Bil. Cybern. 54, 125-132 (1986) Bilgical Cybernetics 9 Springer-Verlag 1986 Cnstraints fr Jint Angle Cntrl f the Human Arm H. Cruse Department f Bilgy, University f Bielefeld, D-48 Bielefeld 1, Federal Republic f Germany Abstract. The targeting mvements f a human arm were examined when restricted t a hrizntal plane. The three jints at shulder, elbw, and wrist are allwed t mve. Thus, the system is redundant and needs cnstraints. A mdel calculatin using a simple frm f cnstraint is fund t describe the experimental results: a cst functin is applied t each jint. The cnstraint cnsists in minimizing the sum f the csts f all three jints. The cst functins might be interpreted as t describing the energy cst necessary t mve the jint and/r represent a mechanism which avids singularities. Intrductin In bth technical and bilgical systems there exist manipulatrs which are able t reach a given bject within their wrkspace in rder t handle it fr a given task. Highly develped examples are rbt manipulatrs (see Luh 1983, fr a summary) and the human arm. In the first case it is interesting t develp laws fr the cntrl f the mvement f the manipulatr. In the secnd case the already existing cntrl mechanisms culd be f interest. This paper deals with the latter prblem. In rder t mve the tip f the manipulatr, the endeffectr, in a given wrkspace, the manipulatr cnsists f links which are cnnected by jints. The usual rbt manipulatr has exactly as many jints as needed t reach the bject with its endeffectr. In cntrast, bilgical manipulatrs as well as recent develpments f artificial manipulatrs may have additinal jints t bring the endeffectr t a given psitin. Such systems have mre degrees f freedm than necessary because fr a given psitin f the endeffectr an infinite number f pssibilities exist fr the cnfiguratin f the links and jints f the manipulatr. These systems pssess redundancy. T be able t chse a definite cmbinatin f jint variables (the jint angles in the case f revlute jints) a redundant systems needs additinal cnstraints. These cnstraints effectively decrease the degree f freedm f such a primarily "undercnstrained" system. In this paper the cntrl f mvement f the human arm is investigated. Three jints, shulder, elbw, and wrist, were allwed t mve, but nly in the hrizntal plane which cntained the shulder jint. The palm had t be held vertically. Thus, three revlute jints culd be used whse axes f rtatin were perpendicular t the hrizntal plane. The psitin f the tip f the endeffectr, in this case a pinter attached t the palm, is determined by tw cartesian crdinates in the hrizntal plane. Therefre, tw jints were sufficient t mve the endeffectr t any pint in the wrkspace. The existence f the third jint prduced an additinal degree f freedm and therefre made the system redundant. The bjective f this paper is t find the nature f the cntsraints in this 3-1inked human arm system. Methds The experiments were perfrmed with 4 subjects (ne female and three males) aged between 23 and 43 years with n physical abnrmalities r disabilities. A rigid pinter was munted at the palm f the right hand s that the effective length f the wrist (wrist jint t end f pinter) was abut the same as that f the frearm. The subject was seated in frnt f a table, high enugh t cnstraint the arm t mve in a hrizntal plane n the surface f the table. During the whle sessin the shulder was fixed by being pressed against a vertical metal ple munted n the edge f the table. Three starting pints (A, B, and C) and 2 target pints were
126 A+ 1 cm _t 4 +B 4 5 6 7 8 9 1 11 12\ 13 ~') 11. 9 ~ ~ " "~ *+C 2 3 In all subjects the mvements were perfrmed in the same tempral series, starting with the starting pint A t target pint 1, then frm B t 1 and frm C t 1. This prcedure was repeated fr all target pints cntinuing with target pint 2, 3 etc. up t pint 2. This was permissible because preliminary experiments shwed that the rder f the target pints des nt influence the results. This methd f measurement is very simple. Hwever, it nly allwed measurement f the static situatin at the end f the mvement, nt the dynamic part, i.e. the angles assumed during the mvement. This will be the subject f a fllwing paper. K.. ~,/ J shulder Fig. 1. Tp view f the experimental arrangement. The 2 target pints are placed n a hrizntal plane in frnt f the subject. The