1994 a New ion Scheme for the Inverse Kinematics Tasks of Flexible Robot Arms (2)

8/6/2019 1994 a New ion Scheme for the Inverse Kinematics Tasks of Flexible Robot Arms (2)

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A new compensation scheme for the inverse kinematics tasks offlexible robot arms

M. M. Svinin M. UchiyamaAero nautics and Space Engineering Dep artmen t, Tohoku University,

Aramaki-aza-Aoba. Aoba-ku, Sendai980, Japan

Abstract

A n i nve r se k inema t i c s p rob l emof f lexible manip-u l a to r s is conside red . Th e p rob l em i sof impor t ancefo r compensa t ion of t ip def lec t ions when the manip-ula tors per for m quasi -s ta tic opera t ions . Based uponh e m a t i c a n d s t a ti c re la t io n sh i ps , a d e f i n i ti o nof theinve r se k inema t i c t a skof f lexible manipula torsis for-

mulated . Specif ic fea turesof the task are conssidered,an,d a conventional approach applicablefo r resolvingthe t a sk is analyzed. To avoid d isadvantagesof th econven t iona l app roach , a n imp l i c i t i t e r at i ve s chemebased upon s tep-by-s tep comp ensat ionfor the relativeelas tic def lec t ions i s proposed. Corresponding conver-gence condi t ionsof the scheme are der ived, and appl i -cabili ty of th,e scheme is verified through simulation,.

1 Introduction

In recent years, t he inverse dynamics of flexible ma-nipulators has been the subject of intensive research[ l , 1. In contrast to the rigid manipulator case, there

ar e a number of difficulties and unclear points (invert-ibility and convergence conditions) in solving the in-verse dyiiamics of flexible manipulators. To make thesituation more clear and to get a deeper insight intoth e inverse task of flexible manipul ators, i t is reason-able at first to study fundamental kinematic aspect ofthe task.

Th e topic considered in thi s paper-inverse kine-mati c ta sk of flexible manipulators-is of great inipor-tance for compensati on of t ip deflections when th e ma-nipulators perform stat ic or quasi-static operations. Inliterature, large number of analytical and numericalmethods have been developed for th e solut,ion of t heinverse kinematic task of rigid manipulators[3, 41. Asfar as flexible manipulators are concerned, the results

(approaches and methods) obtained up to now needan essential revision and reformulation.This paper is organized as follows. Fir st, Sec-

tion 2 gives a definition of th e inverse kinematic ta sk

of flexible manipulator s an d discusses special featuresof the formulation. Next, in Section 3 , computa-tional aspects of the task are considered and con-ventiona l approach-based upon the Newton-R aphsonmethod-to solution of th e tas k is analyzed. Havingidentified disadvantages of this conventional approach,an implicit iterative scheme based upon step-by-stepcompensa tion of relative elastic deflections is proposed

in Section 4. Also in this section, convergence condi-tions for the scheme are derived, and applicability ofth e scheme is verified through simulat ion study. Com-parative analysis of the convergence rate is given inSection 5 . Finally, conclusions are presented in Sec-tion G.

2 Task formulation

Let P E 3 be the set of Cartesian coordinates de-scribing position and orientation of t he manipulatorsend-effector with respect to the inertial frame. Th emanipulator is considered as an open kinematic chaincomposed of elastic links. It is also supposed t ha t a

constant e xterna l force and moment, combined in t hevector @ E %, are applied to th e gripper. The vec-tor s of rigid (joint angles) and elastic variables ofthe manipulator are denoted respectively by q E 3an d 7 E g m . Usually, the elements of the vector 77are introduced on the basis of different approxima-tions such as assumed modes, finite elements, or Ritz-Kantorovich expansions, with different implications onmodel complexity and accuracy.

