Human Placement for Maximum Dexterity

Transcription

1 Human Placement for Maxmum Dexterty Karm Abdel-Malek and We u Department of Mechancal Engneerng The Unversty of Iowa Iowa Cty, IA Tel. (319) amalek@engneerng.uowa.edu wyu@engneerng.uowa.edu Jerry Duncan Human Factors/Ergonomcs Deere & Company Techncal Center 33 Rver Drve Molne, IL Tel DuncanJerryR@JDCorp.deere.com Placement n ergonomc desgn s the problem concernng the specfcaton of the poston of a human wth respect to a pre-exstng work envronment. In an assembly lne, for example, t s advantageous to poston a worker n such a manner to maxmze hs/her dexterty, mnmze the person s stress on each jont, and maxmze ther reach. Ths paper presents a rgorous mathematcal approach that utlzes knematc formulatons from robotcs and optmzaton theory to defne the placement problem. The concept underlng ths approach s that the ergonomc desgn process s ndeed an optmzaton problem wth many parameters. Whle only dexterty s presented n ths paper, the formulaton s broadly applcable and can be generalzed to the ergonomc desgn process for any objectve functon or combnaton thereof. A measure of dexterty s developed and examples are llustrated. The work s a part of a long term vson to establsh a fundamental formulaton for ergonomc desgn. Keywords: Human placement, dexterty measure, reachablty, and ergonomcs. 1

2 Introducton In ths paper, we characterze the ergonomc desgn process as an optmzaton problem wth many varables. We further beleve that n order to acheve a rgorous approach to ths teratve process, the problem must be defned n terms of a multdscplnary approach and as a result mathematcally obtan defntons for the varous cost functons that represent the measures of human performance. Ergonomc desgn has tradtonally depended on emprcal data and rules of thumb as evdenced by the many works that address the ergonomc desgn process. For example, a method for the determnaton of ergonomc parameters that relate people to objects n space was proposed by Costa, et al. (1997). The authors state that mathematcal models of human movements are complex to defne and hard to solve and suggest the use of Artfcal ntellgence n Neural Systems (ANS) as an approach to the problem. Indeed, collecton of data for smulaton of human movement has been done by many researchers (e.g., Cungrad, et al. 1998), but there has never been a fundamental rgorous approach to the problem. The study of human moton has led desgners to produce ergonomcbased workstatons, by addressng work postures, work heght, adjustable chars, foot/hand use, gravty, momentum wthn normal work area all n an expermental manner (Kong 199), whch s costly and s dffcult to use n a desgn scenaro. The proposed work wll provde a vable venue to address such ssues. More rgorous work has recently appeared that has employed a mathematcal model of lmbs as four-lnk systems consstng of trunk, upper arm, lower arm, and hand, beng 2

3 regarded as a redundant manpulator wth a total of eght degrees of freedom (DOF) (Jung, et al. 1992; 1997; Jun and Kee 1996). The authors stated that nverse knematcs were used for solvng the system and that the jont range avalablty was used as a performance functon n order to guarantee local optmalty (Jung and Park 1994). Inverse knematcs s an expresson gven to the mathematcs used n calculatng jont varables gven the poston and orentaton of the end-lnk (e.g., hand). Whle nverse knematcs of an eght DOF system s not only dffcult to obtan, but s also unrelable because of the many redundant solutons that may arse. Indeed, the concept of nverse knematcs n ergonomc analyss and desgn should only be consdered for nonredundant systems. We frst defne the placement problem, ntroduce the needed mathematcs, and then demonstrate ts applcablty to ergonomc desgn. Problem Defnton We wll frst descrbe the problem n rather general terms, followed by a more rgorous approach quantfyng our models. Consder the desgn of an assembly lne n a manufacturng envronment where t s requred to poston a number of operators, each to acheve a specfc task. In order to maxmze the output effcency, t s necessary to place the operators wth respect to the assembly lne whle maxmzng ther dexterty. It s also possble that the desgn would depend on a dfferent maxmzaton/mnmzaton functon such as effort, stress, reachablty, repettve stran, or force. The formulaton developed n ths paper wll lmt tself to the development of a measure for dexterty as the drvng cost functon (so-called n the feld of optmzaton) to optmze the ergonomc placement of a person n a work envronment. 3