target pints 18, 19, and 2 appear twice, abve and belw the number. Usually the upper pints are used except fr ne subject, the results f which are shwn in Fig. 3. In this case the lwer pints are used. Tw extreme arm psitins are shwn fr target pint 6. These are given by the gemetrical limitatins. The three measured angles are marked marked n the table (Fig. 1) and culd be seen by the subject. The subject was asked t mve the tip f the pinter frm each f the three starting pints t each f the 2 target pints. Thus a ttal f 6 mvements were perfrmed. Befre the sessin the axes f rtatin f the three jints were marked by black dts glued t the skin. The psitins f these dts were bserved using a vide camera viewing the table frm abve and a mnitr whse screen was cvered by a transparent millimeter grid. After the pint reached the target pint, the crdinate values f the jints were read frm the mnitr and prcessed by cmputer in rder t calculate the angle values. The angles were defined as shwn fr tw examples in Fig. 1. The shulder angle was defined t be zer when the upper arm was psitined n the line cnnecting bth shulder jints. Mvement in the anterir directin is psitive. Due t the limited exactness f measurement and f slight mvement f the skin carrying the dts the angles culd be measured within a range f + 2 ~ Befre the experiment began the subject was allwed t perfrm a series f mvements between all target pints t becme familiar with the situatin. During the experiment the subject was asked t make the mvements in a "cmfrtable" way. In particular there was n special request cncerning the speed f the mvement. Results Gemetrical Cnstraints When cnsidering the cnstraints f a manipulatr, tw different srts f cnstraints have t be distinguished: a) cntrl cnstraints which, as described abve, are nly due t redundant manipulatr and decrease the degree f freedm and b) gemetrical cnstraints which are due t bth redundant and nnredundant manipulatrs. The gemetrical cnstraints are thse which determine the bundary f the wrkspace. They are given by the gemetrical prperties f the manipulatr, i.e. the lengths f its links and the extreme angle values which can be adpted by the different jints. Figure 1 as an example shws the tw extreme psitins f the arm f a given subject fr the target pint 6. This figure shws that fr a given target pint and given link lengths each angle can nly adpt values within a definite range. As these ranges depend upn the psitin f the target pint, Table 1 gives the ranges f the three angles fr each target pint fr ne subject as an example. It can be seen that in spite f the gemetrical cnstraints there is still a cnsiderable range f pssible mvement fr each jint. Hwever, in this subiect the target psitins 19 and 2 are bviusly at the bundary f the wrkspace: the ranges f pssible mvements are in the rder f the exactness f measurement. This means that in this case the experimental results are cmpletely determined by the gemetry f the system. Whereas Fig. 1 shws tw extreme psitins in the wrkspace (cartesian) crdinates and Table 1 gives the ranges f the pssible jint angles, Fig. 2 demnstrates fr three selected examples (target pints 4, 9, and 12) the gemetrically pssible jint values in jint space crdinates. The values f the three jint angles are presented here in tw diagrams, each using the elbw angle as abscissa and the wrist and shulder angle as rdinate, respectively. This diagram is used latern t shw the experimental data.
127 Table 1. The maximum ranges fr the three angles at wrist (abve), elbw (middle), and shulder (belw) derived as if nly gemetrical cnstraints existed. The values are given fr each target pint in degrees. They are taken frm the subject whse data are shwn in Fig. 4 92 8 72 1. 75 2. 62 3. 56 24 17 14 9 99 98 4. 66 5. 83 6. 79 7. 76 8. 78 22 34 31 88 26 99 73 44 29 35 1. 77 ll. 63 12. 41 13. 27 14. 35 38 44 33 19 19 65 3 19 5 7 16. 59 17. 24 18. 19 19. 4 2. 8 49 49 2 4 4 79 9. 6 16 64. 