Realistic Posture Prediction

Transcription

1 Realstc Posture Predcton K. Abdel-Malek W. Yu M. Jaber 2 Department of Mechancal Engneerng and Center for Computer Aded Desgn (CCAD) The Unversty of Iowa Iowa Cty, IA Tel. (39) Fax. (39) amalek@engneerng.uowa.edu 2 Schools of Aeronautcal, Industral, and Mechancal Engneerng Ryerson Polytechnc Unversty Toronto, ON M5B 2K3, Canada Tel: (46) Ext mjaber@acs.ryerson.ca Abstract Ths paper presents an effcent numercal formulaton for the predcton of realstc postures. Ths problem s defned by the method (or procedure) used to predct the posture of a human, gven a pont n the reachable space. The exposton addresses () the determnaton whether a pont s reachable (.e., does s t exst wthn the reach envelope) and (2) the calculaton of a posture for a gven pont and a gven orentaton. Whle many researchers have used ether statstcal models of emprcal data or nverse knematcs for posture predcton, we present a method based on knematcs for modelng, but one that uses optmzaton of a cost functon to predct a realstc posture. We llustrate the methodology and accompanyng expermental code through a planar and a spatal example. Keywords: posture predcton, nverse knematcs, ergonomcs, human modelng

2 Introducton Human modelng and smulaton has been extensvely used n recent years (wth the appearance of commercal codes) to study the functonalty of humans n ther work envronment wth the am to desgn for ergonomcs. Posture predcton s an mportant aspect of dgtal human smulaton. To explan the meanng of posture predcton, consder a gven a pont n space (e.g., a target), t s requred to determne the jont angles that confgure the arm (and torso f need be) that allow a specfed pont on the ndex fnger to touch the target pont. Smlarly, f a box s located n a gven poston and orentaton, what s the confguraton of the torso, shoulder jont, and arm that would be assumed by a person to lft ths box. Therefore, posture predcton represents algorthms that enable the estmaton of human jont angles for a gven posture. In applyng these methods to any partcular desgn problem, the prmary attempt s to establsh the boundary condtons of the workplace, and predct realstc postures and feasble arm reaches. A proactve, rather than reactve, approach allows desgners to ncorporate reachable features nto desgns that mnmze the rsk of njury before a person ever physcally encounters the product or workplace (Feyen et al. 2). We beleve that dgtal human smulaton s an effectve method for ergonomc desgn. There has been two schools of thought regardng posture predcton. The frst, perhaps the more tradtonal, uses anthropometrcal data, collected from performng thousands of experments by human subjects, or smulaton usng three-dmensonal computer-aded man-modelng software [see for nstance; Porter et al. (99) and Das and Sngupta (995)], whch were statstcally analyzed to form a predctve model of posture; e.g. 2

3 regresson models. Ths school of thought s referred to as emprcal-statstcal modelng. These models have been mplemented n varous smulaton software systems wth some varatons as to the method for selectng the most probable posture. Among the emprcal-statstcal modelers were Beck and Chaffn (992), Zhang and Chaffn (996,997), Das and Behara (998), and Faraway, et al. (999). Early work that rased concern regardng the use of anthropometrcal data was reported by Bonney et al. (98), where they ndcated that very few surveys take suffcent measurements to defne an accurate 3D human models. They suggested that ergonomsts should take more comprehensve measurements to maxmze the potental applcaton of ther data. The second school of thought often used bomechancs as a predctve tool, on a posture that has not been observed but has been estmated as a lkely posture for a task (Tracy, 99). Ths school mathematcally modeled the moton of a lmb wth the goal of formulatng a set of equatons that can be solved for the jont varable. In the feld of knematcs, ths problem s called nverse knematcs. Among the researchers who belong to ths school of modelng are Kee et al. (994), Jung et al. (995a), Jung and Kee (996), Jung and Choe(996), Kee and Km (997), and Wang (999). Whle some researchers that belong to one school of modelng (n partcular Beck and Chaffn 992) cautoned that the nverse knematcs algorthm s not necessarly correct for predcton of posture because of ts theoretcal foundaton others (Faraway, et al. 999) have also argued that postures that are modeled usng angles between body segments, rather than jont coordnates, may volate task constrants. On the other hand, those that belong to the nverse knematcs school of modelng, state that most exstng 3