4 Consder the reach envelope (shown n Fg. 1a) of a human that s known n closed form (.e., the equatons of the envelope are known). Also consder a number of targets that must be touched by the human n the work envronment, whch wll be called target ponts, as shown n Fg. 1b. Intal confguraton Confguraton after placement Reach envelope Target ponts Fg. 1 (a) A human and the reach envelope (b) The reach envelope postoned to nclude three target ponts and the correspondng poston of the human It s requred to poston and orent the human n such a manner to touch all three ponts whle maxmzng the dexterty at each pont. To acheve ths task, we wll manpulate the reach envelope and specfy ts poston and orentaton, whch wll be characterzed by the sx coordnates w = x y z T α β γ. The frst three coordnates dentfy the envelope s poston and the last three ts orentaton. In order to formulate the problem as an optmzaton problem, we defne a functon (also called a cost or objectve functon) as f = Dexterty( w ) as a functon of the w varables 4

5 (also called desgn varables), whereby ths functon must be mnmzed or maxmzed subject to some constrants. These constrants characterze the target ponts fallng wthn the reach envelope, the jonts beng wthn ther ranges of moton, and the person beng fnally n the uprght poston. Once defned n ths manner, an teratve numercal optmzaton algorthm can be called upon to make the necessary calculatons. Plan of Work In order to address the above problem n a mathematcally rgorous manner, we have to formulate the problem as an optmzaton algorthm (Arora 1989), whereby the desgn varables characterze the poston and orentaton of the person and the desgn functon s a quantfyng measure for dexterty. Therefore a measure for dexterty must be developed. Furthermore, constrants mposed on the human s fnal poston nclude the followng: (a) Target ponts must be wthn reach (.e., nsde the reach envelope of the person) and (b) Target ponts are not on the boundary of the workspace envelope, (c) Jont ranges of moton must be consdered, and (d) The fnal poston of the human wthn a fnte space. Modelng Scheme In order to obtan a systematc representaton of the workspace produced by the moton of a pont of nterest (typcally called a pont on the end-effector), we wll use the Denavt- Hartenberg method adopted from the feld of robotcs (Denavt and Hartenberg 1955; Abdel-Malek, et al. 1997; 1999). Consder Fg. 2 where three consecutve lnks are shown. 5

6 Lnk m+1 Lnk m Lnk m-1 Fgure 2 Defne the jont reference frames for the D-H representaton Let -1 and represent fxed axes at ether end of lnk -1, about whch or along whch lnks -1 and move, respectvely (as shown n Fg. 2). Let axes -1 be defned from -1 to and perpendcular to both. Let -1 represent the unque axs that together wth -1 and -1 completes a rght-hand Cartesan coordnate system. Let represent a vector from O -1 parallel to. Let -1 represent a vector from O parallel to -1 as llustrated n Fg. 3. θ -1 Jont Jont -1 α d O -1 a O -1 Fg. 3 The relaton between two consecutve coordnates 6

7 The followng four ordered operatons completely specfy the confguraton of the frame coordnate system relatve to the frame (-1) coordnate system: (a) A constant twst of α degrees about axs -1 of -1 nto, here, α s the angle from -1 to, wth transform matrx T a : cosα -snα 1 T a = snα cosα! 1 " $ # s parallel to (b) A constant dsplacement of b unts along -1 from -1 to, wth transformaton (1) matrx T b gven: T b =! 1 b " $ # (2) (c) A rotaton of θ degrees about of -1 nto, here θ s the angle from -1 to, -1 s parallel to -1, wth transformaton matrx T c gven as: T c =! cosθ -snθ snθ cosθ 1 1 (d) An offset of d unts along from the -1 - ntersecton to O, wth transformaton matrx T d as: T d =! d 1 " $ # " $ # (3) (4) 7