6 25 Experimental Data As will be shwn the results btained frm the fur subjects differed cnsiderably. Therefre it was nt pssible t calculate mean values fr all data and the results are presented fr each subject separately. Fr each f the 2 target pints three measurements were perfrmed as each target pint was apprached frm ne f the three starting pints A, B, and C. Mean values were calculated frm these three measurements and are presented fr the fur subjects in Figs. 3-6 in the jint space crdinates as described abve. There appears t be an bvius difference between the subjects. The strategies which are used by the fur subjects might represent fur different states f a cntinuum and the results are arranged t fllw an intuitive rder: When mving frm Figs. 3-6, the wrist angle seems t be held mre and mre rigid at apprximately -19 ~ unless the gemetrical cnditins cnstraint the system t mve als the wrist jint. One feature shuld be mentined which seems t be cmmn t all subjects althugh the verall results t- O 1L,..- 14.~ r,1 \\ 7 9 1 12 1 ~ 5 I I I l l I I I I I I I I 8 LO I i I i i 1 i i i,p4 g-. r 13 2O L C~ 2 16,,\ "",k4 r- -2-4 12 i I I I I I I I I I I I 1 Z,O 6 8 1 14 elbw angle [] Fig. 2. Three examples f jint angle ranges. The jint angles which were pssible if nly gemetrical cnstraints were existing are shwn fr the target pints 4, 9, and 12. The values are shwn in jint space crdinates: abscissa is elbw angle, rdinate is wrist angle in the upper diagram, and shulder angle in the lwer diagram -2-4 -6 18~. 14 i! i I i I i i I 4 6 8 1 elbw angle [~ Fig. 3. Results fr subject (a). The data are presented in jint space crdinates as explained fr Fig. 2. The mean values are given as clsed circles and are numbered accrding t the crrespnding target pint. The mean value f thse target pints which lie n hrizntal rws in Fig. 1 are cnnected by thick lines. The three small dts cnnected by thin lines with the mean values shw the individual values btained by appraching every target pint frm each f the three starting pints
128 IO', t'- E~ 14 6~11 7,,. 17 -"-"-"--'~ 14 t- O 2 14 ~,6 2 e3 5 ~"x -e 12! L,~,4"2O ~7 1 17 i I l I i I i qo -5 t- ul 8 4 2-2 -4-6 84 I!i ~ 1717 ~~9 2~ ~ / 'S-' 2 r 1 e~2 3 t/.,~ ~;_.~J" -6 -I 1,? 4 6 8 1 1 i! i I i I i i 4 6 8 1 elbw ngle [] Fig. 4. Results fr subject (b). The data are presented in jint space crdinates as explained fr Fig. 2. The mean values are given as clsed circles and are numbered accrding t the crrespnding target pint. The mean values f thse target pints which lie n hrizntal rws in Fig. l are cnnected by thick lines. The small dts cnnected by a thin line with the crrespnding mean value represent the results btained by the mdel calculatin are different. The data f Figs. 3, 5, and 6 shw a sudden increase f wrist angle fr target. In Fig. 4 this is true fr target 2 instead. As this subject had smaller link lengths target pint 2 in this subject might crrespnd t target pint f the ther subjects. Fr ne subject, nt nly the mean values but als the individual measurement values are shwn in Fig. 3. As the deviatins are abut the same in all fur subjects, fr the sake f clarity the individual values are mitted in the ther figures. The standard deviatins cllected frm all subjects were 1.2 ~ fr the shulder angle, 4-2.4 ~ fr the elbw angle, and 4-3.8 ~ fr the wrist angle. The results shw that the deviatins are much smaller than the pssible range due t the gemetrical cnstraints (see Fig. 2). In ne subject the experiment was repeated after three weeks but with a different rder (i.e. beginning with target pint 2) and n significantly different results were fund. elbw angle ["] Fig. 5. Results fr subject (c). The data are presented in jint space crdinates as explained fr Fig. 2. The mean values are given as clsed circles and are numbered accrding t the crrespnding target pint. The mean values f thse target pints which lie n hrizntal rws in Fig. 1 are cnnected by thick lines D these final jint angles depend n the directin by which the target pint is apprached? T clarify this questin the deviatins f all individual measurements frm their mean values were summarized fr all fur subjects but were separated by refering t their starting pint being either A, B r C. The frequencies f these deviatins are shwn in Fig. 7a-c, respectively. Only wrist angles were cnsidered here because they shwed the largest deviatins. The results shw that there is indeed a significant difference between the results f A and C (U-test, p 4.1%). Hwever, the abslute values f the deviatins were very small (A: median: - 2 ~ B: ~ C: -1~ Mdel Data The prblem addressed in this paper deals with the questin f whether it is pssible t find rules which frm cnstraints fr the redundant system under view. In the literature cncerning the cntrl f rbt manipulatrs several simple methds have been previusly prpsed (see Discussin). Hwever, it might well be the case that in the bilgical system relatively cmplicated strategies are used. The bvius dif-