4 human models have not fully utlzed anthropometrc data due to the generalzed formalsm of data manpulaton, whch may result n serous problems when a system s upgraded or when a specfc populaton of operators s consdered (e.g., see the objectorented anthropometrc work by Jung and Kang 995 and Jung, et al. 995). The concept of cost functon for posture predcton s not new. Based on an earler study (Jung, et al. 992), Jung, et al. (994) proposed the concept of usng a psychophyscal cost functon to defne a cost value for each jont movement angle and, subsequently, solve the redundancy problem n human movement. In developng the psychophyscal cost functon they ntegrated the psychophyscal dscomfort of jonts and the jont range avalablty concept; a concept that has been used to predct the arm reach posture for redundant arm manpulaton n robotcs. Jung, et al. (994) results showed that the postures predcted by the psychophyscal cost functon closely smulated human postures and the predctablty was more accurate than that by the bomechancal cost functon. Jung et al. (995a) also states that most exstng reach posture predcton models have used heurstc methods, whch provde only the range of feasble postures, not always ensurng the actual posture that a person naturally takes. As a result, an approxmate analytcal reach predcton algorthm, was developed (Jung et al. 995a) where the Devat and Hartenberg (D-H) notaton was used to represent human moton., Subsequently, Jung and Kee (996) and Jung and Choe (996) developed a regresson model to predct the perceved dscomfort wth respect to the jont movement. Ther reportedly demonstrated that humans adopt postures of mnmum dscomfort among all feasble body confguratons. Smlar results were reported by Dysart and Woldstad (996) who used 4

5 three separate models and objectve functons to predct the postures of humans performng statc sagttal lftng tasks. The models used a common nverse knematcs characterzaton to represent mathematcally feasble postures, but explore dfferent crtera functons for selectng a fnal posture. Dysart and Woldstad (996) results showed that the frst objectve functon (mnmum total torque) was more accurate. Also usng the concept of nverse knematcs, Kee and Km (997) proposed an approxmate algorthm to generate the workspace ncludng foot and trunk moton. More recent results have focused on a combnaton of methods such as both rule-based emprcal and optmzaton to address the posture predcton problem (Wang 999). Indded, Wang showed that the nverse knematcs problem s ll-posed because of the redundancy of the human arm. Wang s algorthm, as compared to the generalzed nverse knematcs algorthm, can handle the non-lnearty of jont lmts wth no need for nverse matrx calculaton. Thus, avodng the stablty and convergence problems, whch often occur near a sngularty of the Jacoban. Other researchers adopted dfferent approaches to predct posture than those aforementoned. The emergence of Artfcal Neural Networks models to provde more accurate predctons over the standard statstcal models (Eksoglu, et al. 996; Jung and Park 994; Hestenes 994). Problem Defnton In ths paper, we explore the use of recent computatonal schemes and genetcs algorthm to ntroduce a thrd approach to posture predcton. Before proceedng, we defne the problems that wll be addressed. () Gven a target pont n space, s ths pont reachable? 5

6 (2) Gven a target pont n space, what are the jont angles that would confgure the lmb such that a partcular pont on the lmb wll reach the target pont, (3) Gven a target pont n space and gven an orentaton at the target pont (e.g., orentaton of the ndex fnger), t s requred to estmate the posture of the lmb that would yeld ths poston and orentaton of the end-effector, and (4) Gven a target pont and a complete descrpton of the orentaton of the end-effector (.e., three vectors are also gven), t s requred to estmate the posture of the lmb that would yeld ths poston and orentaton of the end-effector. The problem s llustrated n Fg. where an ntal posture s shown n Fg. (a). A pont s specfed and Posture I (Fg. b) s calculated n order to reach the pont. However, because of the redundancy n the upper extremty, there are an nfnte number of postures that can be made to reach the pont. Another less realstc posture s shown n Fg. c. Pont Intal Posture Posture I Posture II (a) Intal posture (b) A calculated realstc posture (c) A less realstc calculated posture Fg. Predctng a human posture In ths paper, we present solutons to the above four problems. The answer to the frst queston wll be explored usng our recent results n the closed form determnaton of the workspace envelope (presented n an accompanyng paper). The remanng three 6

