Multi-objective Optimization for Upper Body Posture Prediction

Transcription

1 Mult-objectve Optmzaton for Upper Body Posture Predcton Jngzhou ang *, R. Tmothy Marler, HyungJoo Km, Jasbr S. Arora, and Karm Abdel-Malek ** Vrtual Solder Research Program, Center for Computer-Aded Desgn, The Unversty of Iowa, 111 Engneerng Research Faclty, Iowa Cty, IA The demand for realstc autonomous vrtual humans s ncreasng, wth potental applcaton to prototype desgn and analyss for a reducton n desgn cycle tme and cost. In addton, vrtual humans that functon ndependently, wthout nput from a user or a database of anmatons, provde a convenent tool for bomechancal studes. However, development of such avatars s lmted. In ths paper, we captalze on the advantages of optmzaton-based posture predcton for vrtual humans. We extend ths approach by ncorporatng mult-objectve optmzaton (MOO) n two capactes. Frst, the objectve sum and lexcographc approaches for MOO are used to develop new human performance measures that govern how an avatar moves. Each measure s based on a dfferent concept wth dfferent potental applcatons. Secondly, the objectve sum, the mn-max, and the global crteron methods are used as dfferent means to combne these performance measures. It s found that although usng MOO to combne the performance measures generally provdes reasonable results especally wth a target pont located behnd the avatar, there s no sgnfcant dfference between the results obtaned wth dfferent MOO methods. omenclature = generalzed coordnates (jont angles) = neutral poston x( ) = poston vector of the end-effector f( ) = vector of objectve functons f ( ) = objectve functon (human performance measure) F = aggregate objectve functon g ( ) = neualty constrants h ( ) = eualty constrants DOF = number of degrees of freedom w = vector of jont-dsplacement weghts γ = vector of dscomfort weghts λ = mn-max parameter * Research Engneer, Center for Computer Aded Desgn, Tel: , jyang@engneerng.uowa.edu. PhD Canddate, Department of Mechancal and Industral Engneerng. Post Doctoral Fellow, Center for Computer Aded Desgn. Professor, Department of Cvl and Envronmental Engneerng, AIAA Senor Member. ** Assocate Professor, Department of Bomedcal Engneerng. 1 Amercan Insttute of Aeronautcs and Astronautcs

2 I. Introducton HE development and use of vrtual humans (computer smulatons of humans) have recently ganed momentum Tfor engneerng product desgn and bomechancal studes. In terms of engneerng, t s well recognzed that computer-based smulatons and prototypes can save tme and money. However, actual human nteracton plays a key role n prototype development, and although vrtual prototypes have been used extensvely n the desgn and manufacturng processes, vrtual humans that nteract wth such prototypes have not yet been leveraged thoroughly. Thus, avatars that act effectvely as real humans can provde a key element n engneerng desgn and analyss. In terms of bomechancs, understandng posture and musculoskeletal moton s crtcal n the analyss of jonts and extremtes. To ths end, vrtual humans can provde a convenent and relable means for studyng movement and musculoskeletal functons. Most of the methods for modelng human posture and moton take one of the followng approaches. One may use classcal anmaton based on ether expermental data or user-manpulaton of avatars. However, ths approach lacks autonomy; a separate pcture or anmaton must be created for each posture and/or moton. In fact, we dstngush between anmaton and smulaton, suggestng that vrtual humans should be smulatons that nherently and ndependently demonstrate movements, reactons, and decson-makng capabltes, whch appear natural and approprate gven any general scenaro. Another approach for modelng humans nvolves nverse knematcs, whch entals solvng a system of euatons to determne parameters for the human model that defne poston and moton. However, ths approach can be prohbtvely slow, especally wth a larger number of degrees of freedom (DOF). Recently, an optmzaton-based approach has surfaced. Ths approach entals optmzng objectve functons that represent human performance measures, such as jont dsplacement, dscomfort, etc. These performance measures govern how the avatar moves. Ths approach ensures autonomous movement regardless of the scenaro. In addton, t can be mplemented n real tme. Despte ts advantages, development of the optmzaton-based approach s n ts nfancy, used prmarly wth robotcs. Conseuently, t has only been appled to systems wth relatvely few degrees of freedom, although t can provde computatonally effcent models of more complex potentally redundant systems such as human skeletons. In addton, the advantages of mult-objectve optmzaton (MOO) have not yet been exploted. Thus, we use MOO n two capactes: 1) the development of new human performance measures and 2) the combnaton of dfferent measures to model more accurately, how humans move. In order to aggregate multple measures, we propose a MOO-based approach for posture predcton. Integratng the dscplnes of mult-objectve optmzaton and human modelng yelds exctng results for both felds. A. Lterature Revew Although there s ample lterature concernng moton predcton for humans and robots, there s lmted materal concernng human posture predcton. Most currently avalable methods are lmted n terms of the autonomy they afford the avatar and n terms of the complexty of the human models that are used. In addton, the use of optmzaton-based posture predcton s also lmted, and the use of mult-objectve optmzaton (for the development of human performance measures or for posture predcton) has not yet been addressed. Most classcal anmaton nvolves emprcal-statstcal modelng usng anthropometrcal data. These data are collected ether from thousands of experments wth human subjects, or from smulatons wth three-dmensonal computer-aded human-modelng software (Porter et al 1990; Das and Sngupta 1995). The data are then analyzed statstcally n order to form predctve posture models. These models have been mplemented n smulaton software along wth varous methods for selectng the most probable posture gven a specfc scenaro (Beck and Chaffn 1992; hang and Chaffn 1996; Das and Behara 1998; Faraway et al 1999). Although ths approach s based on actual human data and thus need not be verfed n terms of realsm, t nvolves a tme-consumng data collecton process often reurng thousands of human subjects. The nverse knematcs approach to posture predcton, whch uses bomechancs and knematcs as predctve tools, has receved substantal attenton. Wth ths approach, the poston of a lmb s modeled mathematcally wth the goal of formulatng a set of euatons that provde the jont varables (Jung et al 1992; 1995; Kee et al 1994; Jung and Choe 1996; Wang 1999; Tolan et al 2000). However, as suggested earler, ths approach s restrcted to relatvely smple models. In the feld of robotcs, predctng a posture s not a consderaton. However, t s mportant to predct the path of a pont on a robotc arm. To ths end, consderable research has been conducted wth optmzaton-based path plannng of robot manpulators wth mnmum travelng-tme as a cost functon to predct the path of a manpulator (Ln et al 1983, Shn and Mckay 1986, Chen 1991). Some authors consder mult-objectve optmzaton but only n so much as they smply add two dfferent objectve functons. Weghts may be used n a weghted sum, but the 2 Amercan Insttute of Aeronautcs and Astronautcs