Having introduced t he rigid and elastic coordi-nates, the static equations of the manipulator can berepresented in th e following m atr ix form:

G4(q, 7) + JgT(q, T I ) @+ Q = 0, (1 )Gq(q9 7) + J ; (q , + OQ = CV . (3 )

Here, Q E SRn is a vector of generalized driving forces;G, E 92 an d G, E R are respectively vectors ofthe generalized gravitational forces for the rigid a nd

1050-4729/94 03.00 0 1994 EEE315

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elastic subsystems; Jq E RpBxnnd J,, E ?RpBxmrerespectively the manipulators rigid and elasticJacobians; C E Rmxm s the combined stiffness matri xof the elastic links; D E Smxn s the case-dependentmatr ix of th e boun dary conditi ons of t he elastic links.

Using th e forward kinematics relationships, P maybe expressed as a nonlinear funct ion of q and 9

p = q q , ). (3)

From the viewpoint of kinematics, flexible manip-ulators are redundant ones, with q playing the roleof redundant coordina tes. However, we cannot spec-ify their ar bit rary values, since these elastic coordi-nates are dependent on t he rigid coordinates q an don the applied forces a. Considering th e sta tic equa-tions (1,2), this dependence can be written down inth e following form:

G, - D G , + ( J : - DJ,T)@= Cq, (4 )

9 = H ( q ,a). (5 )

(6)

or, more generally,

Substituting (5 ) into (3), we finally get

p = F(n , H ( 4 , a>).Now, the inverse kinematic task of flexible manip-

ulators can be formulate d as follows. Given desiredvalues of P and a*, it is required to define corre-sponding values of q* satisfying equation (6). It isan importan t fea ture of thi s definition t ha t th e pos-sible solution q depends not only on P but on @ *as well. So, when designing Cartesian-level force an dmotion planning agents for constrained manipulations,this featu re should be taken into account.

Considering equation (G) we have to pay atten-

tion t o the following fact. An analytical solution ofthis equation can hardly be obtained even for those

or, in compact form,

6P = { J q h 9) + J, ,(q, v )J h ( 4 , a)) 6q, (9)where J h = 8 H / 8 q E R m x n s the Jacobian of equa-tion ( 5 ) ; Jq and J, are the rigid and elastic Jaco-bians defined in the previous section.

Having defined the Jacobian of equation (G), wecan construct th e iterative procedure in a conventionalmanner [5 ] in which main ste ps of th e procedu re mightbe fixed as follows:

a Set k = 0 and initial configuration q ( O ) .

a Set k = k + 1 and sequentially compute 9(k)H ( q ( ) , @ * ) ,P(k) F ( q ( k ) , q ( k ) ) , p ( k ) = P -P ( k ) , $k) = {Jik)+ Jp ) r)}#p(k) ) .

0 Set q(k+) = q ( k ) + Aq(k) if the conditionIlAq())JI E is not satisfied.

Here, in the procedure, superscripts taken in paren-theses are used t o designate the iteration number; E isa small positive constan t; symbol # denotes th e gen-eralized inversion operation.

Th e conventional it erative scheme outlined her e ismore or less general. Hence, it is difficult to analyzeth e convergence conditions. Indeed, t he fixed pointequation q = z ( q ) [GI for the scheme under considera-tion is defined as follows:

manipulators whose kinemat.ic schemes admi t a n an-

their equivalent rigid stru ctu res. Therefore, we needto construct an appropriate computational iterativescheme to solve eq uation ( G ) .

alytical representation of the illverse kinematics for where11 11dexlotes a matrix norm being associated

with th e corresponding vector lormaAs can be seen, the condition (12) depends on how

close the initial configurat,ion is to t he t arg et one a ndcannot be inte rpre ted more definitely. However th isis not a main drawback of the scheme, since one may

3 Conventional approach hope that the condition (12) would be satisfied underth e small displacements assumpt,ion.

Th e scheme being analyzed in t his section is basedupon differential kinematic and st ati c of tile

variations of the vector p % a function of independentvariations S q . It follows from (3,5) ha t

