Biol. Cybern. 61, 89-101 (1989) Biologicl Cybernetics 9 Springer-Verlg 1989 Formtion nd Control of Optiml Trjectory in Humn Multijoint Arm Movement Minimum Torque-Chnge Model Y. Uno*, M. Kwto**, nd R. Suzuki*** Deprtment of Biophysicl Engineering, Fculty of Engineering Science, Osk University, Toyonk, Osk, 560 Jpn Abstrct. In this pper, we study trjectory plnning nd control in voluntry, humn rm movements. When hnd is moved to trget, the centrl nervous system must select one specific trjectory mong n infinite number of possible trjectories tht led to the trget position. First, we discuss wht criterion is dopted for trjectory determintion. Severl reserchers mesured the hnd trjectories of skilled movements nd found common invrint fetures. For exmple, when moving the hnd between pir of trgets, subjects tended to generte roughly stright hnd s with bell-shped profiles. On the bsis of these observtions nd dynmic optimiztion theory, we propose mthemticl model which ccounts for formtion of hnd trjectories. This model is formulted by defining n objective function, mesure of performnce for ny possible movement: squre of the rte of chnge of torque integrted over the entire movement. Tht is, the objective function C T is defined s follows: Cr= 2 i= t \ dt ] dt, where z i is the torque generted by the i-th ctutor (muslce) out of n ctutors, nd t: is the movement time. Since this objective function criticlly depends on the complex nonliner dynmics of the musculoskeletl system, it is very difficult to determine the unique trjectory which yields the best performnce. * Present ddress: Deprtment of Mthemticl Engineering nd Informtion Physics, Fculty of Engineering, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113 Jpn ** Present ddress: ATR Auditory nd Visul Perception Reserch Lbortories, Cognitive Processes Deprtment, Twin 21 Bldg. MID Tower, 2-1-61 Shiromi, Higshi-ku, Osk, 540 Jpn *** Present ddress: Deprtment of Mthemticl Engineering nd Informtion Physics, Fculty of Engineering, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113 Jpn We overcome this difficult by developing n itertive scheme, with which the optiml trjectory nd the ssocited motor commnd re simultneously computed. To evlute our model, humn hnd trjectories were experimentlly mesured under vrious behviorl situtions. These results supported the ide tht the humn hnd trjectory is plnned nd controlled in ccordnce with the minimum torquechnge criterion. I Introduction In order to control voluntry movements, the centrl nervous system (CNS) must perform complex informtion processing. We propose computtionl model of voluntry movement s shown in Fig. 1, which ccounts for Mrr's (1982) first level for understnding complex informtion processing systems, i.e., computtionl theory. The model proposes tht the following three computtionl problems re solved t different levels in the CNS: (1) determintion of desired trjectory, (2) trnsformtion of visul coordintes of the desired trjectory to body coordintes nd (3) genertion of motor commnds (e.g. torques) to relize the desired trjectory. Consider thirsty person reching for glss of wter on tble. The gol of the movement is moving the rm towrd the glss to reduce thirst. First, one desirble trjectory in the tsk-oriented coordintes must be selected from out of n infinite number of possible trjectories, which led to the glss whose sptil coordintes re provided by the visul system (step 1 in Fig. 1). Second, the sptil coordintes of the desired trjectory must be reinterpreted in terms of corresponding set of body coordinte, such s joint ngles or muscle lengths (step 2 in Fig. 1). Finlly, motor commnds, tht is muscle torque, must be