129 22 e starting frm A 2 1&/~6 7 2 eell\ e'e~s/ 9 ILL-.5 9?. 9 1 _e C O 14 19~11 16 ~' 12 I i i i i I ) s i-i I I I I starting frm B 4 2 5 el \,23 ID "6 1 / 1 C D -2-4 -6 i 4O i 7 9 P 8 i~e~e i i i i i i i 6 8 1 elbw angle [~ Fig. 6. Results fr subject (d). The data are presented in jint space crdinates as explained fr Fig. 2. The mean values are given as clsed circles and are numbered accrding t the crrespnding target pint. The mean values f thse target pints which lie n hrizntal rws in Fig. 1 are cnnected by thick lines ferences between the subjects might fr example suggest that the individual apply very different strategies. Or it might be that ne individual uses different strategies fr target pints lying in different regins f the wrkspace. A simple methd t cnstraint a redundant rbt manipulatr has been prpsed by several authrs (fr summary see Hllerbach and Suh 1985; Liegis 1977); a similar methd was applied t the cntrl f the bdy psitin f a six-legged animal walking ver uneven surfaces (Cruse 1976). It is assumed that a cst functin is attached t each jint. Mving a jint away frm a minimum cst psitin in any directin increases the cst f this jint. The csts f all three jints are summed up t give the ttal cst value. The psitin with the minimum ttal cst value is be chsen. (Fr pssible interpretatins f such a cst functin see Discussin.) This mdel which intrduces a cnstraint by assuming a minimum ttal cst cnditin was applied t the data. Fr this purpse the frm f the three cst E c" 1 lq I -12-8 -4 I I IL-J~t i i starting frm C l 4 8 12 deviatin f actual wrist angle frm mean value [] Fig. 7. The deviatin f wrist angles frm the mean value. The diagram shws the difference between the individual value f the wrist angle and the mean value btained frm the three measurements when the target pint is apprached frm each f the three starting pints A, B, and C. (a)when starting frm A; (b) when starting frm B; (c)when starting frm C functins (shulder, elbw, wrist) were varied in rder t find ut whether a set f cst functins culd be fund which prduced arm psitins similar t thse fund in the experiments. The cst functins applied cnsisted f the sum f tw expnential functins with different sign fr the argument. Eventually the steepness was increased by adding an additinal expnential functin r a linear functin nt it. The mrphlgical jint limits were fitted t the functin by an additinal strng increase in the cst. The deviatin between the mdel and the experimental data f all subjects is shwn in Fig. 8 fr each angle. The standard deviatins are 2.6 ~ fr the shulder angle, 4-3.6 ~ fr the elbw angle, and 6.3 ~ fr the wrist angle. These results were btained by
13 "6.13 1 r-,,,i, [q~ shulder angle ~:,3 S.D = 2,6 c 1 E L ] elbw angle u ~=,5 5 S.D. = 3,6 r" 1 wrist angle ~~~,, ~ ~, ~:,g 5 S.D. : 6,3 q 1, p//p, -12-8 -4 /+ 8 12 28 difference between experimental data and mdel results [] Fig. 8. The differences between mdel and experimental data. The differences were cllected fr all fur subjects and shwn separately fr the three jint angles: (a) shulder, (b) elbw, and (c) wrist jint angle a~'~......./ 1- b) '~ ]~-... c) '~ 1/ d} '~ 1 "2~r~... / ]... ii... -6 4* *2 2 4 /4 6O 8 1 1!4 2 shulder ngle I~ elbw ngle ['I wtis~ cngle [~ Fig. 9a-d. The cst functins fr shulder, elbw, and wrist jint as assumed fr the mdel calculatin, am shw the fur sets f cst functins applied fr the fur subjects whse data are given in Figs. 3-6, respectively. The rdinates describe the cst value in relative units using the cst functins presented in Fig. 9. The deviatin between mdel and experimental data was nearly twice the standard deviatin f the experimental data which were 1.2 ~, 2.4 ~, and 3.8 ~, respectively. Fr visual inspectin f the deviatin between experi- mental results and mdel data, the latter are als shwn fr ne subject in Fig. 4. It might well be pssible that a mre sphisticated apprximatin prcess wuld prvide a better fit t the data: the shape f the cst functins were changed intuitively accrding t visual inspectin f the deviatins btained by the frmer