From the viewpoint of control applicability, themain disadvantage of the scheme under consideration

alytical expIeSSiOilS for the elements of t he J h matrix

require a lot of cumbersome computations for whichspecial rather sophisticated computer-oriented meth-

bp = J,(qr 9 ) q + J , ( Q , 7) b, (7 ) ods need t o be developed. Indeed. even the ta sk ofb q = J h ( q , q ) bq , (8) computation of the vector-function H ( q ,@) from the

manipulator. At th is point, let us firstly define small is t he necessity to ConlPute nlatmrix J h . Apparently, an-

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equations (4 ) is not trivial one and is roughly as diffi-cult as t he task of dynamic model computation. Thus,the level of complexity of the computation of the Jhmatrix is comparable with that of the linearized dy-namic equations.

All this, taken together, makes it extremely diffu-

cult, if not inapplicable, under the current state-of-artof co mputer hardware, to use the above-described con-ventional scheme for manipu lator s with flexible links.Thus, special methods for numerical solution of theinverse kinematic task of flexible robot arms need tobe invented.

4 Compensation scheme

To overcome disadvantages of the above-describediterative computational scheme, another approachbased upon the ideas of compensation control of flex-ible manipulators [7, 8 , 91 is proposed in this section.

4.1 Description of the scheme

First of all, let us assume that 3q = qrlgld suchthat F ( q , i g i d , O ) = P'. This means tha t for the givenP * there exists an inverse kinematic solution for theequivalent "rigid" manipulator.

Let us set t he initial configuration q ( O )= Qrlg ld . Dueto th e exte rnal force loading, in thi s configuration wehave the following elastic deflections

q ( 0 )= H ( q ( O ) ,@*). ( 1 3 ) .

To compen sate for thes e initial deflections, we ap-ply small corrections. Th e corrections are defined fromthe requirement such th at the resulting errors on thelevel of end-effector are zero. Thus, from equation (7 )we get

= -J,#(q''),O) J , (q( ' ) ,O) q ( O ) . (14)

Next, for th e new configuration q( ' ) = q ( O )+Aq(O) , heelastic deflections are q ( l ) = H ( q ( ' ) ,a*) .Note tha t wecannot compensate for the absolute errors (differencebetween the current deflections and the reference de-flections tha t we are looking for) induced by t he elas-tic coordinates. Yet, it does not mean t ha t we cannotcompensate for the relative errors which in our caseare the difference between the current and the previ-ous deflections.

Following this logic, we define A q ( l )as q ( l )- 7 CO).Linearizing (3 ) in th e vicinity of th e operati ng point

{ q ( l ) , ( O ) } ,from the compensation condition A P = 0we compute

= - # ( & l ) , q ( O ) )J , ( & l ) , q ( o ) ) AV ( ' ) , (15)

and define q ( 2 ) = q ( ' ) + Aq( ' ) for the new operatingpoint { q ( 2 ) , ( ' ) } .

In general, th e proposed iterative scheme is fixed asfollows:

As can be seen, the main iteration formula of thecomputational scheme makes use of the informationabout two previous stat es of th e system:

Therefore, the scheme can be regarded as a second-order difference approximation. In this case, the cor-responding fixed point equabion cannot be representedin the explicit analytical form. As such, the directanalysis of the convergence conditions is no t possible.

