas second-order ndamped inear system [3wherea2: A(0)-ta1e;no. (2

The caseof A(e) : A(0) results n @2 kp.

C. The AlgorithmThe algorithm for identifying mass, center of mass, a

moment of inertia through natural oscillation is presentedthis subsection.For mass and center of mass. he identificatiprocedure s the following.

1) Select initially large values for the mass fr, and cenof mass I to initialize the dynamics parameters of tmulti-link rigid body.

2) For each of the oint, send he torque c correspondingthe Coriolis and centrifugal forces and the gravitatioterms.

3) Let each ink oscillate from a predefined nitial displament, to help create consistency of results. The systwill start to oscillate at a certain frequency co.

4) Measure the frequency of oscillation or.13


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, r - {3-DOF

Fig. 4. The frame assignment or I-DOF,2-DOF, and 3-DOF manipulators.

5) Adjust the values of the dynamics parameters i, and 7to minimize the frequency of oscillation ar.

6) Repeat Steps2-5 until the minimum frequency of oscil-lation ar is found such that its corresponding it, and fare the best estimates.

In identifying moments of inenia i, the same algorithmas stated above will be used, except that f is the unknownparameter instead of ft and L Then instead of minimizingo, the corresponding objective value of the frequency ofoscillation will be a2 : kp .

The large values of the initial dynamics parameters arerecommended to have an initially large solution space toexplore. A predefined initial displacement is used to createconsistent results in measuring the frequency of oscillation.The predefined value of this initial displacement should beabout *15 degrees and with alternating opposite signs onconsecutiveoints. This setupwill help minimize the displace-ment for each oint such that the assumptionon linearity ofthe manipulator responsewill hold. In addition, this will alsominimize the contribution of Coriolis and centrifugal forcessuch that the accumulateddisplacementof each oint remainssmall.

To measure the frequency of oscillation crr,normally thenumber of periods of oscillation is countedwithin a predefinednumber of update cycles. An incremental step is performedwithin the searchspace of the dynamics parameters o findthe combination of values that correspond o the desired al.The dimension of the searchspace ncreasesas the number ofdynamics parameters o be identified increases. ncrementallychanging he values of the dynamic parametersand measuringthe corresponding requency of oscillation are repeatedlyper-formed. For higher degreesof freedom, it is highly possiblethat the identified parameterswill result in a combination ofpossible range of values. Each of the possible combinationscan be checked ater against the robot response n the actualmanipulatorcontrol.

),- *

IV. RESULTS AND ANALYSISSimulations are performed to test the proposedalgorithm

in identifying the dynamicsparametersof l-,2-, and 3-DOFsmanipulators. Open dynamics engine (ODE), an OpenGLprogam integrated with C code, is used as the simulationplatform. This simulation platform requires the actual phys-ical model for the system to move in a virtual world. Thedynamics parameters are then provided to create a model forthe simulation. Then the conesponding orques are sent to thejoints using estimateddynarnicsparametervalues.

In this section, he range of estimatedvalues correspondingto the desired at will be presented or each of the simulatedmanipulator. The frame assignment s shown in Fig 4. Thesimplified symbolic dynamics model of l-,2-, and 3-DOFsmanipulators shownhere are expressedn terms of massesandcenter, ncluding the inertias. Correspondingmass factors areused to vary the possiblerange of values for the mass, centerof mass, and inertia parameters.The following symbol sim-plificationsare used Si...n: sin(lf Q) and Cr...n: cos(lf 0;).

A. One Degree of FreedomThe simplest case of I-DOF is shown to test the sensitivity

of the system to numerical errors. Given a mass rzr and linklength L, a mass factor fm1 and fm2 are multiplied to thegravitational and inertial terms, respectively.