calculatin. In particular, there seems t be ne effect which is nly qualitatively described by the mdel calculatin. This is the strng increase f the wrist angle at the target psitin, mentined earlier. Particularly fr the data shwn in Fig. 6 this psitin shws a large deviatin. Althugh it cannt be ruled ut that a further effrt t change the shape f the cst functins wuld imprve the mdel, anther pssibility has t be taken int accunt. It might well be that the cst functin f a jint is nt simply a functin f the angle f this jint, but may depend, at least t sme extent, als n the psitin f the ther jints. Such a mdel was indeed fund t imprve the results, but was nt cnsidered further because the minr imprvements did nt justify the increases in cmplexity. Such a change f the mdel will nt be intrduced until experiments in prgress prvide further infrmatin as t whether the cst functins are dependent n ther jint angles. Thus, at the present time the mdel given is cnsidered t prvide a sufficient first rder descriptin f the data. Discussin The cntrl f a redundant manipulatr requires mre cmputatinal csts cmpared t a nn-redundant manipulatr. Nevertheless, many bilgical and sme rbt manipulatrs are cnstructed t have redundancy. Redundancy prvides several advantages (see Yshikawa 1985; Hllerbach and Sub 1985; Salisbury and Abramwitz 1985): First, redundancy can help t avid bstacles. In a nn-redundant manipulatr an bstacle in the wrkspace might prevent the manipulatr frm mving the endeffectr alng a specific path. With additinal jints such bstacles culd be avided withut perturbing the path f the endeffectr. Secnd, redundancy can help t avid s called singularities. Singularities are psitins in the wrkspace where the pssible directins f mvements f the endeffectr are limited. This means that the degree f freedm is decreased fr such psitins. This is immediately bvius fr singularities, which appear at the uter bundary f the wrkspace. This uter bundary cannt be changed by additinal jints. Hwever, "inner" bundaries may als ccur, fr example, with a tw-link manipulatr whse links have different lengths. In this case the endeffectr cannt reach int a circle arund the central jint. This type f singularity r thse prduced by limited jint
131 ranges culd be avided by additinal jints. Secnd, additinal jints can increase the wrkspace. Third, redundant systems permit the ptimizatin f trques arund the different jints (Hllerbach 1985). If ne jint, e.g. the wrist jint can prduce nly a small trque cmpared t the ther jints, then an alignment f the wrist with the vectr f an external frce acting n the endeffectr wuld minimize the trque develped by the wrist jint. Only a redundant system wuld allw fulfillment f this additinal cnditin. Thus, redundant manipulatrs are in general superir t nn-redundant nes and it is f interest hw the mvement f a bilgical redundant manipulatr is cntrlled, i.e. what srt f cnstraints are develped. The experimental results presented in Fig. 7 shw that in ur paradigm, as a first rder apprximatin, the jint angles at the end f the mvement d nt depend n the directin f the mvement and are nly cntrlled by the psitin f the target pint. This result rules ut a pssible strategy described by Krein (1985), the s called "reach hierarchy" algrithm. Fllwing this prpsal t slve the prblem f redundancy, first nly the prximal link, i.e. the upper arm shuld mve as lng as the target gets in the wrkspace f the subsystem "frearm and hand," than the frearm mves until the target is in the wrkspace f the hand and finally the hand mves t the target. Using this strategy the final jint angle wuld depend n the psitin f the starting pint. The same was true when applying the strategy that the ttal amunt f energy spent fr the mvement shuld be minimized. Neither strategy was used t define the principal rules fr the cntrl f arm mvement in ur paradigm. The results are well described if we assume that a cst functin exists fr each jint. The additinal cnstraint necessary fr the cntrl f jint psitin in the redundant system is given by the cnditin that the cmbinatin f angles is chsen which prduces the minimum ttal cst. Tw pssible, nt necessarily exclusive, interpretatins f these cst functins will be mentined. First, the cst functin might describe the physilgical cst, i.e. the amunt f energy necessary t hld the jint in a given