8 Note that the four parameters θ, d, a, α completely defne the relaton between any two consecutve frames. These values are entered n a table, whch s typcally known as the DH Table. The overall Denavt-Hartenberg coordnate transformaton matrx from frame coordnate system relatve to the frame - 1 coordnate system s then gven by: T- 1= TaTb Tc Td =! cosθ -snθ b cosα snθ cosα cosθ -snα -d snα snα snθ snα cosθ cosα d cosα 1 Smlarly, for an n-dof model of a lmb, the global jont and end-effector frames usng Eq. (5) are restated usng n-homogeneous transformaton matrces n T,T,T,T, L,T -. The transformaton matrx from the end-effector frame to n 1 global frame s then obtaned by pre-multplyng each matrx n seres as: = - n 1 T q T T 2 1 T 3 n ( ) ( q1) ( q2) 2 ( q3) L Tn 1( q n ) (6) T where q = [ q1... q n ] are the generalzed coordnates (jont varables) of the lmb and where the resultng transformaton matrx T R n n n =! " $# " $ # n contans the ( 3 3 ) R rotaton matrx and ( 3 1) n poston vector, whch represents every pont that can be touched by the ndex fnger (or any other specfed part). Therefore, the reach envelope s descrbed by the vector (5) n x= [ x y z] = ( q) (7) where x: R n R 3 s a smooth vector functon defned as a subset of the Eucldean space. However, the boundary of the reach envelope s not yet known. In order to determne the boundary, we apply a rank defcency condton to obtaned from dfferentatng the poston vector n. The so-called Jacoban s n as follows: 8

9 x& = q q& (8) where &x represents the absolute velocty of the hand and &q represents the vector of jont veloctes. Therefore, the Jacoban q q = relates both veloctes. It was shown by Abdel-Malek, et al. (1997; 1999) that consecutve applcaton of a rank defcency condton on the Jacoban q yelds sngular sets denoted by s, whch defne the boundary to the reach envelope. The benefts of ths method s two fold: (a) The reach envelope boundary s determned n closed form. (b) The reach envelope boundary s exact. Surface patches on the boundary of the reach envelope of human lmbs are delneated n closed form by substtutng the sngular sets nto the equaton for the reach envelope. () ( u) = ( s, q) (9) where u are the remanng varables (recall that s s a set of constants). These () are ndeed surface patches n closed form and characterze the boundary of the reach envelope (Abdel-Malek, et al. 2). For example, f the base coordnates are embedded n the shoulder, and we are seekng the reach envelope of the tp of the fnger wth respect to the shoulder, then the surface patches are obtaned and shown n Fg. 4. 9

10 WCS q 1 x(q) Pont s p Fg. 4 The reach envelope of the upper extremty Consder a number of target ponts p ( j ) ; j = 1,..., l, located n space, whereby t s requred to place the human such that these target ponts are wthn reach yet maxmzng the dexterty functon (to be developed n the followng secton). To ascertan that target ponts p ( j) are nsde the reach envelope, the absolute value of the dstance between a target pont and the boundary should be greater than a specfed value ε (a specfed tolerance). Ths wll guarantee that the target ponts are located nsde of the reach envelope but not on boundary. In order to track the poston and orentaton of the envelope, we shall use a set of 6 generalzed coordnates w = x y z w w w T α β γ, where a poston vector v( xw, yw, zw ) wll be used to track the poston and a rotaton matrx R( α, β, γ ) wll be used to track the orentaton. The dstance between all target ponts p ( j) and all surface patches () ( u ) should be greater than a specfed mnmum value such as 1

11 ( p j ) ( ) ( ) ( u, w) ε where j = 1,...,l and = 1,..., m (1) j () () where ( wu, ) = R( αβγ,, ) x ( u) + v( x, y, z ) (11) w w w and where R s the rotaton matrx that wll be used to orent the reach envelope, v s the poston vector that wll be used to locate the envelope, and where ε j > are specfed constants. If a target pont satsfes both condtons of Eq. (4) and (5), then ths pont s nternal to the reach envelope (.e, have placed the envelope n a confguraton such that all ponts can be reached. In order to move the human to a new poston, we wll move the reach envelope towards the target ponts, subject to the followng constrants: (1) Reach envelope at least coverng the target ponts (shortest dstance between the target ponts): g () mn p ( q, w) G β for = 1,...,l (12) where G( wq, ) = R( α, β, γ ) F( q) + v( x, y, z ) w w w and β s a very small postve number and subject to the ranges of moton or jont lmts as L U q ˆq ˆ q for k = 1,..., n (13) k k k (2) Embeddng the target ponts nsde the reach envelope (a mnmum dstance between target ponts and surface patches). ( ) ( ) ( u, w) ε for = 1,..., m and j = 1,...,l, k = 1,...,( l m) (14) j g k p j where ε j s the depth of the target pont nsde the reach envelope. There are l+ ( l m) + n total number of constrants. We have now defned all constrants but have 11