90 step 5 -trjectory determintion.coordintes trnsformtion -genertion of motor commnd step 4 9 coordintes [ t~nsformtlon i.genertion of motor commnd step 1 step 2 step 3 Fig. 1. A computtionl model for voluntry movement generted to coordinte the ctivity of mny muscles so tht the desired trjectory is relized (step 3 in Fig. 1). It must be noted tht we do not dhere to the hypothesis of the step-by-step informtion processing (i.e. step 1~2~3) shown by the bottom line of this figure. Rther, our model indictes tht there re other informtion processings (step 4 nd step 5 in Fig. 1) which relize the desired trjectory. In step 4, the motor commnd cn be obtined directly from the desired trjectory represented in the tsk-oriented coordintes: tht is, the two problems (coordinte trnsformtion nd genertion of motor commnd) re simultneously solved. Further, in step 5, the motor commnd is clculted directly from the gol of movement: tht is, the three problems (trjectory formtion, coordinte trnsformtion nd genertion of motor commnd) re simultneously solved. We will minly discuss the first problem (trjectory determintion) out of the three computtionl problems nd develop n lgorithm tht corresponds to step 5 in Fig. 1. The other two problems (coordintes trnsformtion nd genertion of motor commnd) hve been discussed in erlier ppers (Kwto et l. 1987, 1988, b). In this pper, the term "trjectory" refers to nd of movement: the is sequence of positions tht the hnd follows in spce, nd the is time sequence of movement velocity long the. Erly studies of the motor control hve concentrted on single-joint rm movements (e.g. Polite nd Bizzi 1979; Bizzi et l. 1984). Wheres, recently, severl studies hve been reported regrding the kinemtic nd dynmic spects of multijoint rm movements. For multijoint rm movements, there exist new control problems tht do not exist in the single-joint cse (Hollerbch nd Flsh 1982). Even two-joint movement is vstly more complicted thn single-joint movement becuse of the presence of interctionl forces (e.g. Colioris forces, rection foces nd centrip- etl forces). When the hnd of the multijoint rm is moved from one position to nother, there re n infinite number of possible s which led to the finl position. Wht strtegy does the CNS use to determine desired trjectory? In wht coordintes frme is the trjectory plnned? Morsso (1981) provided experimentl dt which suggests tht the desired trejectory is first plnned t the tsk-oriented (visul) coordintes. He mesured humn two-joint rm movements restricted to n horizontl plne, nd found the following common invrint kinemtic fetures. When subject ws instructed merely to move his hnd from one visul trget to nother, his hnd usully moved long roughly stright with bell-shped profile. Morsso lso reported tht, in contrst to the simple hnd profile, the ngulr positions nd velocity profiles of the two joints (shoulder nd elbow) were widely different ccording to the prts of the work-spce in which movements were performed. These results provide strong support for the hypothesis tht rm movements re plnned in terms of the hnd kinemtics t the tsk-oriented coordintes rther thn joint rottions t the body coordintes. Abend et l. (1982) investigted not only stright s but lso curved s. When subject ws sked merely to move his hnd from one trget to nother, his hnd ws roughly stright nd the ssocited hd single-peked profile, tht ws entirely consistent with Morsso's experiment. In contrst to the point-to-point movements, when the subject ws instructed to move his hnd while voiding n obstcle or long self-generted curved, the hnd ppered to be composed of series of gently curved segments nd the profile hd often severl peks. In this cse, the curved usully contined distinct curvture peks which were temporlly ssocited with vlleys in the hnd profile. In order to ccount for these kinemtic fetures, Flsh nd Hogn (1985) proposed mthemticl
91 model, "minimum jerk model". The minimum jerk model is formulted by defining the following objective function, mesure of performnce for ny possible movement: squre of the jerk (rte of chnge of ccelertion) of the hnd position integrted over the entire movement. Flsh nd Hogn showed tht the unique trjectory which yields the best performnce ws in good greement with experimentl dt in some region of the work-spce. Their nlysis ws bsed solely on the kinemtics of movement nd independent of the dynmics of the musculoskeletl system. On the other hnd, considering the dynmics of the rm, few reserchers proposed severl performnce indices for trjectory formtion, though their studies were restricted to single-joint