3 weghts serve only as scalng factors; there s no ndcaton of preference between the two objectves. MOO s not thoroughly exploted n terms of potental methods or n terms of theoretcal analyss of the results. For nstance, hao and Ba (1999) propose a optmzaton approach wth load or torue as an objectve, and they use multobjectve optmzaton to combne these objectves. Saramago and Steffen J. (1998, 2000) present a mult-objectve optmzaton soluton to the problem of movng a robot manpulator. They optmze the travelng tme and mnmze the mechancal energy of the actuators, consderng dynamcs and collson avodance of movng obstacles. Saramago and Ceccarell (2002) proposed a smlar mult-objectve optmzaton approach wth payload constrants. Because much of the focus for moton predcton has been on robotcs, lttle work has been conducted wth the development of objectve functons that are talored to human posture. Abdel-Malek et al (2001b, 2001c) study sngle-objectve optmzaton-based human posture predcton usng genetc algorthms, and M et al (2001) extend ths work to real-tme smulaton. However, no other work concernng optmzaton-based human posture predcton has been conducted, and no work concernng multobjectve optmzaton wth posture predcton s avalable. B. Overvew of the Paper Based on the above-mentoned defcences n the current state of the art, we pursue the followng objectves n ths paper: 1) Incorporate MOO n optmzaton-based posture predcton algorthms; 2) Usng MOO, develop new human performance measures that more accurately smulate how humans move; 3) Compare the performance of basc MOO methods; 4) Evaluate the concept of basng human moton on multple performance measures smultaneously. Before provdng a bref revew of key concepts assocated wth MOO, we present an overvew of the human modelng method that s necessary for the analyss. ext, the general optmzaton formulaton used for posture predcton s dscussed, varous human performance measure are explaned, and the MOO methods for combnng these measures are summarzed. Vsual and numercal results are shown, frst by usng each human performance measure ndependently, and then by combnng them wth MOO. These methods are compared n terms of ther computatonal performance and n terms of the realsm of the conseuent postures. II. Overvew of Vrtual Human Model Essentally, the human body s modeled as a knematc system, a seres of lnks connected by revolute jonts that represent musculoskeletal jonts such as the wrst, elbow, or shoulder. Our approach entals fndng the rotatonal dsplacement of these jonts necessary to optmze one or more objectve functons that represent human performance measures. In ths secton, the fundamentals of the knematc model are presented. In order to represent gross moton, a basc model for the upper body s developed that ncorporates the torso, spne, shoulders, and arms. The rotaton of each jont n the human body s represented by a generalzed coordnate, as shown for the seres of lnks n Fgure 1. Global coordnate system Local coordnate system x ( ) 1 Target pont n End-effector Fgure 1. General Knematc Model. 3 Amercan Insttute of Aeronautcs and Astronautcs