12 not provded for a cost functon to optmze. We wll address ths task n the followng secton. A Measure of Human Dexterty In ths secton, we defne a cost functon that s based on maxmzng the dexterty at target ponts. Indeed, to mathematcally formulate ths problem, t s necessary to use a dexterty measure at specfc target ponts and that s a functon of the desgn varables w. Such a measure must account for the ranges of moton for each jont. Because of the need for an analytcal expresson that can be used n the proposed optmzaton approach, we defne a new dexterty measure. Because human jonts are constraned, we must characterze each jont lmt by an L U nequalty constrant n the form of q ˆq ˆ q. In order to nclude ranges of moton n the formulaton, we have used a parameterzaton (see Appendx A) to convert nequaltes on q to equaltes q = L( λ ), where the new varables are defned by l = λ1, λ2,..., λn T n R. For the hand at a gven locaton x h (.e., for the hand at a n specfc poston that can be reached), - x = must be satsfed. Moreover, the parameterzed constrants of the ranges of moton must also be satsfed as L- q=. Therefore, the general constrant can be obtaned by augmentng both equatons to obtan h the ( n + 3 ) constrant vector as * Gq ( ) =! n ( q) - x ( ) q L l " h = $# ( n+ 3) 1 (15) 12

13 * where the augmented vector of generalzed coordnates s q = [ x T q T l T ] T. By T T T defnng a new vector z= [ q l ] (nput parameters), the augmented coordnates can be parttoned as * The set defned by G( q) q * = x T z T T (16) s the totalty of ponts n the reach envelope that can be * touched by the hand. The so-called extended Jacoban of G( q) dfferentatng G wth respect to z as G z =! " $# s obtaned by q (17) I L l whch s an ( n+ 3) ( 2 n) matrx, where q = qj s a ( 3 n ) matrx, I s the ( n n) dentfy matrx, and L l = λ s an ( n n) dagonal matrx wth dagonal Λ elements as Λ λ ) = b cosλ. We defne G z as the augmented Jacoban matrx. j Snce the extended Jacoban G z nherently combnes nformaton about the poston, orentaton, and ranges of moton of the hand, t s a vable measure of dexterty. Furthermore, because of the smplcty n determnng an analytcal expresson of G z, t s well-suted as a cost functon for an optmzaton problem. We defne the dexterty measure as T D = GG z z (18) Note that the measure characterzed by Eq. (18) takes nto consderaton all ranges of moton and sngular orentatons for a gven arm, lmb, or any seral chan. 13

14 The Placement Problem as an Optmzaton Algorthm Gven l target ponts p () ( x, y, z ) for = 12,, L, l defned n space, we ntroduce a () dexterty measure at each pont denoted by D( p ), where t s necessary to place a human to acheve maxmum dexterty at each target pont. Suppose the vector descrbng the pont on the ndex fnger of a human arm s gven by q ( ) and the boundary envelope of workspace are determned as closed-form surface patches denoted by ( j ) ( ( j u ) ), j= 12L,,, l. A mathematcal model of the placement of the robot base subject to maxmzng the dexterty at specfed target ponts s characterzed by the followng optmzaton problem. Cost functon l () Maxmze f ( w) = Ê ω D( w, p ) = 1 (19) Subject to the followng constrants: (1) Target ponts are nsde the workspace volume P - G ( w, u ) ε (2) () ( j) () where = 12,, L, land j= 12,, L, m. (2) Target ponts are not on the boundary of the workspace envelope () mn P - ( w, q) ˆ δ, where = 1, 2, L, l (21) where δ, = 12L,,, l, ε j, = 12L,,, l, and j= 12,, L, m are postve constants. (3) Ranges of moton are mposed L U q ˆqˆ q (22) j 14