movements. Nelson (1983) computed the trjectories which minimized vrious mesures of physicl cost (for exmple, movement time, mximum force, impulse, energy etc.), nd compred the resulting trjectories with ech other. Hsn (1986) gve criterion function which ws the time integrl of the product of muscle stiffness nd squre of the time differentil of the equilibrium trjectory. But it is not cler whether their nlyses cn be pplied to multijoint rm movements. Presuming tht the objective function must be relted to the dynmics, we (Uno et l. 1987) proposed the following mesure of performnce index: sum of squre of the rte of chnge of torque integrted over the entire movement. Here, let us cll this model "minimum torque-chnge model". Regrding the movements which were exmined by Morsso (1981) nd Abend et l. (1982), the hnd trjectories predicted by the minimum torque-chnge model re in firly good greement with those of the minimum jerk model. However, under severl behviorl situtions, the predictions of these two models re quite different. These two models re investigted in detil nd compred with ech other on the bsis of our experimentl dt bout humn plnr rm movements. 2 Minimum Torque-Chnge Model Skilled movements re in generl extremely smooth nd grceful. Hogn (1984) proposed single orgnizing principle to predict the qulittive nd quntittive fetures of single-joint forerm movements, ssuming tht mximizing smoothness my be equivlent to minimizing the men-squre jerk. Here, jerk is mthemticlly defined s the rte of chnge of ccelertion. Flsh nd Hogn (1985) generlized this orgnizing principle to multijoint motion, using dynmic optimiztion theory. Dynmic optimiztion requires the definition of objective function (criterion function), which is generlly expressed s time integrl of performnce index. Tking ccount of the kinemtic fetures of the motion nd the suggestion tht movements re plnned in terms of hnd trjectories rther thn joint rottions, Flsh nd Hogn dopted the Crtesin jerk of the hnd s the performnce index. In moving from n initil to finl position in given time ts, the criterion function to be minimized is expressed s follows: Cd= 1 t f ~(d3x~2 (d3y~2~ 2 o [kd~-/ + k, df~/j dt. (2.1) Here, (x, y) is the Crtesin coordintes of the hnd position. The criterion function determines the form of the movement trjectory. The methods of vritionl clculus nd optiml control theory (Bryson nd Ho 1975) were pplied to find mthemticl expressions for x(t) nd y(t), which minimize the criterion function Cs. If the boundry conditions t the onset nd termintion of the movement re given, the criterion function Cs determines the form of the hnd trjectory completely. Assuming the movement to strt nd end with zero velocity nd ccelertion, the following expression for hnd trjectory re obtined: x(t) = Xo + (Xo - xy) (15z 4-6z s - 10z 3) y(t) = Yo + (Yo -- Yl) (15z 4-6z 5 -- 10z3), (2.2) where z = t/ty, (xo, Yo) is the initil hnd position t t=0, nd (xy, yy) is the finl hnd position t t--ty (Flsh nd Hogn 1985). One cn esily see tht the derived from (2.2) is stright line between the initil nd the finl positions nd the ssocited profile is bell-shped. This model predicted nd reproduced the qulittive fetures nd the quntittive detils of the humn hnd trjectories between two trgets which re locted pproximtely in front of the body (Flsh nd Hogn 1985; Fig. 3). Furthermore, the minimum jerk model successfully reproduced curved movement through certin vi-point s well s stright movement between two points (Flsh nd Hogn 1985: Figs. 5-7). Since x(t) nd y(t) depend only on the initil nd finl positions of the hnd nd movement time, the optiml trjectory is determined only by the kinemtics of the hnd in the tsk-oriented coordintes nd is independent of the physicl system which genertes the motion. In this wy, the minimum jerk model is consistent with the hypothesis tht the desired trjectory is first plnned t the tsk-oriented (visul) coordintes. However, it seems very strnge tht the optiml trjectory of our voluntry movement is determined perfectly independent of the dynmicl quntities such s rm length, pylod, motor commnd, torque or externl force etc.