4.2 Convergence analysis

The convergence analysis presented in this sectionis based upon the linear model for resulting correc-tions and errors (we denote them by 6, keeping sym-bol A for finite differences). Also, while computingJacobians, we neglect small changes of the manipula-tor's configuration. Thus, from now on, in our nota-tions we assume that Jq = Jq (q ( ' ) ,O) , , , = J , (q(O),O),

from themain formulae of the considered compensation schemewe have

J h = J h ( q ( O ) , @ * ) .Taking into account that 6q(O) =

Combination of (17) and (18) gives

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It can also be represented in terms of the initial cor-rection and error,

6 q ( k )= ( - J , # J&)k 6 q ( O ) , (21)

6 7 p = ( - - J h J , # J , ) k 6q(O). (22)

Now, the corresponding convergence conditions un-der which limk,, 116q(k)ll= 0 an d limk,, 116q('))11 =0, follows from the theorem of compressing mapping[ lo]. They are respectively

llJ,#JqJhll 5 1 , an d llJhJ,#J,ll 5 , (23)

where 11. 11denotes a matrix norm being associatedwith the corresponding vector norm.

It is interesting to note th at under the assumptionsmade on the stage of the convergence analysis we canobtain an analytic al representation for th e limit pointsq* = limk,, q ( k )and q * = limk,m q( ' ) . Indeed, byconstruction we have q* = q ( O ) 6 q ( k )and v * =CEO ~ ( ~ ) .aking into account that 6$') = q ( O )

and S q ( O )=

- J ,# J ,7 l (O) , on the bases of (21,22) wehave the following expressions:00

Q * = {I 3- C ( - l ) y J h J , # J , ) k } 7 + O ) , (24)k= 1

00

q* = q ( O ) - { I + x(- l )k(J ,#J ,Jh)k}J ,#Jq ')(25)

If the conditions (23) are satisfied, th e series in (24,25)converge, and we finally get

k = l

q* = {I + JhJ ,#J ,} - ' ?,I@ ), (26)q* = 9") - {I + J , # J q J h } - ' J , # J , 77"). (27)

It is important to note that the iterative schemeproposed does not directly make use of the resultingerrors A P ( k ) = F ( q ( k ) , ( ' ) ) )- P* eached on the end-effector level. To complete our analysis, l et us considerthe sequence of the above-mentioned errors under theassumptions made in this section.

Remembering that P* = F ( q ( ' ) , O ) , we define6P@) = F ( q @ ) ,p ) F(q(O), ) = J* q @ ) - q ( ' ) ) +

(28)

J q 7 p or, after some transformations,

bp(k++ ' )= 6 p ( k )+ J q b q ( k ) + J,6q(k+l),

Next, on the bases of (17,18) we have

6P(k'+1) 6 P ( k ) {I + J , J ~ J , # ) J , ~ V ( ~ ) .29)Taking into account that 6P(O,

=J,6$o), analysis of

(29) gives

6P(k ) = ( - J q J h J , # ) k 6P(O). (30 )

r"

3 m

fi

Om

/ x0

Figure 1: Flexible robot in constr ained motion

The corresponding convergence condition under which

limk,, IIsP(')IJ=

o isllJqJhJ,#Il 5 1 . (31)

To satisfy the conditions (23,31) simultaneously, wefinally require tha t

IlJVll' IlJhll ' llJ,#III . (32)The condition derived completes the convergence anal-ysis. Note th at thi s condition does not depend onthe elastic coordinates and is defined by the appliedforces, th e manipulator's stiffness paramete rs, and bythe rigid coordinates specified in an equivalent "rigid"configuration.

4.3 Simulation

To validate the applicability of the compensationscheme, the simulation study being described belowis conducted. The simulation employs a planar three-link manipulator whose model paramet ers ar e chosento be in correspondence with those of the ESA-SMSspace manipulator [ l l ] . They are: I1 = 12 = 3 m , 63 =0.5m,ml = m2 = 10kg,m3 = SOkg,q = r2 =1.5m, r3 = 0.25m - hich are respectively lengths,masses and center of masses of the links.

First two links of the manipulator are modeledas identical flexible homogeneous cylindrical beams,with area moment of inertia about the bending axisand Young's modulus of elasticity as I, = 0.73631 x

m4, and E = 1.22231 x loQN/m2, respectively.The par ameters a re chosen from th e requirement suchthat maximum end-point transversal deflection is notmore tha n m under statically applied force of 1 N .