15 (4) The moton s wthn a fnte space L U w ˆwˆ w (23) Because ths s a mult-objectve optmzaton problem, we wll assgn the dexterty at each pont a weght so as to transfer the problem nto a classcal optmzaton problem where w, = 12L,,, l are weghts at each pont. We now ntroduce the general teratve numercal algorthm for solvng the optmzaton problem n Fg. 5. Input to the algorthm s a complete defnton of the human (.e., the DH Table). For all practcal purposes, the dmensons and jont ranges of moton of an upper extremty s enough to manpulate the placement. Also as an nput, we defne n terms of equatons, the surface patches that characterze the boundary of the reach envelope. These surfaces are defned wth respect to the coordnate system n the shoulder. We also defne the cost functon as a measure of dexterty and defne the constrants for ths optmzaton problem. The numercal algorthm teratvely moves the reach envelope towards ncludng the target ponts wthn the envelope yet attemptng to maxmze the cost functon (.e., dexterty). The algorthm wll stop when all of the tolerances are satsfed. 15

16 Defne target ponts Defne human (dmenson and ranges of moton) Reach envelope has been dentfed Move boundary of reach envelope Defned by the sx generalzed coordnates w that characterze ts poston and orentaton w = w + w Iteratve algorthm to move the workspace Cost functon Maxmze Dexterty Constrants (no need for nverse knematcs) Iterate Satsfes Tolerance Stop Fg. 5 Algorthm for Placement Smple Example Consder a model of the arm that s restrcted to move on a planar surface (e.g., the surface of a table). Ths example wll be used to llustrate the theory and wll be followed by a more realstc model of the upper extremty. There are three target ponts, namely, (1) T P = [14 1], P (2) = [1 1 T 5], and P (1) = [1 1 T 75], whch must be touched by the human hand wth the ablty to reach these ponts from many drectons (.e., maxmzng the dexterty of the upper lmb at these ponts). 16

17 The arm shown n Fg. 6 s modeled as three revolute jonts and restrcted to planar moton (e.g., on the surface of a table). Fg. 6 Model of the arm wth three revolute jonts The DH table s readly determned and presented n Table 1. Table 1: DH Table θ d α a 1 q q q 3 1 Substtutng each row nto Eq. () and performng the multplcaton yelds the coordnates of the tp of the ndex fnger are gven by 4 cos q1 n ( q) = 4sn q1 + 2 cos( q + 2sn( q q + q 2 2 ) + cos( q ) + sn( q q + q q3) + q ) 3 (24) For smplcty, ranges of moton on each jont are defned as π 3 q π 3 ; = 1,2, 3. Results of the reach envelope determnaton yeld the followng boundary curves (note that curves are generated because we have restrcted the arm to planar movement. The boundary curves are defned by the followng sets: x ( π 3, q2,); q2 [ π 3, ], x ( π 3, q2,); q2 [, π 3], x q,,); q [ π 3, 3 ] x π 3, π 3, q ); q [ x ( 1 2 π ( π 3, π 3, q3); q3 [, π 3] x ( 3 3 π 3,] ( q1, π 3, π 3); q1 [ π 3, π 3] 17

18 and x q, π 3, π 3); q [, 3]. ( 1 1 π Substtutng the sngular sets nto Eq. () yelds equatons of curves shown n Fg., whch s the reach envelope as shown n Fg. 7. y x Fg. 7 The exact reach envelope of the arm (restrcted to planar moton) As a result of the teratve algorthm, the sx desgn varables representng the poston and orentaton of the reach envelope are calculated as [ x y ] T = [ ] T w = α and shown n Fg. 8. The measure of dexterty at each pont s calculated as (1) D( P ) = (2) D ( P ) = (3) D ( P ) =

19 Note that any confguraton that would have ncluded the three ponts s a soluton, however, the soluton calculated usng ths method yelds the poston of the arm that would maxmze the dexterty at all three ponts Fg. 8 The ntal and fnal poston of the arm. Example: A Realstc Model of the Arm Consder the upper extremty shown n Fg. 9 and modeled as a total of 9DOF (5DOF n the shoulder, 1DOF n the elbow, and 3DOF n the wrst). Note that we have accounted for two translatons of the shoulder jont and three rotatonal motons. Also note that n order to allow for realstc moton of the shoulder, we have coupled the translatonal and rotatonal jonts by a lnear equaton such that the behavor of the moton of one jont s dependent upon the other. 19