92 On the bsis of the ide tht the criterion function must be relted to some physicl vribles concerning the dynmics of the controlled object, we exmined few kinds of performnce indices (e.g. energy, torque, movement time etc.). As result of these investigtions, we propose the minimum torque-chnge model, which is formulted by the following performnce index: T? 10 cm I I T3 T4 T8 T+I T+6 Z L2 CT=,~1 dt. (2.3) Here, z i is the motor commnd (torque) fed to the i-th ctutor (muscle) out of n ctutors. The criterion function C w is the sum of squre of the rte of chnge of torque integrted over the entire movement. One cn esily see tht the two objective functions C s nd CT re closely relted, becuse ccelertion is loclly proportionl to torque t zero. 3 Predictions of Minimum Torque-Chnge Model To compute the optiml trjectory predicted by the minimum-torque-chnge model, the dynmics eqution of the musculoskeletl system must first be specified, becuse the criterion function CT depends on the dynmics of the controlled object. But, it is very difficult to describe the musculoskeletl system exctly becuse it is n extremely complex system. Hence, for simplicity, we use the following dynmics eqution of two-joint robotic mnipultor illustrted in Fig. 2 insted of the rel musculoskeletl system. Z 1 = (11 + 12 + 2M2L1S 2 cos 0 2 q- M2(11)2)0.1 +(12 + M2L1S2 COS02)0.2 - MEL1S2(201 + 02)02 sin0z + b101 Z 2 ~--" (12 + M2LIS 2 cos 02)0.1 -t- 120" 2 + M2L1S2(O1) 2 sin02 + b202. (3.1) Here, M i,.li, S i, nd I i represent the mss, the length, the distnce from the center of mss to joint, nd the rotry inerti of the link i round the joint, respectively, bi nd z i represent the coefficients of viscosity nd the ctuted torque of the joint i. The joint ngle 01 nd 02 re defined s indicted in Fig. 2. The links 1 nd 2 correspond to the upper rm nd the forerm, nd the joints 1 nd 2 correspond to the shoulder nd the elbow. The joint 1 (shoulder) ws locted t the origin of the X- Y coordintes. The vlues of these physicl prmeters re given in Tble 1. The vlues of Mi, Li, nd S~ were estimted from mesurement of the humn rm. The vlue of 12 ws ssumed to be 0.1 kg- m 2 which ws typicl vlue for humn forerm rottion bout the elbow joint (Cnnon nd Zhlk 1982), nd 11 ws estimted tking ccount of the geometricl feture of the upper Fig. 2. A two-joint robotic mnipultor which moves within horizontl plne. The origin of the X - Y coordintes represents the loction of the joint 1 (shoulder). X nd Y xes represent the side direction nd the front direction of the body. T1 ~ T8 re the trget positions. See Tble 1 for vlues of the physicl prmeters of the mnipultor Tble 1. Vlues of physicl prmeters of the two-joint mnipultor shown in Fig. 2 Prmeter Link 1 Link 2 Mi (kg) 0.9 1.1 Li (m) 0.25 0.35 Si (m) 0.11 0.15 Ii (kg. m 2) 0.065 0.100 bi (kg. mz/s) 0.08 0.08 rm. The vlue of b i ws ssumed s 0.08 kg. mz/s, which ws in the rnge 0.02,-~ 0.2 kg. m/s estimted for monkeys (Hogn 1984). We confirmed tht the clculted trjectories were bsiclly the sme for the rnge of b i from 0.02 to 0.2 kg. m2/s. Since the dynmics of the multijoint robotic mnipultor is nonliner s shown in (3.1), the problem to find the unique trjectory which minimizes CT is nonliner optimiztion problem. Owing to the nonlinerity, it is much more difficult to clculte the unique trjectory which minimizes CT thn C. Consequently, it seems impossible to obtin the nlyticl expression for the solution of this problem, such s solution (2.2) in the minimum jerk model. However, using n itertive lerning scheme, we cn compute the optiml trjectory for the minimum torque-chnge model; mthemticlly, the itertive lerning scheme cn be regrded s Newton-like method in functionl spce. The detils of this itertive method re given in Sect. 5. We will describe the trjectories derived from the minimum torque-chnge model for vrious movements, while compring them with the predictions of the minimum jerk model. T1