4 Each generalzed coordnate s assocated wth a local coordnate system. n R s the vector of n generalzed 3 coordnates n an n-dof model and represents a specfc posture. x( ) R s the poston vector n Cartesan space that descrbes the locaton of the end-effector as a functon of the generalzed coordnates, wth respect to the global coordnate system. Thus, the poston of the end-effector s defned as follows: ( ) ( ) ( ) x x( ) = y z (1) An end-effector s the end-pont n a seres of lnks such as an arm. We are concerned wth fndng the values of the generalzed coordnates when the poston of the end-effector s constraned wth respect to the global coordnate system. The Denavt-Hartenberg (DH) method (Denavt and Hartenberg, 1955) s used to determne x for a gven. The DH-method provdes a matrx notaton and approach for relatng the poston of a pont n one coordnate system to another coordnate system, by usng a unue transformaton matrx. Such an approach s useful wth knematc systems n whch a seres of components are connected by jonts. A local coordnate system and a local transformaton matrx are assocated wth each jont, descrbng ts confguraton wth respect to the prevous jont and coordnate system. Multple transformaton matrces can be combned to determne the poston of any pont on the knematc system wth respect to any local coordnate system or wth respect to a global coordnate system, based on all of the jont dsplacements. Ths approach has been used for modelng human bomechancs, knematcs, and dynamcs (Jung et al, 1995; Abdel-Malek et al, 2001a; ang et al, 2003). In ths report, the DH-method s used to descrbe a seres of lnks that lead from the wast of a human model to the upper extremtes. Detals concernng the method are gven by Marler (2004). Wth ths study, a 21-DOF model for the human torso and rght arm s used, as shown n Fgure 2. Wrst Elbow Shoulder Clavcle Spne Global Reference Frame Fgure DOF Knematc Model. The dstances between the jonts are represented by L, and the axes of rotaton are ndcated by z where z corresponds to through 12 represent the spne. 13 through 17 represent the shoulder and clavcle. 18 through 21 represent the rght arm. The end-effector s the tp of the ndex fnger, and the set of ponts that can be contacted by the end-effector s called the reach envelope. The detals of ths model are provded by Farrell and Marler (2004). Although ths study focuses on upper-body posture predcton, Fgure 2 represents just one part of a complete vrtual human model called Santos, whch s llustrated n Fgure 3. III. Overvew of Mult-Objectve Optmzaton In ths secton, the fundamentals of mult-objectve optmzaton are revewed. The general MOO problem s posed as follows: 4 Amercan Insttute of Aeronautcs and Astronautcs

5 Fnd: DOF R (2) to mnmze: f( ) f1( ) f2( ) f k ( ) subject to: g ( ) 0 = 1,2,, m h ( ) = 0 j = 1,2,, e j = where k s the number of objectve functons, m s the number of neualty constrants, and e s the number of DOF k eualty constrants. E s a vector of desgn varables. f( ) E s a vector of objectve functons DOF 1 f ( ) : E E. The feasble desgn space s defned as Π = { g j ( ) 0, j = 1, 2,..., m ; ( ) = = } k and h 0, 1,2,..., e = f R such that f = f( ), Π. Ponts n the feasble crteron space that can be determned usng a specfc method are called attanable. The pont n the crteron space where all of the objectves have a mnmum value smultaneously s called the utopa pont f. In general, f s unattanable; t rarely s possble to fully optmze each ndvdual objectve functon ndependently and smultaneously, whether the problem s constraned or not.. The feasble crteron space s defned as { } T Global Reference Frame Fgure 3. Santos, a Complete Vrtual Human. The dea of a soluton for (2), where multple objectves may conflct wth one another (e.g., what mnmzes one functon may ncrease another), can be unclear. Conseuently, the dea of Pareto optmalty s used to descrbe solutons for MOO problems. A soluton pont s Pareto optmal f t s not possble to move from that pont and mprove at least one objectve functon wthout detrment to any other objectve functon. Based on ths defnton, the mnmum of a sngle objectve functon s Pareto optmal f t s unue. Alternatvely, a pont s weakly Pareto optmal f t s not possble to move from that pont and mprove all objectve functons smultaneously. Typcally, there are nfntely many Pareto optmal solutons for a MOO problem. Thus, t s often necessary to ncorporate user preferences n order to determne or select a sngle sutable soluton. Wth methods that ncorporate a pror artculaton of preferences, the user ndcates the relatve mportance of the objectve functons or desred goals before runnng the optmzaton algorthm. Dfferent methods allow one to artculate preferences n dfferent 5 Amercan Insttute of Aeronautcs and Astronautcs