It fm, ImrLz)Bt : fr u i mlgL 51. (23)

An initial 15 degrees displacementwas used to perturb thesystem and let it achieve natural oscillation. The results areshown in Table I for the mass and center of mass and Table IIfor the moment of inertia. TABLE

MAss AND CENTER OF MASS IDENTIFICATION FOR 1-DOF

1.0201.0101.0051.004l 003t.oo21.0011.0000.999

8. 76. 95.85. 55 . 14. 84 )4 . 1(falls under gravity)

The correct values of the dynamics parameterscorrespondto fmy: fmz: 1.00. Table I shows consistently hat asthe estimated parameters start from large values and de-crease owards the correct values, the frequency of oscillationdecreases.And at 0.001 difference of the mass factor, adifference in the frequency of oscillation is achieved. Theminimum frequency of oscillation is shown to be 4.1 periodsper 5,000 updatecycles.A mass actor 6:0.999 will causthe system to fall due to gravitational force. Table Il showsconsistently hat as the mass actor increases, hat is, the inertiaestimatesbecome arger than the actual values, the frequencyof oscillation increases.It can be safely assumed that the

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difference in the frequency of oscillation is more elaboratewhen the mass actor changes o around5Vo.ltis alsopossibleto increase he number of sampling updates n order to makemore elaboratedifferences n the frequencyof oscillation. Ananalytical sensitivity analysis on the mass and center of massidentification effor as compared to the inertia identificationerror will be shown after this section.B. Two Degreesof Freedom ,

The symbolic form of the components of inertia A(0),Coriolis and centrifugal forces c(9, 0) and gravitational termsg(0) of the 2-DOFs manipulator is shown in (24).

TABLE IIIMAssAN DCENTERFMASSDENTIFICATIONOR -DOFSat

fmt'.f*z Link I, Link 25. 44.74.54. 54.4r ' , )4.03. 6( falls undergravity )

TABLE yINERTIA IDENTIFICATION FO R 2-DOFS

atf^2, f^t Link 1. Link 2

TABLEIIINERTTADENTIFICATIoNoR1-DOF

f*z Link 11.00l . 0 1t.o21.031.041.051.061.071.081.091 . 1 0

35.5035.5736.0036.0036.1036.5036.6236.6037.0037.0037. t3

1.0051.0041.0031.0021.0011.000o.9990.998

1.01t.o21.03t.o41.051.061.071.081.091 t0

35.6136.0036.0036.1436.5036.5036.6637.0037.0037.17An fmalmll?+ fmtmzl](t +Cz)Atz : A21 fm3m2t?11+C21Azz : fm3\m2L2cr -fm3lm2Lz4z(24t*qz)Szc2 : frylm2LzS2Qlsr : f ^t(L*t -fm2)Lgs1*m2lm2LgSp

(24)

Ez : fm2)m2LgSpIn the symbolic model shown, the mathematical model

of the inertias were simplified such that only the principalmoment of inertia about the joint axis is considered. Theother two principal moments of inertia and all the Broducts ofinertia were assumedzero.However, in the simulation modelthis is not true. as the ODE simulation mimics the real-worldphysical system these values can only be assignedrelativelysmall values and not completely zeros.

For each of the umped nertia parameters,mass actors z 1,fmz, fmz, and fma are used. Note that the dynamics modelcan be identified:jndividually without the lumped symbolicmodel. However, the value of the lumped model only servedto show the limits on the productsof the identified parameters.In a sense t is the lumped model that is identified such thatany values of m1 and L would suffice to model the dynamicsparametersas ong as the product of their values correspondingto the actual physical model and is correct to within a desiredaccuracy.This precision defines he range ofpossible identifiedvalues.

Table III shows the experimentalresults for 5900 updatesof a two degreesof freedom robot with varying mass factors

f6 and f*2.The frequency of oscillation is recorded for thedifferent values of the mass factors. The robot falls under theinfluenceof gravity when the mass actors are 0.002 below theassignedphysical mass actor of 1.0. The minimum frequencyof oscillation applies when all mass factors are 0.001 below1.0. The frequenciesof oscillations for both links decreasedaccordingly as estimated values decreased owards the actualphysicalvalues.