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10.80.60.40.2

0-0.2-0.4-0.6-0.8

f i N l lJr711 IlJhll l lJ ,#II- l o z N 3 - 5 0.295 N 0.634- lo3 N 10 - 15 0.462 N 0.868

5 - 3 . 1 0 3 ~ > 1

Compensation abo ut 41 axis [deg],

0 0.5 1 1.5 2 2. 5 3

f z

-500 N

-400 N

-300 N

-200 N- 1 o o N

YIml

Compensation abou t 92 axis [deg]1 1 I I I I I I f z

-500 N-400 N-300 N

-0 .5 -200 N-1 - 1 o o N

-1 .50 0.5 1 1.5 2 2.5 3 Y[n1]

Compensation abou t 43 axis [deg]2.5

21. 5

10.5

0-0.5

-1-1.5

-2

f z-500-400

-300-200-100

Figure 2: Joint-level corrections

The assumed modes method [12] is used to obtainthe stat ic equations of the manipulator. Each of theelastic links is assigned 5 modal coordinates, with themodal forms being of th e clamped-free type. Duringthe simulation, numerical values of 77as a function of4 an d iP are calculated iteratively from equation (4 ) .The elements of the matrix J h , used only for the es-timation of t.he criterion (32), are defined numericallyon the basis of the numerical differential approxima-tion.

In the simulation, the manipulators end-effectoris commanded to perform a straight motion (Fig-ure 1) from the initial lower position Po =(3.5m1 O,O, 0, 0,O). to the final upper position Pf =(3.5 m, 3 m, O,O, 0, O)T. As far as the external force is

concerned, t,he simplest law with the constant vec-to r iP = ( f z , O , O , O , O , O ) is chosen. The example istypical for constrained motions - he manipulators

end-effector moves along a wall, with a desired forcecontact being kept with the constraint surface.

The reaction force f i is changed in the discreterange: -{ 1,2,3,4,5} x lo2 N. The simulation resultsare depicted in Figure 2. They show the joint-levelcorrections of the rigid coordinates as functions ofthe variable coordina te y. These corrections are t,o beapplied for th e compensation for the deviations of theend-effectors trajectory due t o t he initial deflections.

As far a s th e convergence conditions are concerned,the y are checked in th e course of t he simulation un-der different values of f z . Th e results-the number ofiteration N and the values of the criterion (32)-aresummarized in the Table 1. As can be seen, th e com-pensation procedure becomes divergent in the range offz 5 -3. lo 3 N for which the assumption of small elas-tic displacements of th e beams is not valid for systemunder consideration.

5 Comments on the convergence rate

The scheme outlined in the previous Section makesuse of the compensability equation ( Jq6q+ J,6v = 0 )(91 which deals with the deviations and correctionson the level of generalized coordinates of the manip-ulator. Due to this fact, the scheme can be classi-

fiedas

a generalized-coordinate-level-based compensa-tional scheme. Th e scheme exploits differential kine-matic model of the manipulator (matrices J p an d J q )and does not use the differential static model (matrixJ h ) . The latter is the main advantage of the schemebecause definition and computation of matrix Jh ar eextremely difficult. However, it is quite obvious tha tthe convergence rate of t he scheme is slower than th atof th e scheme based on th e Newton-Raphson method,and this is the price for not using the full set of deri w-tives.

Next step towards simplification of the iterativeschemes and reducing the computational difficultiesmight have been based upon neglecting the informa-tion about the rigid and elastic Jacobians. In

this connection, the idea of using the direct Cartesian-coordinate-level-based compensat ion of the t ip deflec-tions may be considered. The key point is to use infor-

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mation about the equivalent rigid kinematic modelof th e manip ulato r only, without taking t he matricesJq nd J,, into consideration.