7 questons wll be addressed usng a novel cost functon optmzaton analyss and solved usng genetcs algorthms (GA). We wll frst present the background materal necessary for the analyss. We then present an effcent method for answerng whether a gven pont s reachable n the workspace. Cost functons are then derved that wll be used to predct realstc (sometmes called naturalstc) postures. Before proceedng, t s mportant to note that although our exposton wll be focused on the arm, the mplementaton of ths mathematcal formulaton s vald for any part of the human body. For example, one may elect to model the arm and the torso as an open knematc chan. Modelng and Ranges of Moton Introduce the modelng method here. Put a nce pcture as well of the arm and two lnks. The D-H representaton provdes a systematc method for descrbng the relatonshp between adjacent lnks. The 4 4 transformaton matrx descrbng a transformaton from lnk (-) to lnk for a revolute jont s T cos q = sn q cosα sn q cosα cos q snα snα sn q snα cosq cosα a cosq a sn q d () where θ, depcted n Fg., s the jont angle from x to the x axs, d s the dstance from the orgn of the (-) th coordnate frame to the ntersecton of the z axs wth the x, a s the offset dstance from the ntersecton of the z axs wth the x axs, and α s the offset angle from the z axs to the z axs. The poston vector of a pont of 7

8 nterest on the end-effector of a human artculated model (e.g., a pont on the thumb wth respect to the shoulder) can be wrtten n terms of jont coordnates as x = F( q) (2) where q³ R n s the vector of n-generalzed coordnates, and F( q ) can be obtaned from the multplcaton of the homogeneous transformaton matrces defned by the D-H representaton method (Denavt and Hartenberg 955) as n- Tn = T T2... Tn =! R ( q) F( q) n " $# (3) where R s the rotaton matrx and T s the homogeneous transformaton matrx relatng bodes to j. j j In order to extend the formulaton to nclude ranges of moton n the form of nequalty mn max constrants such as q q q (e.g., the range of moton of the elbow s o o approxmately ˆq ˆ 5 ), a transformaton s ntroduced as max mn where a = ( q + q ) q = a + b sn λ (4) max mn 2 and b = ( q q ) 2 are the md pont and half range of the nequalty constrant and λ s a slack varable (.e., we have converted the nequaly to an equalty). The poston constrant functon s then wrtten n terms of the extended vector l = λ λ λ 2... n T such that x F ( l) = F( q( l)) (5) = 8

9 To study the boundary of ths functon, we use the mplct functon theorem to determne the reach envelope (Abdel-Malek, et al. 997; 999; 2 and Abdel-Malek and Yeh 2). Indeed, f we dfferentate Eq. (4) wth respect to tme to obtan the velocty, the rght hand sde wll comprse two matrces and the jont velocty vector as x& = F q l q & l (6) where q l = q λ j s a dagonal matrx, l & = dl dt s the vector of jont veloctes, and F q s the Jacoban defned by F q = F( q) q (7) Usng the Implct Functon Theorem, we were able to delneate sngular surfaces that are on the boundary of the reachable workspace,.e., the reach envelope. Ths s accomplshed by studyng the rank defcency of the Jacoban. Defne the subvector of q as a set of constant generalzed coordnates p j m ³ R where m n, and q= u p. Sngular sets p of the manpulator can be obtaned from studyng the dmenson of the null space of F T q ( q), defned by Abdel-Malek and Yeh (997) as the set > q C (8) m S= p ³ R ;dm T Null ( F ( q )), q= u p for some constant p where u³ R n-m s the vector of generalzed coordnates that are not n p. On a sngular surface, the term Fq q l l qo, o s rank-defcent. Therefore, a boundary s dentfed when the rows of F q q s are dependent. As a result, the sets of sngular values p are dentfed and substtuted ntof( q ) to yeld sngular surfaces on the boundary of the reach envelope, whch we wll denote by Y( u ). 9

10 Is a pont reachable? Ths problem s of sgnfcant mportance n human moton smulaton because t s extensvely used n the desgn of ergonomc workspaces and layouts. For example, the user of a human modelng and smulaton code may nqure whether a gven pont can be touched by a dgtal person! Commercal systems address ths queston by frst generatng all possble curves by drvng the jonts of the lmb n queston through all possble ranges. The result s a combnaton of curves that are then overlad wth a mesh. The pont s then vsually checked whether t s enclosed by ths fcttous surface. Ths queston s readly and exactly answered usng our formulaton by mathematcally studyng the exstence of a pont nsde the reach envelope. Because the above formulaton yelds a complete exact representaton of the boundary to the reach envelope n closed form, t s possble to mathematcally determne whether a gven pont (e.g., a button), curve (.e., a trajectory), or an object (e.g., a lever) are nsde the reach envelope. The parametrc surface patches on the boundary of the reach envelope are descrbed by a number of Y( u ) as llustrated n Fg. 2. A pont s reachable f t exsts nsde the reach envelope. Therefore, t s now necessary to establsh whether a pont s nsde the reach envelope. For a number of surface patches, a pont (e.g., pont p n Fg. 2) s nsde the envelope f the ray cast from that pont ntersects an odd number of tmes wth the surface patches. If ntersects an even number of tmes, t s outsde (.e., not reachable).