6 ways, but the most common approach s to have the user set parameters such as weghts. Alternatvely, preferences may not be avalable, the decson-maker may not know or cannot concretely defne what he/she prefers, or the problem may be purely mathematcal. Thus, some methods nvolve no artculaton of preferences. Although the exact soluton pont provded by such methods s somewhat arbtrary, these types of methods can provde useful benchmark results for mult-objectve analyss. IV. Mult-objectve Posture Predcton In ths secton, we formulate the posture predcton optmzaton problem. In dong so, MOO s used n two capactes. Frst, t s used to develop new human performance measures. It s then used to combne these measures, whch serve as multple objectve functons, n a fnal optmzaton problem. Detals concernng the methods dscussed n ths secton are gven by Marler and Arora (2004), whereas bref overvews of the methods are gven throughout ths secton. A. Desgn Varables and Constrants As suggested earler, the desgn varables for the fnal MOO problem are the generalzed coordnates, whch ndcate the rotaton of the jonts n unts of degrees. The vector represents the conseuent posture. The frst constrant, called the dstance constrant, reures the end-effector to contact the target pont. In U addton, each generalzed coordnate s constraned to le wthn predetermned lmts. represents the upper lmt L for, and represents the lower lmt. These lmts ensure that the vrtual human does not assume a poston that s completely unrealstc gven the nature of actual human jonts. B. Human Performance Measures 1. Jont Dsplacement The frst performance measure represents jont dsplacement; t s based on the weghted sum method for multobjectve optmzaton. Ths method always provdes a Pareto optmal soluton and entals mnmzng the followng aggregate objectve functon: k = 1 ( ) F = w f (3) 2 where w are postve weghts used for a pror artculaton of preferences. In general, they ndcate the relatve mportance of the objectve functons. The value of each weght s only sgnfcant relatve to the other weghts and relatve to the value of ts correspondng objectve functon. The detals of the jont dsplacement functon are explaned as follows. Let be the neutral poston of a jont measured from the startng home confguraton. The home confguraton s characterzed by = 0, and the neutral poston represents a relatvely comfortable poston. Then, conceptually, the dsplacement from the neutral poston for a partcular jont s gven by. However, to avod numercal dffcultes and nondfferentablty, the terms ( ) are used. Each of these 21 terms (one for each degree of freedom) can be treated as an ndvdual objectve functon, whch are combned usng a weghted sum. Because some jonts tend to be actvated more than others, the scalar weghts w are ntroduced to stress the mportance of partcular jonts. The cumulatve jont dsplacement s modeled usng the followng weghted sum: Jont dsplacement The values for the weghts are gven n Table 1. DOF ( ) = ( ) = 1 2 f w (4) 6 Amercan Insttute of Aeronautcs and Astronautcs

7 Table 1: Jont Weghts for Jont-Dsplacement. Jont Varable Jont Weght Comments 1, 4, 7, 10,,, ,,, , 15, Used wth both postve and negatve values of When > 0 When < 0 5 Used wth both postve and negatve values of 75 Used wth both postve and negatve values of 1 Used wth both postve and negatve values of 17 18, 19, 20, When > 0 When < 0 1 Used wth both postve and negatve values of For ths model, the neutral poston s chosen based on observaton of the sknned model n Fgure 3 rather than a skeletal model lke the one shown n Fgure 2. The resultng vector s defned as = 0; = 1,...,12,19,20 (5) =- 15.0, = 20.0, = 100.0, =- 10.0, =- 80.0, =- 35.0, = Ths generally represents a posture wth the arms straght down, parallel to the torso. Wth the jont dsplacement performance measure, the avatar s poston gravtates towards the neutral poston. 2. Delta-Potental-Energy In ths secton, we dscuss a potental-energy functon, whch s ndrectly based on the weghted sum method for MOO. However, n ths case, the weghts are based on the mass of dfferent segments of the body. Wth the prevous functon, the weghts are set based on ntuton and expermentaton, and although the postures obtaned by mnmzng jont dsplacement are realstc, there are other ways to assgn relatve mportance to the components of the human performance measure. The dea of potental energy provdes one such alternatve. We represent the prmary segments of the upper body wth sx lumped masses: three for the lower, mddle, and upper torso, respectvely; one for the upper arm; one for the forearm; and one for the hand. We then determne the potental energy for each mass. The heghts of these masses, rather than the jont dsplacements for the generalzed coordnates, provde the components of the human performance measure. Mathematcally, the weght (force of gravty) of a segment of the upper body provdes a multpler for movement of that segment n the vertcal drecton. The heght of each segment s a functon of generalzed coordnates, so, n a sense, the weghts of the lumped masses replace the scalar multplers, w, whch are used n the jont dsplacement functon. If the potental energy functon were used drectly, there would always be a tendency to bend over, thus reducng potental energy. Conseuently, we actually mnmze the change n potental energy. Each lnk n a segmented seral chan, as depcted n Fgure 4 (e.g., the forearm), has a specfed center of mass. 7 Amercan Insttute of Aeronautcs and Astronautcs