93 T~] e~"... /+T5 ~1 // ""...:3,.,~...'.]'Z":.--_~, T6 T2 C T1 d b (time) 0 Y ~" /--~.. X "1 \' 'SOOmsec ~ i... -'Z'=.~:.,',jF: "7, ~ i ~T6 'S0(]msec (time) Y d A b e C / ',, '500msec 9,. // \,, '5oo~ e c ] //2' ~ i':.7 ~OOm~ f %oo~j f When the strting posture ws stretching n rm in the side direction nd the finl position ws pproximtely in front of the body, the hnd s of the two models were quite different, while the hnd profiles were similr. In the minimum jerk model, the hnd is lwys stright, becuse the hnd trjectory is determined only by the hnd kinemtics nd its shpe is invrint with respect to the region of the work-spce. On the other hnd, in the minimum torque-chnge model, the hnd ws gently convex curve s shown in Fig. 4A-. As seen from comprison of roughyl stright s in Fig. 3A- with curved in Fig. 4A-, the shpe of the derived from the minimum torque-chnge model chnged in ccordnce with the region of the workspce where the movement ws executed. In the lrge movement from the strting posture with stretched rm, the strt point is on the boundry of the workspce, where the dynmics of the rm is very different from tht in front of the body. This is the min reson why the shpe of the shown in Fig. 4A- differs from the shpes of the s shown in Fig. 3A-. o x B Fig. 3A nd B. Free movements between two trgets locted pproximtely in front of the body. A Hnd trjectories predicted by the minimum torque-chnge model, shows the five hnd s(h: T3--* T6, e: T2--* T6, d: T1 ~T3,e: T4~T1, f: T4-* T6). The origin represents the loction of the joint 1 (shoulder). Figure h~f shows the corresponding hnd tngentil profiles long the s. B Hnd trjectories observed in humn rm movements. Four trils re depicted for ech movement. The figure formt is the sme s A. shows the hnd s nd Fig. h ~ f shows the corresponding profiles T8 %. 9...,.. ~",,. '.~ o ' 50Omsec X 77 I (time) A b For unconstrined horizontl movements between two trgets locted pproximtely in front of the body, the minimum torque-chnge model predicted roughly stright hnd s s shown in Fig. 3A-, though they were lwys not completely stright (for exmple, the hnd leding from the trget T 2 to T 6 ws slightly convex); the ssocited profiles were singlepeked nd bell-shped s shown in Fig. 3A-b to A-f. These predicted trjectories were in good greement with the experimentl dt reported by Morsso (1981 ) nd Abend et l. (1982), nd hence coincided with the predictions of the minimum jerk model. However, the trjectories derived from the minimum torque-chnge model were quite different from those of the minimum jerk model under the following behviorl situtions, some of which hd not been exmined in pst experiments. + 10cm +. T8 9 '"~":%j, B Fig. 4A nd B. Lrge free movements between two trgets (T7--*T8); the strting posture is stretching n rm in the side direction nd the end point is pproximtely in front of the body. A Hnd trjectory predicted by the minimum torque-chnge model, shows the nd h shows the corresponding profile. B Observed hnd trjectories for the seven subjects, shows the s nd h shows the corresponding profiles b 500 rnsec (time)
94 For horizontl rm movements which hd to trvel between two trgets pssing through specified point ( vi-point), both the models predicted curved hnd s with single-peked or double-peked profiles. It depended on the loction of the vi-point whether the hnd profile hd single pek or two peks. In both the models, if the vi-point ws locted ner to the line connecting the initil nd the finl trgets, the hnd profiles were single peked; on the other hnd, if the vi-point ws locted further wy from the line connecting two trgets, highly curved movements were produced nd the hnd profiles were double-peked. Furthermore, ccording s the curvture of hnd becme lrger, the vlley in the double-peked profile tended to be deeper. In this cse, the pek in the curvture,,0c. Y ol x A b... ;, \ s k ' ' ''(time) "~'- 500reset ~,~ C/~ '500msec b /"~ p, _ ":~" b ~...... % 8... [~ pt h 500 msec g B Fig. 5A nd B. Free movements pssing through vi-point, P1 or P2. P1 nd P2 re locted symmetriclly with respect to the line connecting T3 nd T5. A Hnd trjectories predicted by the minimum torque-chnge model, shows the convex (b: T3 P I ~ T5) nd the concve (e: T3 ~P2~ T5). Figures b nd e show the corresponding profiles (dotted curves) nd curvture profiles (solid curves). B Hnd trjectories observed in humn rm movements. Four trils re depicted for ech movement, shows the hnd s, nd figures b nd e show the corresponding profiles (dotted curves) nd curvture profiles (solid curves) ws temporlly ssocited with the vlley in the profile (see Fig. 5A-c). However, when the vi-point ws locted t certin distnce from the line connecting two trgets, the two models predicted quite different trjectories. Consider two subcses, with identicl strt nd end points, but with mirror-imge vi-points (see Fig. 5A-). Tht is, the strt point T3 nd the end point T5 re the sme for these two subcses, but the two vipoints P1 nd P2 re locted symmetriclly with respect to the line connecting the common strt nd end points. Here, the vi-point P1 is locted further wy from the body thn the line T3 T5 nd the vipoint P2 is locted nerer to the body thn the line T3T5 s shown in Fig. 5A-. If one notices invrince of the criterion function Cs under trnsltion, rottion nd turning up, it is esy to see tht the minimum jerk model predicts identicl s (with respect to turning up) nd identicl profiles for the two subcses. On the other hnd, the minimum torque-chnge model predicted two different shpes of trjectories corresponding to the two subcses; for the movement pssing through the vi-point P1, convex curved ws formed nd the ssocited profile hd only one pek (Fig. 5A-b); in contrst, for the movement pssing through the vi-point P2, concve ws formed nd the ssocited profile hd two peks (Fig. 5A-c). Furthermore, the convex (T3 ~P1 ~ T5) nd the concve (T3 ~P2~ T5) were not symmetric with respect to the line T3T5. In short, the trjectory derived from the minimum jerk model is determined only by the geometric reltion mong the initil, finl nd intermedite points, wheres, the trjectory derived from the minimum torque-chnge model depends not only on the reltion mong these three points but lso on the rm posture (in other words, the reltive loction of the shoulder for the three points). We found the difference between the predictions of the two models not only for the bove free movements but lso for the constrined movements in which n externl force cted on the rm. Consider tht subject is told to move his hnd between two trgets while resisting ginst the force of spring, one end of which is ttched to his hnd nd the other end is fixed t some position. The minimum jerk model lwys predicts stright nd bell-shped profile regrdless of the externl force, becuse the minimum jerk trjectory is determined independent of the dynmics of the controlled object. On the other hnd, in the minimum torque-chnge model, the trjectory ws influenced by the externl force. Figure 6A shows the predicted trjectories for the free movement (b) nd for the movement constrined by the spring with 70 N/m spring constnt (c). While the ws stright nd the
95 IOcm T4 %-"--.C i U",t t b "4 71 j P" "., Y \,, x 500 mser (time) 0 r ~ 10cm' T4.,~.;... C ~ A 0 X '5(time/00 msec' B ::_-:J i'.! C '50 O reset'... }' '500 reset' tl_ Fig. 6A nd B. Free movements between two trgets (b: T4--. T6) nd constrined movements in which spring force cts on the hnd (e: T4-*T6). A Hnd trjectories predicted by the minimum torque-chnge model, shows the hnd of the free movement b nd the hnd of the movement influenced by the spring e. Figure b nd e shows the corresponding profiles, B Hnd trjectories observed in humn rm movements. Four trils re depicted for ech movement, shows the hnd s nd, b nd e show the corresponding profiles profile ws bell-shped for the free movement (Fig. 6A-b), the ws curved nd the profile ws not necessrily bell-shped for the constrined movement (Fig. 6A-c). The mgnitude of spring force which cted on the hnd chnged with the loction of the hnd; for the movement shown in Fig. 6A-c, the mximum of the spring force ws 10.4N nd the minimum of tht ws 3.3 N. This mgnitude of spring force ws much smller thn the limit vlue of the rel musculoskeletl system which ws observed experimentlly by Cnnon nd Zhlk (1982). Consequently, the externl force did not give overlod to the musculoskeletl system. Furthermore, for the movements executed between two trgets in verticl plne under the effect of grvity, the minimum jerk model predicts stright hnd s with bell-shped profiles, which is the sme result s the model predicted for the point-topoint movements in horizontl plne. This is becuse trget,f.