20 q 4 q 2 y o q 3 q 5 q 1 x o q 6 q 7 q 9 (θ) q 8 Fg. 9 Model of the upper extremty The D-H parameters for ths arm are presented n Table 2. Table 2: DH Table for the arm θ d a α 1 π 2 q 1 -π 2 2 -π 2 q 2 π 2 3 +q 3 π 2 4 π 2+q 4 π 2 5 +q 5 2 cm -π 2 6 +q 6 cm π 2 7 +q 7 -π 2 8 q 8 -π 2 -π 2 9 +q 9 2

21 Ranges of moton for ths arm are as follows (note that the frst two jonts are translatonal): -38. ˆq ˆ38. cm; -38. ˆq ˆ38. cm; -π 2ˆq ˆπ 2; 1-11π 8 ˆq ˆ2π 3; -π 2ˆq ˆπ 2; ˆq ˆ5π 6; -π 3ˆq ˆπ 3; 4 -π 9ˆq ˆπ 9; and -π ˆq 9 ˆ It s requred to place the human such that the followng three target ponts are touchable and dexterty s maxmzed. (1) T P = [1 1 5], P (2) = [1 1 T 5], and (1) T P = [1 1 75]. The poston of the ndex fnger x( q)[ 1] =- 2ccs + 2ss + 1c( - ccs + ss) - 1ccs + 5( c( c( c( -ccs + ss )- cc s ) + ( cs + css ) s ) + (-ccc -( ccs + ss ) s ) s x( q )[2] =-2css - 2cs + 1c (-css -cs ) - 1c ss + 5( c ( c ( c (-css -cs 1 3) - css 2 1 4) + (- cc 1 3+ sss 1 2 3) s5) + (-ccs ( -css cs 1 3) s4) s6 x( q )[] 3 = 2cc ccc ss ( c6( c5( ccc ss 2 4) -css 2 3 5) +- ( cs -ccs) s) where c = cos q, s = sn q, and q = q q 1 7 T.... Note that the frst two jonts are coupled and therefore the model s reduced to 7DOF. The boundary surfaces are delneated and shown below n Fg. 1 (a total of 22 boundary surfaces): 21

22 Fg. 1 Surface patches These surface patches are combned, the 9DOF reach envelope s calculated, and shown n Fg

23 Fg. 11 Reach envelope of the upper extremty As a result of the teratve algorthm, the sx desgn varables representng the poston and orentaton of the reach envelope are calculated as w = [ ] T The measure of dexterty at each pont s maxmzed and ts value s (1) D( P ) = (2) D( P ) = (3) D( P ) = The ntal and fnal confguratons of the reach envelope of the arm are shown n Fg

24 Fg. 12 Intal and fnal confguratons of the reach envelope Conclusons A rgorous mathematcal formulaton for placement of humans n a work envronment whle maxmzng dexterty has been ntroduced. Ergonomc desgn has tradtonally been dependent upon emprcal data and rules of thumb, manly because of the many varables assocated wth the desgn process. We beleve ths approach s an ntal step towards makng the ergonomc desgn process more rgorous and ntroducng wellestablshed methods from optmzaton. It was shown that optmzaton algorthms can be used n an teratve manner to calculate the optmum poston and orentaton of a human whle consderng a multtude of constrants. Furthermore, t was shown that 24