Table IV showsthe experimentalresults for 20000 updatesof a two degreesof freedom robot with two mass factors fm3and ma being varied. The frequency of osr:illation s recordedfor the different values of the mass factors. A mass factor of1.0 corresponds o the correct natural frequency a2 : kp. Thefrequenciesof oscillations for both links increasedaccordinglyas the mass actor were increased.Again around 5Vo differencein the mass factor created an elaborate difference in therecordedperiod of oscillation.

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A n :A n :

.f y * (* t * 4m2 7m3)? + f m5 mz 2m3) 2 2+ fmam3L2(Q+Czz)t2, : 1m6 (mz 4mz)L2 f m5l(m2+ 2ry) ?c2* fmalm3L2(2Cz*Czz) tA31 f mafm3L2 Z+ 3Ct 3Czz)A32: fmafm3L2(2+3q)f*sl@z + 4ry) L2 f mq3mtL2zfmalryL2- f msL@z* 2ry)L2Q2(2h+ 4z) z- Jma)ryl? (42Q4 * 2Q2 q) s3+ @z+ Qt)Qh I q2+ 4z)szz)"f*si@z+2ry)L24ls2-t f malry L2 - qz zq, 2Qz qz)*4?szt)7m4ryr2((4 * 4z)zz+ 47szz). fq i@r *2(m2*ry))Lgs1+ fm2|(m2-t2m3)LgSp

t fm3)m3LgSp3f*zi@z*2m3)Lgsp

TABLE VMAssANDCENTERFMAss DENTIFICATIoNoR3.DoFs

fm1, fm2, fm 3 ^AnAzzAzzAn

c l

1.005l.()ol1.0031.0021.00r1.000o.9990.998

4.O3.93. 95 - t3. 23 .13. 0(falls under gravity)

c Z :

g 2 :* fm3)m3LgSp3g3 : fm3)m3LgSp3 (2s)

C. ThreeDegreesof FreedomThe symbolic full dynamics model of a 3-DOFs is shownin (25).Th e link masses remr,m2,andm3 for l inks I to 3

and have equal link lengths of I-Mass factors fmt, fmz, and m3 are used or the massandcenterof mass dentification, while massfactors from fma tofm7 are used for the lumped inertia parametersdentification.It is assumed hat the principal inertias of each ink about thejoint is dominant, suchthat the other principal inertiasand theproductsof inertia are assumed ero.The inertiasare expressedas massesand centersof mass.The valuesof fm1 to fm3 arevaried to determine he systemresponseat minimum a;, whilethe values of fma to f-m7 are varied to determinethe systemresponsen finding a2 : kp.Tables V and VI show the experimentalresponseof the 3-DOF manipulator. n the experiment, he natural oscillation isachievedat slow speedand the entire systemmoves n unison.This is helpful but not necessary, n achieving well-definednaturaloscillations.The inertias were also expressedn termsof massesand centersof mass,and the dominant parametersare gravitational and not the Coriolis and centrifugal forces.As in the previous results, there is a distinctive decreaseofthe period of oscillation as the mass factors fm1, fm2, andfm3 decrease owards the correct value. At mass factors of0.998 the systemfell under gravity. The minimum frequencyof oscillation is achieved at a mass factor value of 0.999.The system can be consideredas having a tolerance imit of+0.001. Such tolerance value is also demonstrated n mass