8 Local coordnate system- h r Ar 0 ' ero potental energy plane 0 Ar y 0 x 0 z 0 Fgure 4. Illustraton of the Potental Energy of the Upper Body. The vector from the orgn of a lnk s local coordnate system to ts center of mass s gven by r, where the subscrpt ndcates the relevant local coordnate system. In order to determne the poston and orentaton of any part of the body, we use the transformaton matrces ( 1) A, whch are 4 4 matrces that relate local coordnate system- to local coordnate system- 1. Conseuently, r s actually an augmented 4 1 vector wth respect to local coordnate system, rather than a 3 1 vector typcally used wth Cartesan space. g = [ 0 g 0 0] T s the augmented gravty vector. When the human upper body moves from one confguraton to another, there are two ' potental energes, P whch s assocated wth the ntal confguraton and P whch s assocated wth the current confguraton. Therefore, for the frst body part n the chan (the lower torso), the potental energes are ' T 0 ' T 0 ' T 0 ' 1 ' P1 = mg Ar and P = mg Ar The potental energes for the second body part are P2 = m g A 2 1 A2r 2 and T 0 1 P = m g A A r. The potental energes for the th ' T 0 ' 1 ' T 0 1 body part are P = mg A1 Ar and P = mg A A r. In 1 Fgure 4, h s the y-component of the vector 0 ' 1 1 ' 0 A Ar 1 1 A Ar. The fnal objectve functon, whch s mnmzed, s defned as follows: κ Delta potental energy = 1 ' ( ) = ( ) 2 ote that (6) can be wrtten n the form of a weghted sum as follows: where ( mg ) 2 represent the weghts and ( h ) 2 f P P (6) κ 2 2 Delta potental energy ( ) = ( ) ( ) = 1 f mg h (7) act as the ndvdual objectve functons, κ = 6 s the number of lumped masses. In ths case, the ntal poston s the neutral poston descrbe n relaton to jont dsplacement. Wth ths performance measure, the avatar agan gravtates towards the neutral poston. However, horzontal moton of the lumped masses has no affect on the objectve functon. 3. Dscomfort The dscomfort human-performance-measure s based on the lexcographc method for MOO. A pror artculaton of preferences s used wth ths method, as t was wth the weghted sum, but preferences are artculated n a slghtly dfferent format. Rather than assgn weghts that ndcate relatve mportance, one smply prortzes the objectves. Then, one objectve at a tme s mnmzed n a seuence of separate optmzaton problems. After an objectve has been mnmzed, t s ncorporated as a constrant n the subseuent problems. The soluton to ths method s Pareto optmal, f t s unue. 8 Amercan Insttute of Aeronautcs and Astronautcs

9 The concept behnd ths new dscomfort measure s that groups of jonts are utlzed seuentally. That s, n an effort to reach a partcular target pont, one frst uses one s arm. Then, f necessary, one bends the torso. Fnally, f the target s stll out of reach, one may extend the clavcle jont. The lexcographc method for MOO s desgned to ncorporate ths type of preference structure. However, solvng a seuence of optmzaton problems can be tme consumng and mpractcal for real-tme applcatons such as human smulaton. The weghted sum method can be used to approxmate results of the lexcographc method f the weghts have nfntely dfferent orders of magntude (Mettnen, 1999; Romero, 2000). Ths results n the weghts shown n Table 2. Table 2: Jont Weghts for Dscomfort. Jont Varable Jont Weght Comments 1,, Used wth both postve and negatve values of 13, Used wth both postve and negatve values of,, 1 Used wth both postve and negatve values of Although weghts are used here, they do not need to be determned as ndcators of the relatve sgnfcance of ther respectve jonts; they are smply fxed mathematcal parameters. In addton, the exact values of the weghts are rrelevant; they smply have to have sgnfcantly dfferent orders of magntude. ote that some of the weghts n Table 1 (used wth jont dsplacement) are dscontnuous, and ths s because movement n varous drectons can result n dfferent degrees of acceptablty. These dscontnutes can lead to computatonal dffcultes. However, wth ths dscomfort objectve, such dscontnutes are avoded. The weghts n Table 2 are used n a functon that s based on (4) wth the neutral poston defned as shown n (5). However, pror to applyng the weghts, each term n (4) s normalzed as follows: norm U L = (8) norm Wth ths normalzaton scheme, each term ( ) 2 acts as an ndvdual objectve functon and has values between zero and one. Generally, ths approach works well but often results n postures wth jonts extended to ther lmts, and such postures can be uncomfortable. To rectfy ths problem, extra terms are added to the dscomfort functon such that the dscomfort ncreases sgnfcantly as jont values approach ther lmts. The fnal dscomfort functon s gven as follows: f = + G QU + G QL (9) 1 ( ) DOF norm γ ( ) Dscomfort G = 1 U ( ) 5.0 QU = 0.5Sn U L L ( ) 5.0 QL = 0.5Sn U L where G QU s a penalty term assocated wth jont values that approach ther upper lmts, and G QL s a penalty term assocated wth jont values that approach ther lower lmts. γ are the weghts defned n Table 2. U U L L U L vary between zero and Each term vares between zero and G, as ( ) ( ) and ( ) ( ) one. Fgure 5 llustrates the curve for the followng functon, whch represents the basc structure of the penalty terms: Amercan Insttute of Aeronautcs and Astronautcs