11 2 x ξ (5) ξ (2) p -2 ξ (4) ξ (3) ξ () y 2 3 Fg. 2 Determnng whether a pont s reachable Cost Functons If after determnng whether a pont s reachable, the user requres a predcton of a posture at that pont, the analyss becomes more dffcult due to the nature of the redundancy nherent n the human body. It s evdent that the moton subtended by the human arm to reach a specfc target s drectly dependent upon the arm s ntal posture (.e., ntal condtons). A person wll usually reach towards a target usng the least moton of the jonts possble. A person wll also usually avod exertng unnecessary energy aganst gravty (.e., humans do not lke to mantan ther arms up n the ar). Therefore, we propose drvng functons on the moton characterzed by mnmzng the dsplacement from the ntal posture and mnmzng the potental energy exerted towards reachng the target, subject to the knematc constrants

12 dscussed above and each jont s range of moton. Such cost functons can be developed for dexterty, stress, potental and knetc energy, and force. As we have explaned n the Introducton, n order to determne the nverse knematcs of a redundant knematc chan, t wll be necessary to select the most sutable soluton from a famly of solutons. The choce of solutons must be made subject to a naturalstc moton (and to a certan extent) based on the motor control ablty to choose a correct posture. We therefore propose the use of cost functons to drve the arm s posture from ts ntal poston to one that s most lkely to be adopted by a person. It must be noted that these cost functons wll be used to drve the arm and not to optmze for the best soluton. Consder a cost functon that measures the level of dscomfort from the most neutral poston of a gven jont. Let q N be the neutral poston of a jont. Then the dsplacement from the neutral poston s gven by q N - q. Because the dscomfort s usually felt hgher n some jonts versus others, we also ntroduce a weght functon w to stress the mportance of one jont versus another. The total dscomfort of all jonts s then characterzed by the functon n N d Ê = f = w q - q (9) where w s a weght functon assgned to each jont for the purpose of gvng mportance to jonts that are usually more affected than others. 2

13 For a second cost functon, consder the potental energy exerted by a lmb. Each lnk (e.g., the forearm) has a specfed center of mass. The vector from the orgn of the lnk s coordnate system to the center of mass s gven by r, where smlar superscrpt and subscrpt ndcate that the vector s resolved n the lnk s coordnate system as llustrated n Fg. 3. A r r Fg. 3 Illustratng the potental energy of the forearm The total potental energy f P s the sum of all ndvdual potental energes P. For a gravty vector gven by g = -g T, the total potental energy of the arm s gven by f = P = -mg( A r) P Ê n n Ê3 8 () = = 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 3

14 analytcal expresson that can be used n the proposed optmzaton method, 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 = Ψ( λ ), where the new varables are defned by l = λ, λ2,..., λn T n R. For any admssble confguraton (.e., for the hand at a specfc poston that can be reached), the followng ( n + 3 ) augmented constrant equatons must be satsfed * Gq ( ) =! n P ( q) - x ( ) q Y l " = $# ( n+ 3) () * 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 (2) s the totalty of ponts n the reach envelope that can be * touched by the hand. The so-called extended Jacoban of G( q) s obtaned by dfferentatng G wth respect to z as G z =! Pq I " (3) Y l $# 4

15 whch s an ( n+ 3) ( 2 n) matrx, where P q = P q j s a ( 3 n ) matrx, I s the ( n n) dentfy matrx, and Y l = λ s an ( n n) dagonal matrx wth dagonal elements Ψ j as Ψ λ ) = b cosλ. We defne G z as the augmented Jacoban matrx. Note that the Jacoban was obtaned from dfferentatng the poston vector P as follows: x& = P q q& (4) where &x represents the absolute velocty of the hand and &q represents the vector of jont veloctes. Therefore, the Jacoban P P q q = relates both veloctes. 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 f x = GG T z z (5) Note that the measure characterzed by Eq. () takes nto consderaton all ranges of moton and sngular orentatons for a gven arm, lmb, or any seral chan. We shall use these cost functons to drve the arm towards the target. However, an arm consderng only the glenohumeral jont has seven DOF. If we consder the coraco clavcular jont then t can be modeled wth at least 9 DOF, where we have taken nto account only two translatonal drectons of the moton of the shoulder. Wth ths many 5