TABLE VIINERTIADENTTFtcATtoNoR3-DOFsatfma,f m5,fm6Jm7 Link I, Link 2,Lir/r-3

1 .01r.o21.031.041.051.06l 0 71.081.091 . 1 0

35.6236.0036.0036.t436.5036.5036.663't.N37.0037.18

factor values of 1.004 and 1.003 where the correspondingperiods remain the same. For practical purposes, t can bestated that the tolerance value is at i0.002. In the inertiaidentification, there is a distinctive difference in the periodof oscillationwhen the mass actorsf mq, -fms, f m6, and m7differ by around 5Vo. Again, increasing the sampling updatecycles can make the difference n the frequencyof oscillationsmore elaborate or the samemass actor range.The initial values were purposely chosen to be not toofar from the real values (at a maximum of l\Vo) to showthe sensitivity of the system to numerical errors when theestimatedvaluesare getting closer to the physical values.Thisis important because his will show the degree of precisionin the proposed method's estimation ability. Because theproposedmethod converts he identificationprocedure nto anoptimization computation, convergence rom a given set ofinitial valueswill be dependenton the optimization procedureused. One optimization procedure that may be used is theprobabilistic gradient descent method that will perform apredefinednumberof random walks upon convergence.n thisway, the computation can increase he probability of gettingout of local minima.

In most cases, t is hard to quantify the acceptableolerancevalues of an identified parameter because, to the best ofour knowledge, there is no such quantification study. Thecorrectnessof the identified value is only checked hrough theaccuracyof the robot response.But even with the quantifica_tion of the accuracyof robot response, here s no set standardon how much is the tolerance limit to be considered ,.bestperformance"given set parametersas speed, ask requirement,and load. Thus the successof most dynamics identification

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method normally is based on response of the manipulatorwhere the identification is being implemented. And this isdependent on what type of task the manipulator is requiredto perform, and is generallynot applicable to all systems.

V. SpNstrwtrv ANaI-vsIs rIdentifying the inertia parameters, together with the mass

and center of mass, can create different errors in the torqueresponsedue to inaccuracy n.the identified parameters.Thissection will show sensitivity to parameter errors in terms ofgravitational terms as compared to inertial terms, and howboth terms affect the error in torques. For simplify, sensitivityanalysis is performed for I-DOF manipulator.

Given a computed torque control where the dynamics pa-rameters are correctly modeled under proportional control isexpressedas

t : - i k r eg * f i (26)where ee - 0a- 0. Then, given a perturbation in the gravi-tational model, 69, the resulting perturbed computed torquewill be

r + 6rs -i kre6 (g+6s ) (27)such that, the torque perturbation divided by the computedtorque is

future direction of this work is to implement the method ona physical robot manipu lator.where the friction parameterinfluence the results. The ultimate goal of this experimentaprocedure s to implement it on a humanoid robot moving ina full-dynamicswhole-bodycontrol [31].

VII. AcTNowLEDGEMENTThe authorswould like to acknowledge he researchsuppor

given by the University Research Council of De La SalleUniversity - Manila.

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6g: - .-I kpes frOn the other hand,given a perturbation n the inertia model,

6i, the resulting perturbedcomputed torque is,

6rr (28)

(2e)(30)

and

Clearly, (30) showed hat an error in the inertia identificationhas ess effect than an error in gravitationalterm identificationas shown n (28). This is because 30) is dependeriton the errorterm ee which can becomevery small as the gain compensatesfor the error. Compared o (28), the error in gravitationll termcan never be compensated.

VI. CONCLUSION AND FUTURE WORKThis work showed a new method in identifying the mass,

center of mass, and inertia parametersof a rigid-body rnanip-ulator through natural oscillation ar. A mathematical proof ispresented o support he validity of the proposed rlgrilithm.The correct values of the dynamics parameterscorre:'pond othe minimum frequency of oscillation r.o for the mass andcenter of mass identification, and az : ftp tbr the inertiaidentification. Results sensitivity and range of the identifiedexperimentalvalues were shown for one. trvo. and three de-greesof freedommanipulators, s well as sensitivityanalysison errors of identified inertia and gravitational paranieters.The

r * 5 q : - ( i + 6 i ) k r e s * p

6q _ -_6i koeet - I koes*S

17


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