10 ( 0.5 ( ) 1) 100 Q = Sn r + + (10) Q r Fgure 5. Graph of Dscomfort Jont-Lmt Penalty Term. r represents ether ( U ) ( U L ) or ( L ) ( U L ). Thus, as Fgure 5 llustrates, the penalty term has a value of zero untl the jont value reaches the upper or lower 10% of ts range, where ether U U L or ( L ) ( U L ) 0.1. The curve for the penalty term s dfferentable, and ( ) ( ) 0.1 reaches ts maxmum of hgh functon-values. 6 G = 1 10 when x = 0. The fnal functon n (9) s multpled by 1 G to avod extremely C. Posture Predcton Formulaton Gven the above-mentoned desgn varables, constrants, and human performance measures, the optmum posture for the 21-DOF system shown n Fgure 2 s determned by solvng the followng MOO problem: DOF Fnd: R (11) to mnmze: Jont dsplacement, Delta-potental-energy, and Dscomfort dstance = x x ε subject to: ( ) end-effector target pont L U ; = 1,2,, DOF where ε s a small number that approxmates zero. All optmzaton problems are solved usng the software SOPT (Gll et al, 2002), whch uses seuental uadratc programmng (Arora, 2004). After solvng (11) by usng each performance measure ndependently (wth sngle-objectve optmzaton), we use three dfferent approaches to MOO. Each of these methods nvolves no artculaton of preferences. Ths s because the ntent here s smply to nvestgate the advantages and/or dsadvantages of ncorporatng multple human performance measures smultaneously. Whle there are many dfferent approaches for MOO wth no artculaton of preferences (Marler and Arora, 2004), we consder fundamental methods that tend to be less demandng computatonally and lend themselves well to real-tme smulatons. Frst, we consder the objectve sum method, whch smply nvolves usng (3) wth all of the weghts set to one. Then, we use the mn-max method wth whch one mnmzes the followng functon: F = max f 1 k (12) Because (12) can nvolve potental dscontnutes, t s reformulated wth an addtonal desgn varable λ and addtonal constrants, as follows: 10 Amercan Insttute of Aeronautcs and Astronautcs

11 Fnd: λ, (13) to mnmze: λ subject to: f ( ) λ 0; = 1, 2, 3 We refer to the addtonal constrants as functon-constrants. Although the mn-max method may yeld non-pareto optmal solutons n some cases, t always provdes weakly Pareto-optmal solutons. Fnally, the global crteron method s used, wth whch the followng aggregate functon s mnmzed: 1 p k p ( f ) = 1 (14) F = where p s a postve number. In ths case, we use p = 2. p generally ndcates the amount of emphass that s placed on mnmzng the objectve functon wth the hghest value. When p = 1, (14) reduces to an objectve sum, and when p =, (14) reduces to the mn-max method. Ths approach always yelds Pareto-optmal solutons. V. Sngle-objectve Optmzaton Results In ths secton, numercal and vsual results are presented wth each performance measure used ndependently. These results correspond to Santos s posture when he touches the two target ponts shown n Fgure 6. Target 1 Target 2 Fgure 6. Target Ponts. The lghter colored sphere s target 1, and the darker sphere s target 2. Target 1 s located at ( 38,39,34), and target 2 s located at ( 36, 6, 26). The pont ( 0,0,0 ) s located at Santos s hp. In antcpaton of usng mult-objectve optmzaton, the objectve functons are normalzed such that they all have values between zero and one. In ths way, no sngle objectve functon domnates the aggregate functons used for MOO. The normalzed objectves are also used for sngle-objectve optmzaton, so the sngle-objectve results are comparable to those obtaned wth MOO. The feasble space for the problem n (11) s varable, dependng on where the target pont s located. Conseuently, the absolute maxmum and mnmum, consderng all possble target 11 Amercan Insttute of Aeronautcs and Astronautcs