~ ~ i ~ a ~ ,he idea cm be described as follows. Letus denote F,(~) F ( ~ ,), with ~,-i(p) being theinverse kinematic mapping for th e equivalelltmanipulator. As has already been mentioned, th e goalof the task is to find out q* such that

developed. Th e scheme exploits the knowledge of th erigid and the elastic Jacobians of the manipula-tor and is based upon stepby-step compensation ofelastic deflections. In addition to this, an appropriateconvergence criterion has been outlined for the itera-tive scheme. Th e applicability of the computational

scheme has been validated under simulation.

F ( q * ,H ( q * , *)) = P*. (33) References

This god, however, can be reformulated indirectly inth e following form-to find ou t in the Cartesian spaceth e equivalent rigid configuration P,* = F,.(q*,0)with q* satisfying equation (33). Having reformulatedthe goal, it is easy to dra w the corresponding iterativescheme which solves th e problem:

[I ] F. Pfeifer and B. Gebler, A Multistage Approachto the Dynamics and Control of Elastic Robots, inProc . IEEE In t . Conf. o n Robotics and Automation,Philadelphia, April 1988, Vol. 1, pp. 2-8.

[2] E. Bayo, P. Papadopoulos, J. Stubbe, and M . A .Serna. Inverse Dvnamics and Kinematics of Multi-

Pji+l) = p p + (P * - F ( q ( ) ,Tp)), (34)q(+) = y ( p p ) ) , p = p * . (35)

As can be seen,all

that we need in this scheme isthe ability to measure/compute Cartesian coordinatesof the end-effector (forward kinematic task of the flex-ible manipulator), an d the ability to solve the inversekinematic task of the equivalent rigid manipulat or.As far as th e convergence rate is concerned, it is cer-tainly slower than that of the generalized-coordinate-based compensation scheme, since the derivatives-J q ,J,,, Jh-are not used in th e scheme at all.

Finally, considering futu re research, it is worthwhilet o mention th e use of t he neural-network-based learn-ing control schemes to resolve the contradiction be-tween the convergence rate and the number of avail-able derivatives. Th e importa nce of these schemes isseen mainly in th eir ability t o learn th e extremely com-

plicated nonlinear mapping (33) and to achieve thereal-time computational capability.

6 Conclusions

An inverse kinematic problem for flexible manipu-lato rs has been investigat,ed n this paper. Special fea-ture of th e problem is th at the la tter requires account-ing not only for the kinematic considerations but alsofor the s tat ic ones. An analytical solution of the taskcan hardly be o btai ned and special numerical methodsneed to be considered. It has been shown tha t t he useof conventional approaches based upon the Newton-Raphson method requires a lot of cumbersome compu-tati ons in this partic ular case. Therefore, they cannotbe recommended for application t o t he flexible manip-ulators. Specific features and computational aspectsof inverse kinematic task have been analyzed, and anon-conventional implicit iterative scheme has been

Link Elastic Robols: An Iterative Frequency DomainApproach, Int. Journal of Robotics Research, 1989,

(31 M. VukobratoviC and M. IiirCanski, Kinemat ics and

Dajectory Synthesis of Manipulation Robots, Scien-tific Fundamentals of Robotics, Vol. 3, Springer-Verlag, Berlin, 1986.

(41 K.S. Fu, R.C. Gonzalez, and C.S.G. Lee, Robotics:Control, Vision, and Intelligence McGraw-Hill, NewYork, 1987.

[5] A.A. Goldenberg, B. Benhabib, and R.G. Fenton, Acomplete generalized solution to the inverse kinemat-ics of robots, IEEE ournal of Robotics and Autom a-tion, 1985, Vol. RA-1, No. 1, pp. 14-20.

[SI J.B. Scarborough, Numerical Mathematical AnalysisBaltimore: The Johns Hopkins Press, 1958.

[7] Z.H. iang, M. Uchiyama, and I

Documents

1994 a New ion Scheme for the Inverse Kinematics Tasks of Flexible Robot Arms (2)