psitin. The sense f applying the minimum cst cnstraint wuld then be t minimize muscle energy. Secnd, the cst functin might be cmpletely independent f the actual physilgical csts but might represent a functin defined within the nervus system. In this case the sense f the applicatin f the cst functin might be t keep the jints away frm extreme angles near the jint limits. This avids the singularities described abve. Further experiments must be perfrmed t distinguish between these tw pssibilities. Finally, I wish t cnsider hw the three angle values which give the minimum ttal cst might be determined. One pssible methd crrespnds t that used in this mdel calculatin: a three dimensinal space is cnstructed using the three angles (shulder, elbw, wrist) as crdinates. Every pint in that space crrespnds t ne arm psitin. T each f these pints a ttal cst value is assigned. When a target pint is given, a path in this three-dimensinal space can be calculated which crrespnds t all arm psitins which allw the arm t reach the target pint (see Fig. 2). Then the minimum fr all cst values f the pints lying n this path has t be calculated in rder t btain the actual jint angle values. These three values might then be used as cmmands t the individual jints. This methd, intuitively, seems unlikely fr the human brain because much cmputatin wuld be necessary. Hwever, the calculatin f the three jint angle values need nt be repeated fr each mvement. As an alternative, ne culd als imagine that the angle values fr each target pint were precalculated (fr example during a learning prcess) and were stred in a lk-up table. Althugh this wuld simplify the calculatin prcess, it wuld require much strage space. Particularly, if ne remembers that the real arm can mve in a three-dimensinal space and has mre degrees f freedm than in ur paradigm (where rtatin abut the lng axis f the upper arm and f the fre arm as well as vertical mvements f the upper arm are excluded, nt cunting the hand jints). In the real situatin the required amunt f strage wuld be much higher. This is even mre imprtant when the wrist is prlnged by a pinter as in ur experiment. Each different length wuld require a ttally new set f lk-up tables. Thus, ne is inclined t pstulate ther mechanisms as being mre likely. Pssibly the real methd des nt calculate merely the final angle values. These values might simply result frm the determinatin f the varying angles adpted during the mvement, values which were cmpletely neglected in this paper. Future investigatins f the dynamical behaviur f the arm shuld prvide mre insight int the way in which the final angles are determined. Acknwledgements. This wrk was supprted by the Deutsche Frschungsgemeinschaft (Cr 58/5-1). It was perfrmed in the labratry f Prf. Dr. L. Stark, UC Berkeley, t whme I want t express my warm thanks fr his invitatin and crdial supprt. I als want t thank t Dr. D. Graham fr prf reading the English manuscript. References Cruse H (1976) The cntrl f bdy psitin in the stick insect (Carausius mrsus) when walking ver uneven surfaces. Bil Cybern 24:25-33
132 Hllerbach JM (1985) Optimum kinematic design fr a seven degree f freedm manipulatr. In: Hanafusa H, Inue H (eds) Rbtc research. The 2nd Int. Symp. f Rbtics Research, MIT Press, Cambridge, Mass. Lndn pp 2-222 Hllerbach JM, Suh KC (t985) Redundancy reslutin f manipulatrs thrugh trque ptimizatin. Prceed. f the 1985 IEEE Internatinal Cnference n Rbtics and Autmatin. IEEE Cmputer Sc Press, Silver Spring, Md Krein JU (1985) A gemetric investigatin f reach. MIT Press, Cambridge, Mass Liegis A (1977) Autmatic supervisry cntrl f the cnfiguratin and behaviur f multibdy mechanisms. IEEE Trans Syst Man Cybern 7:868-871 Luh JYS (1983) Cnventinal cntrller design fr industrial rbts - a tutrial. IEEE Trans Syst Man Cybern 13:298-3 Salisbury JK, Abramwitz JD (1985) Design and cntrl f a redundant mechanism fr small mtin. Prceed. f the 1985 IEEE Internatinal Cnference n Rbtics and Autmatin. IEEE Cmputer Sc Press, Silver Spring, Md Yshikawa T (1985) Manipulability and redundancy f rbtic mechanism. Prceed. f the 1985 IEEE Internatinal Cnference f Rbtics and Autmatin. IEEE Cmputer Sc Press, Silver Spring, Md Received: December 24, 1985 Prf. Dr. H. Cruse Fakult/it fiir Bilgie Universitfit Bielefeld Pstfach 864 D-48 Bielefeld 1 Federal Republic f Germany