12 ponts, are used for normalzaton. Each objectve functon has an absolute mnmum of zero, acheved when Santos s postured n the neutral poston. Therefore, each objectve functon s normalzed smply by dvdng by ts correspondng maxmum. The maxmum values for jont dsplacement, dscomfort, and delta-potental-energy are , , and , respectvely. The values for the normalzed objectve functons are gven for target 1 and target 2, n Tables 3 and 4, respectvely. Table 3: Objectve Functon Values for Target 1 Jont Dsplacement Values Dscomfort Values Delta-Potental- Energy Values Mnmzed Dsplacement Mnmze Dscomfort Mnmze Delta-Potental-Energy Table 4: Objectve Functon Values for Target 2 Jont Dsplacement Values Dscomfort Values Delta-Potental- Energy Values Mnmzed Dsplacement Mnmze Dscomfort Mnmze Delta-Potental-Energy Each column of these tables represents values for a partcular objectve functon. Each row represents the desgn pont * when a partcular objectve s mnmzed. For nstance, the frst value n the frst column represents the mnmum value for jont dsplacement. The second value n the frst column represents the value of the dsplacement functon evaluated at the pont that mnmzes dscomfort. The values n Tables 3 and 4 are rather small. Ths s a conseuence of the target ponts that are used n ths study; other target ponts result n hgher objectve-functon values. ote that the absolute values for the objectve functons are not necessarly sgnfcant n terms of uantfyng the concepts that each performance measure represents. Rather, we are concerned wth the change n objectve-functon values as dfferent postures are assumed. For the gven target ponts, none of the optmum ponts * are domnated, whch means there s no row for whch all of the values n that row are greater than the correspondng values n another row. These are conflctng objectves n that what reduces one functon, ncreases at least one other functon. Dscomfort has partcularly hgh values when delta-potental-energy s mnmzed. Ths s because, as we wll show, the energy performance measure results n substantal torso movement, whereas the dscomfort functon ncorporates the dea that such torso movement tends to be more uncomfortable than arm movement. The postures determned when each of the objectves s mnmzed, are shown n Fgures 7 through 9. Clearly, usng dfferent human performance measures as objectves provdes sgnfcantly dfferent results. In fact, although all of these performance measures result n postures that tend to gravtate towards the neutral poston, each has ts own set of advantages and potental applcatons. The jont dsplacement functon provdes a fundamental objectve that yelds reasonable results. It provdes benchmark postures that are acceptable vsually. However, t often results n postures wth the arm relatvely close to the torso. In addton, there can be slghtly more movement n the spne than one mght antcpate. The dscomfort functon corrects these ssues. As dscussed earler, wth the dscomfort functon, the spne only bends f necessary, and ths results n postures that are more realstc especally when the target ponts are n front of the avatar. In essence, the dscomfort functon provdes an accurate approxmaton of the lexcographc approach to mult-objectve optmzaton. It also tends to avod postures where the elbow hugs the torso. Ths tendency results from the penalty assocated wth jonts that are near ther lmts, and t s noted n Fgures 7 and 8 when comparng the postures wth respect to target 1. When the dscomfort functon s used wth target 2, t may appear as f the resultng posture s less natural than the posture acheved wth the dsplacement functon. However, agan, ths has to do wth how jont lmts are ncorporated. Approachng the lmts of a jont can result n dscomfort that s not necessarly vsualzed. 12 Amercan Insttute of Aeronautcs and Astronautcs

13 Target 1 Target 2 Fgure 7. Postures when Jont Dsplacement s Mnmzed. Target 1 Target 2 Fgure 8. Postures when Dscomfort s Mnmzed. 13 Amercan Insttute of Aeronautcs and Astronautcs

14 Target 1 Target 2 Fgure 9. Postures when Delta-Potental-Energy s Mnmzed. Thus, the posture determned wth the dsplacement functon, wth regards to target 2, can actually be more uncomfortable than the posture predcted wth the dscomfort functon. Ths dscrepancy s especally true when the avatar s reured to reach behnd tself or across the torso. As suggested earler, potental energy does not change wth rotaton n the torso. Thus, usng the delta-potentalenergy performance measure ndependently tends to result n postures wth excessve torso rotaton. In addton, t can result n excessve bendng n the wrst. Recall that the orgnal hypothess was that the mass component n potental energy would provde a natural weghtng factor for the dfferent jont values, thus allevatng the need for somewhat ad hoc weghts n the jont dsplacement functon. Although the energy functon does not provde a replacement for the dsplacement functon, t can provde useful results when coupled wth other performance measures, as we wll demonstrate n the next secton. VI. Mult-objectve Optmzaton Results The characterstcs of the dfferent performance measures are combned usng MOO. As suggested earler, we study three MOO methods: the objectve sum method n (3), the mn-max method n (13), and the global crteron method n (14). The objectve-functon values when each of these methods s used are shown n Tables 5 and 6. Table 5: Objectve Functon Values for Target 1 Jont Dsplacement Values Dscomfort Values Delta-Potental- Energy Values Objectve Sum Method Global Crteron Method Mn-max Method Amercan Insttute of Aeronautcs and Astronautcs

15 Table 6: Objectve Functon Values for Target 2 Jont Dsplacement Values Dscomfort Values Delta-Potental- Energy Values Objectve Sum Method Global Crteron Method Mn-max Method As wth the sngle-objectve results, none of the MOO soluton ponts s domnated. The results for the objectve sum and the global crteron are smlar, whch s common wth many MOO problems. When the mn-max method s used, however, all of the objectve functons have the same value. Ths s because all of the functon constrants are actve at the soluton pont. In general, the mn-max method prevents any sngle objectve functon from becomng sgnfcantly larger than the other functons. The postures correspondng to Tables 5 and 6 are shown n Fgures 10 through 12. Compared to the postures wth target 1 provded wth the objectve sum method and the global crteron method, the results wth the mn-max method ndcate a slght ncrease n torso bendng. In turn, ths results n an ncrease n the dscomfort, as shown n Table 5. Wth target 2, the mn-max method results n less bendng of the wrst. Wth the objectve sum method and the global crteron method, excessve bendng n the wrst s a result of the contrbuton of the delta-potental-energy functon. Usng MOO (as apposed to sngle-objectve optmzaton) clearly makes a dfference n the fnal postures. It acts to balance the unue results of the ndependent performance measures. However, the fnal posture s not partcularly senstve to the MOO method that s used. The computatonal performance, n terms of CPU tme and the number of functon calls n the optmzaton algorthm, was smlar for each of these methods. Target 1 Target 2 Fgure 10. Postures wth Objectve Sum Method. 15 Amercan Insttute of Aeronautcs and Astronautcs