25 realstc ranges of moton are consdered n the formulaton and a new performance measure (cost functon) was developed and used to evaluate dexterty at gven target ponts. References 1. Abdel-Malek, K. and eh, H.J., 1997, Analytcal Boundary of the Workspace for General Three Degree-of-Freedom Mechansms, Internatonal Journal of Robotcs Research. Vol. 16, No. 2, pp Abdel-Malek, K., 1996, Crtera for the Localty of a Manpulator Arm wth Respect to an Operatng Pont, IMEChE Journal of Engneerng Manufacture, Vol. 21 (1), pp Abdel-Malek, K., Adkns, F., eh, H.J., and Haug, E.J., 1997, On the Determnaton of Boundares to Manpulator Workspaces, Robotcs and Computer-Integrated Manufacturng, Vol. 13, No. 1, pp Abdel-Malek, K., eh, H-J, and Kharallah, N., 1999, Workspace, Vod, and Volume Determnaton of the General 5DOF Manpulator, Mechancs of Structures and Machnes, 27(1), Abdel-Malek, K. and eh, H. J., (2) "Local Dexterty Analyss for Open Knematc Chans," Mechansm and Machne Theory, Vol. 35, pp Abdel-Malek, K., ang, J., Brand, R., and Vanner, M., (submtted) Understandng the Workspace of Human Lmbs, Internatonal Journal of Ergonomcs. 7. Arora, J.S., 1989, Introducton to Optmum Desgn, McGraw-Hll Book Co., New ork. 8. Char, P.; Chaffn, D.B., Human smulaton modelng - wll t mprove ergonomcs durng desgn?, Proceedngs of the st Annual Meetng of the Human Factors and Ergonomcs Socety. Part 1 (of 2) v Albuquerque, NM, pp Chou, H.C.; Sadler, J.P., 1993, Optmal locaton of robot trajectores for mnmzaton of actuator torque, Mechansm and Machne Theory, Vol. 28, No. 1, pp Cungrad, B.; Costa, M.; Pasero, E.; Maccharulo, L., 1998, Recurrent network for data drven human movement generaton, Proceedngs of the 1998 IEEE Internatonal Jont Conference on Neural Networks. Part 1 (of 3) May v Anchorage, AK, pp Costa, M.; Crspno, P.; Hanomolo, A.; Pasero, E., Artfcal neural networks and the smulaton of human movements n CAD envronments, Proceedngs of the 1997 IEEE Internatonal Conference on Neural Networks. Part 3 (of 4) Jun v Houston, T, pp Denavt, J., and Hartenberg, R.S A knematc notaton for lower-par mechansms based on matrces. Journal of Appled Mechancs, ASME, Vol. 22, pp

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27 29. Papadopoulos, E. and Gonther,., 1995, On manpulator posture for plannng for large force tasks, Proc. Of the IEEE Int. Conf. On Robotcs and Automaton, Nagoya, Japan. 3. Rchards, J. 1998, The Measurement of Human Moton: A Comparson of Commercally Avalable Systems, Proceedngs of the 5 th Internatonal Symposum on the 3-D analyss of Human Movement, Chabanacoque, Tennessee 31. Roth, B., 1991, On the number of lnks and placement of telescopc manpulators n an envronment wth obstacles, Proceedng of the Ffth Internatonal Conference on Advanced Robotcs - '91 ICAR, pp Seraj, H., 1995, Reachablty analyss for base placement n moble manpulators, Journal of Robotc Systems, Vol. 12, No. 1, pp Spvak, M. 1968, Calculus on Manfolds, Benjamn/Cummengs. 34. Tu, Q.; Rastegar, J., 1993, Determnaton of allowable manpulator lnk shapes; and task, nstallaton, and obstacle spaces usng the Monte Carlo Method, Journal of Mechancal Desgn, Transactons of the ASME, Vol. 115, No. 3, pp Vncent, T.L. Goh, B.S. and Teo, K.L., 1992, Trajectory-followng algorthms for mn-max optmzaton problems, Journal of optmzaton theory and applcatons, Vol. 75, No. 3, pp eghloul, S. and Blanchard, M. A, 1997, SMAR: A robot modelng and smulatng system Robotca, 1997, Vol. 15, pp eghloul, S.; Pamanes-Garca, J.A., 1993, Mult-crtera optmal placement of robots n constraned envronments, Robotca, Vol. 11, pt 2, pp hang, et al. (1997) used dynamc human models that are emprcally valdated for a typcal range of drvng tasks and for people of vared anthropometry, gender and age. 39. hang,.; Chaffn, D.B.; Thompson, D., Development of dynamc smulaton models of seated reachng motons whle drvng, SAE Specal Publcatons Progress wth Human Factors n Automotve Desgn: Seatng Comfort, Vsblty, and Safety 1997 v 1242, Detrot, MI, pp Appendx A Ranges of moton are mposed n terms of nequalty constrants n the form of where L U q q q (a.1) = 1,... n. We transform the nequaltes above nto equaltes by ntroducng a new set of generalzed coordnates l = [ λ... λ ] 1 n T such that L U U L q = ( q + q ) 2 + ( q q ) 2 sn λ n = 1,..., (a.2) 27