16 degrees of freedom, t s very dffcult to mplement a closed form nverse knematcs algorthm n an effectve manner. We address ths ssue n the followng secton. Formulaton for Posture Predcton and Constrants The problem s defned as follows: (a) Gven: The coordnates of a pont n space (e.g., the pont p) and the dmensons of a knematc chan are specfed and used as nput. Ths s ndeed the DH data characterzng the defnton of each lnk n the chan and the locaton of jonts, and ther type. Ths chan could be any seres of lnks (e.g., arm, torso, etc.). (b) Cost functon and Constrants Cost functons can be selected and weghted from among a number of functons: n N d Ê = Dscomfort from neutral poston: f = w q - q Potental energy: f = P = -mg( A r) Dexterty: f x P = GG z Ê n n Ê3 8 = = T z Jont ranges of moton are also mposed on ths moton to acheve a gven posture n the L U form of nequalty constrants as q ˆq ˆ q. (c) Requred It s requred to calculate a set of varables q that would allow the gven posture whle optmzng (maxmzng/mnmzng) the gven cost functons and subject to the specfed constrants. 6

17 (d) Why Genetcs Algorthms? A genetc algorthm s a search/optmzaton technque based on natural selecton (Goldberg 989). Successve generatons evolve more ft ndvduals based on Darwnan survval of the fttest. The genetc algorthm s a computer smulaton of such evoluton where the user provdes the envronment (functon) n whch the populaton must evolve. An adaptaton of such algorthm to the problem descrbed heren s shown n Fg. 4. () Genetcs algorthms yeld the global mnmum versus only a local soluton. (2) Genetcs algorthms are usually used when the search space s very large. N L U Input q, q, q ; =,..., 6 Ex.: Defne Cost Functon, Ex.: 6 =Ê N w ( q - q ) Determne the Length of the Chromosome, e.g. 9 Intalze the Populaton Convert all the populaton nto decmal numbers Intalze Process Ftness (Calculate cost functon) Evaluate Calculate the selecton probablty Cross procedure Mutaton procedure Do selecton New generaton Man Genetcs Algorthm Tolerance Fg. 4 Algorthm for predctng a posture usng genetcs algorthms 7

18 Illustratve Example (Arm restrcted to planar moton) Consder, for example, the smplfed planar 3DOF arm shown n Fg. 5. Ths s an llustratve example to demonstrate our formulaton where we have restrcted the arm to planar moton. q 3 z 2 q 2 z q z z z Pont of Interest P z 2 Fg. 5 The upper extremty modeled as 3DOF and restrcted to planar moton The objectve of ths exercse s two fold:. Gven the pont p =[ ], and the followng jont ranges of moton -π 3ˆq ˆπ 3; = 23,,, t s requred to determne whether ths pont s reachable. 2. If ths pont s reachable and t s requred to reach the pont wth the followng orentaton a=[ ], t s necessary to calculate a posture of the upper extremty. The DH table s readly determned and presented n Table. 8

19 Table : DH Table θ d α a q 4 2 q q 3 Substtutng each row nto Eq. () yelds the followng ( 4 4) transformaton matrces 2 T = 3 T =!! cosq -sn q 4cosq sn cos sn " $ # " $ # q q 4 q, cosq snq -snq cosq cosq snq T = 2! cosq -snq 2cosq sn cos sn " $ # 2 2 q2 q2 2 q, Performng the multplcaton and obtanng the poston vector yelds 4 cos q X n ( q) = 4sn q + 2 cos( q + 2sn( q + q + q 2 2 ) + cos( q ) + sn( q + q + q q3) + q ) 3 (6) We shall also mpose ranges of moton on each jont as π 3 q π 3 ; =,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 ( 2 π ( π 3, π 3, q3); q3 [, π 3] x ( 3 3 π 3,] ( q, π 3, π 3); q [ π 3, π 3] 9