16 Target 1 Target 2 Fgure 11. Postures wth Global Crteron Method. Target 1 Target 2 Fgure 12. Postures wth Mn-max Method. 16 Amercan Insttute of Aeronautcs and Astronautcs

17 VII. Concluson A general mathematcal formulaton for predctng human postures has been presented, demonstrated, and augmented wth the use of MOO. MOO has been used to develop new human performance measures and to aggregate these measures n an optmzaton-based posture predcton problem. The vrtual human, Santos, has been used to evaluate dfferent performance measures and to test the applcablty of MOO to posture predcton. Each performance measure s based on a unue premse, and each s most applcable to subtly dfferent scenaros. Whch performance measure s most approprate can depend on where a partcular task s beng completed relatve to the avatar. Jont dsplacement provdes a relable standard. The dscomfort functon takes nto consderaton the dscomfort of havng to move one s torso and/or clavcle, and the dscomfort assocated wth operatng at the lmts of one s range of moton. It provdes the most realstc posture when target 1 s used. Delta-potental-energy ncorporates dffculty assocated wth supportng the weght of dfferent body parts. However, t allows for excessve torso rotaton, and t s most approprately used n conjuncton wth other performance measures. Dfferent MOO methods have been compared for use wth optmzaton-based posture predcton, and although the dfferences n results obtaned wth dfferent MOO methods are subtle, MOO n general provdes consstently reasonable postures. It s partcularly well suted for target ponts located behnd the avatar. In fact, the most realstc posture wth target 2 s provded wth the mn-max method for MOO. In ths study, we presented results for only two target ponts, although other targets have been tested. In general, targets reurng sgnfcant extenson result n smlar postures regardless of the performance measure that s used. Ths s because the prmary dffculty wth such problems s determnng a feasble soluton (one n whch the avatar actually contacts the target pont). Conseuently, there are fewer feasble solutons n terms of potental postures. Targets behnd the avatar depend on vson, as well as dsplacement, dscomfort, or energy. One typcally moves n order to see the target as well as to touch t. Ths s why one may expect substantal twst n the torso for targets behnd the avatar, although such movement may actually be uncomfortable, as suggested n Fgure 8. Thus, t s necessary to develop a performance measure that consders ths, and such work s currently beng pursued. Postures n general depend heavly on the range of moton for each jont, and such data vares wdely from person to person. We have provded a formulaton that ncorporates these jont lmts. In addton, the new dscomfort objectve ncorporates the dscomfort assocated exercsng one s jonts near ther lmts. Acknowledgements Ths research was funded by the US Army TACOM project: Dgtal Humans and Vrtual Realty for Future Combat Systems (FCS). References Abdel-Malek, K., ang, J., Brand, R., and Tanbour, E., 2001a, "Towards Understandng the Workspace of The Upper Extremtes," SAE Transactons-Journal of Passenger Cars: Mechancal Systems, Vol. 110, Secton 6, pages Abdel-Malek, K., u, W., and Jaber, M., 2001b, Realstc Posture Predcton, 2001 SAE Dgtal Human Modelng and Smulaton. Abdel-Malek, K., u, W., M,., Tanbour, E., and Jaber, M., 2001c, Posture Predcton versus Inverse Knematcs, Proceedngs of the ASME Desgn Engneerng Techncal Conference. Arora, J. S., 2004, Introducton to Optmal Desgn, 2 nd ed., Elsever, San Dego, CA. Beck, D.J. and Chaffn, D.B., 1992, An evaluaton of nverse knematcs models for posture predcton, Computer Applcatons n Ergonomcs, Occupatonal Safety and Health, Elsever, Amsterdam, The etherlands, pp Das, B. and Behara, D.., 1998, Three-dmensonal workspace for ndustral workstatons, Human Factors, Vol. 40, o. 4, pp Das, B. and Sengupta, A.K., 1995, Computer-aded human modelng programs for workstaton desgn, Ergonomcs, Vol. 38, pp Denavt, J. and Hartenberg, R.S., 1955, A knematc notaton for lower-par mechansms based on matrces", Journal of Appled Mechancs, Vol. 77, pp Faraway, J.J., hang,.d. and Chaffn, D.B., 1999, Rectfyng postures reconstructed from jont angles to meet constrants, Journal of Bomechancs, Vol. 32, pp Farrell, K., and Marler, R.T., 2004, Optmzaton-Based Knematc Models for Human Posture, Unversty of Iowa, Vrtual Solder Research Program, Techncal Report umber VSR Gll, P., Murray, W., and Saunders, A, 2002, SOPT: An SQP Algorthm for Large-Scale Constraned Optmzaton, SIAM Journal of Optmzaton, Vol. 12, o. 4, pp Jung, E.S. and Choe, J., 1996, Human reach posture predcton based on psychophyscal dscomfort, Internatonal Journal of Industral Ergonomcs, Vol. 18, pp Amercan Insttute of Aeronautcs and Astronautcs

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