( ) = : a torque vector composed of shoulder torque and elbow torque, corresponding to

Transcription

1 Supplementary Materal for Hwan EJ, Donchn O, Smth MA, Shamehr R (3 A Gan-Fel Encon of Lmb Poston an Velocty n the Internal Moel of Arm Dynamcs. PLOS Boloy, :9-. Learnn of ynamcs usn bass elements he nternal moel of the ronmental torque s: ˆ = w ( z ˆ : the expecte ronmental torque. z : a esre state of the lmb (consstn of lmb poston an velocty calculate wth mnmum jerk moel. : a bass element. w : a torque vector compose of shouler torque an torque, corresponn to each bass element. In trann, aaptaton s realze by a tral-to-tral upate of torque vectors follown raent escent: ( z w = η ~, ~ = ˆ ~ : the fference between the actual torque experence urn the movement an the expectaton of torque currently precte by the nternal moel. : the actual torque experence urn the movement. η : learnn rate he bass elements represent the arm s poston an velocty as a an-fel: ( q, = poston, ( q velocty, ( poston, velocty, ( q = k q + b ( = exp σ q : a esre jont poston, vector compose of shouler an jont splacement.

2 : a esre jont velocty, vector compose of shouler an jont velocty. : the preferre velocty. k : a vector compose of raents for shouler an jont splacement. k can be ecompose nto two components: mantue an rectonal unt vector as cosθ k =. k snθ b: ntercept (constant of a lnear functon. σ : wth of Gaussan functon All the parameters were fxe n the follown manner: he recton of postonal raents (θ were unformly strbute from to 35 wth a 5 ncrement. he preferre veloctes ( are unformly tle n D jont velocty space wth a.6 /sec spacn an wth (σ. he rane of preferre jont velocty s [-3 /sec, 3 /sec] an [-65 /sec, 65 /sec] n shouler an axs respectvely. 3 he total number of bass elements was equal to the number of the preferre postonal raents multple by the number of preferre veloctes because we use every possble combnaton of raent an preferre velocty. hus, the total number of bases was 96 (8 postonal raents 87 preferre veloctes. he ntal weht s zero. 5 he learnn rate, η s.. 6 Ranom nose was njecte nto the torque n the smulate system so that movements n the null fel ha the same stanar evaton of p.e. as the subjects movements. he ranom nose use n our smulaton was Gaussan nose wth zero mean an.3 N m stanar evaton. 7 In explorn the parameter space of k an b, we foun that a slope of ra - an.3 ntercept ave a oo ft of nterference as a functon of separaton stance. Smulaton of human arm reachn n D space o smulate human arm reachn, we use the follown moel of the arm s ynamcs that escrbe the physcs of our expermental setup (Shamehr an Mussa-Ival, 99. For every tme step ( ms, we calculate the jont acceleraton ( usn = H ( q where { H ( q + C( q, K ( q q p K v ( ˆ q : a vector compose of shouler an jont splacement. : a vector compose of shouler an jont velocty. + C( q, }

3 H: nerta matrx that vares as a functon of jont splacement. H (q a3 + a = a + a + a + a a a + a C: corols matrx that vares as functon of jont splacement an velocty. - a sn(q C ( q, q = a sn(q shouler K p, K v : spnal an muscle feeback coeffcent. - a sn(q ( shouler + : the actual torque experence urn the movement, followe the rule = J F = J B x J : jacoban matrx that vares as a functon of arm poston - l sn(q J = l shouler shouler - l sn(q + l shouler shouler l sn(q l ˆ : estmate torque an calculate usn = w ( z ˆ. shouler shouler hen, we calculate the jont velocty an poston for the next tme step by nteratn ths acceleraton. he parameters for ths smulaton were K p = [5 6; 6 6] k m /s, K v =.5 K p, l =.33 m, l =.3 m, a =.587 k, a =.3 k m, a 3 =.667 k m, an a =.968 k m. Evence for spatal eneralzaton n the tral-to-tral varablty We hypothesze that the varablty n the center movements as shown n F. a was ue to eneralzaton of errors from nehborn movements. For example, after a movement n the rht spatal locaton, rhtwar forces experence n that locaton shoul eneralze to the center movement, causn a leftwar after-effect. hs pont s llustrate for three consecutve movements n F. s-a. he sequence of movements s center-rht-center. he error n the frst center movement (#3, 9 th movement n the set s small. hs movement s followe by a movement at the rht, where a lare error s experence (#3. hs s followe by another movement at the center (#33, where a lare chane n the opposte recton s observe. It appears that when two movements at the center have an ntervenn movement at ether the left or rht, that ntervenn movement affects the upcomn movement at the center. In the roup wth the larer separaton stance, no such effect s apparent (F. s-b. o quantfy whether there was a consstent pattern to ths nterference between movements, we plotte the chane of error from one center movement to the next as a functon of the number of fel trals between them (F. s-c. For example, when the taret sequence s

4 center-left-rht-left-center, the value on the ornate s (one rht movement mnus two left movements an the abscssa value s the fference n error between the frst an last movements n the sequence. If the force experence at the left or rht nfluences the expecte force at the center, the movements n the center wll show ncrease compensaton for rhtwar force when there are more movements at rht an ncrease compensaton for leftwar force when there are more movements at left. he mantue of the chane n error measures the nfluence of se movements on the center movement. We foun that ths nfluence, as quantfe by the slope of the lnes n F. s-c, s larer n the roup wth the smaller stance. hat s, as the nehborn movements became closer, the effect on the center movement became stroner. In ths analyss, we treate sequence C-L-R-C, C-C, an C-R-L-C as f they woul affect the secon center movement by the same amount norn the temporal effect. o elmnate ths complex temporal-effect, we also the same analyss usn only sequences C-L-L-C, C-L-C, C-C, C-R-C, an C-R-R-C. However, we o not fn any snfcant fference (equatons form lnear reresson: y = -.7x -.3, y = -.8x -.5, y = -.86x -., an y =.6x -.6 for each roup. p.e. of n - p.e. of st center movements. (cm c Perpencular error (cm a Rht b Rht Group Group Group 3 Group y = -.7x -.8 y = -.8x -. y = -.9x -. y =.75x (# of rht - # of left movements n between two center movements Fure s- Measures of nterference. (a Expane vew of movement errors n movements 3 to 33 n F. (a. he arrow ncates chane of expecte force at center after movement 3 (rht movement n F. (a. (b Movement errors n movements 3 to 33 n F. (b. (c Abscssa s the fference between the number of fele movements on the rht an the number of fele movements on the left between two center movements. Ornate s chane of p.e. from the frst

5 movement n the center to the secon movement n the center. he box has lnes at the lower quartle, mean, an upper quartle values. he whskers are lnes extenn from each en of the box to show the rane of the ata. he ray lne supermpose on each plot s a lnear reresson, an the equaton of the lne s shown. A steeper slope ncates a reater nfluence of the ntervenn movements.