20 and x q, π 3, π 3); q [, 3]. ( π Substtutng the sngular sets nto Eq. (6) yelds equatons of curves shown n Fg. 6, whch s the reach envelope. y x Fg. 6 The reach envelope of the upper extremty restrcted to planar moton We now answer the two objectves: (a) To determne whether a pont s reachable, we study the exstence of the pont nsde the boundary to the reach envelope. Ths s a smple problem wth many wellestablshed algorthms. In ths case, the pont s wthn the boundary. To predct a posture at ths poston p wth the gven orentaton, we mplement the mnmum effort cost functon gven by: n fnal Feffort = Ê ( q - q ntal )2 = (7) 2

21 The computed jont angles for the fnal posture are q =. 78 π / π / π / 8 T. The posture s schematcally represented n Fg. 7. Fg. 7 The computed posture A 9-DOF arm wth two coupled jonts Consder now a much more realstc model of the shoulder, elbow, and wrst characterzed by 9 DOFs and shown n Fg. 8. We have modeled the shoulder by fve DOFs (two of whch are translatonal that characterze the coraco clavcular jont), the elbow by a DOF rotatonal jont, and the wrst by 3DOF. In realty, muscles typcally extend over more than one jont, therefore some of the jonts are coupled. Ths effect s especally pronounced n the shoulder between the rotatonal and translatonal moton. To model ths behavor, we propose a relaton that couples the translatonal jonts to the 2

22 revolute jonts. Ths couplng s observed when one rotates the shoulder, whch wll also result n nvoluntary translatonal moton. The problem s now reduced to a 7DOF system, where the moton of the frst and second jonts are modeled as a combnaton of the summaton of the thrd and fourth jonts as q = 6. ( q + q ) (8) 3 4 q = 5. ( q + q ) (9) Two objectves are to be accomplshed (a) Determne whether a pont s reachable and (b) If t s reachable, calculate a posture. The ranges of moton for ths arm are gven as follows: -5." ˆq ˆ5."; -5." ˆq ˆ5."; -π 2ˆq ˆπ 2; -π 8 ˆ ˆ2π q ; -π 2ˆq ˆπ 2; ˆq ˆ5π 6; -π 3ˆq ˆπ 3; -π 9ˆq ˆπ 9; and -π ˆq 9 ˆ q 4 q 2 y o q 3 q 5 q x o q 6 q 7 q 9 X(θ) q 8 Fg. 8 Model of the upper extremty 22

23 The 9 DOF reach envelope s then calculated where surface patches on the boundary are determned n closed form and shown n Fg. 9. X X Z Z Y Y Fg. 9 Reach envelope of the upper extremty 2-4 T The pont n queston has coordnates p = and a requred orentaton defned by a = T. Based on a mnmum effort cost functon, the posture of the arm s calculated as q 3 =. 7 8 / π, q 4 = / π, q 5 = / π, q 6 = / π, q 7 = 2. 8 / π, q 8 = / π, and q 9 = / π. Now consder the same model of the 9DOF arm but wth a dfferent drvng functon (cost functon). The functon used to predct a posture n ths case s dexterty defned by a new measure for dexterty as follows: Upon executng the algorthm, the maxmum dexterty functon s calculated as T D = GG q q = at the followng posture 23

24 q 3 = 5. π / 8, q 4 =. 48 π / 8, q 5 =. 6 π / 8, q 6 = 7. π / 8, q 7 =. 5 π / 8, q 8 =. 9 π / 8, and q 8 = -. 6 π / 8. Conclusons A broadly applcable formulaton and accompanyng expermental code for predctng realstc postures as a result of human moton has been presented and llustrated. Whle many models are avalable especally n the robotcs lterature that predct posture (elsewhere known as nverse knematcs), we beleve that realstc postures can only be obtaned from () Emprcal data and (2) Optmzaton through cost functons. The method presented n ths paper has been shown to yeld realstc posture because t s based on cost functons that characterze human performance measures and that are used as drvng functons n an teratve numercal optmzaton algorthm. The advantage of the method descrbed heren over other exstng methods can be seen by studyng the posture of the arm gven two ntal confguratons. For example, for a specfc target pont, f the arm was ntally at the person s wast, then the calculated posture should be dfferent than that calculated f the arm was ntally at the person s head. Ths s evdently not true for the other two methods (nverse knematcs and statstcally based methods). Moreover, we beleve that the method presented heren s the bass for addressng ssues pertanng to neural commands and the central nervous system, whch s the subject of our current efforts. The long term vson for ths research s to develop rgorous analyss tools to ad n the ergonomc desgn process. 24

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