// /.,.:. strt... Sii-D'-i- tget x I 0 ~1 strt locm i I :..f" %t. :.':./.~. "~.. ' 500msec' (time) Fig. 7. Movements between two trgets within verticl plne. The left figure shows the hnd s nd the right figure shows the corresponding profiles. The origin O of the X--Y coordintes represents the loction of the joint 1 (shoulder). X nd Y xes represent the horizontl direction nd the downwrd direction. The minimum torque-chnge model predicts the curved for the lrge, up nd down movement nd the roughly stright for the smll, front nd rer movements b the minimum jerk model determines trjectories irrespectively of the grvity. On the other hnd, the minimum torque-chnge model predicted curved s for lrge, up nd down movements (Fig. 7-), while it predicted roughly stright s for smll, front nd rer movements (Fig. 7-b); the profiles were bell-shped for both of the movements. Why does the minimum torque-chnge model predict the curved s shown in Fig. 7-? One reson for the curved my be sought in the fct tht the dynmics of verticl rm movement is ffected by the grvittionl force nd the minimum torquechnge trjectory depends on the dynmics. Another reson my be the complicted dynmics of the rm tht rises from the lrge movement; this is the sme reson s for the lrge horizontl free movement between two trgets. Although we do not show simultion results, we exmine lrge downwrd movement for which the strt nd the end points were exchnged from those of Fig. 7-. The predicted ws close to the upwrd shown in Fig. 7-. We lso studied up nd down movement with grsped pylod of order of severl hundreds grm. The ws gin consistently but slightly different from the without pylod. These simultion results suggest the bove second reson for the curved of lrge up nd down movement. Although the shpe of the ws significntly ffected by the externl force exerted by spring s shown in Fig. 6, it ws not so severly distorted by the direction of the up nd down movement or by the mount of pylod. This might be understood tht the grvittionl force did not chnge so much for different
96 postures lthough the spring force drsticlly chnged with the posture. The spring force hd more drmtic effect on the dynmics of mnipultor nd the environment, so its strongly ffected the shpe of the. From the bove results, we cn summrize the differences between the two models s follows. The trjectories derived from the minimum jerk model re invrint with respect to the region of the work-spce nd independent of the externl forces. On the other hnd, the trjectories of the minimum torque-chnge model depend on the region of the work-spce nd re ffected by the externl forces. 4 Experimentl Results of Humn Arm Movements We exmined humn rm trjectories under the vrious situtions for which the minimum jerk model nd the minimum torque-chnge model contrdicted. Referring the pprtus designed by N. Hogn, we mde n pprtus shown in Fig. 8. Our pprtus ws lrger nd the fulcrum position ws different from the originl mnipulndum so tht lrger movements cn be mesured. The procedure of our experiment ws fter those of Abend nd Bizzi (1982) nd Flsh nd Hogn (1985). The subject ws seted nd held the end of two-link mechnicl mnipulndum. The subject ws instructed to move his hnd ccording to illumintion of light emitting diode (LED) trgets mounted on horizontl plexiglss pnel. Since the subject's wrist ws brced nd his rm motion ws restricted to n horizontl plne, his rm hd only two degrees of freedom (i.e. shoulder nd elbow motion). The joint ngles of the mnipulndum were monitored by two P2 T8 F x,y) U Fig. 8. Experimentl pprtus for mesuring rm trjectories in horizontl plne. The subject ws instructed to move the hndle of the two-link mnipulndum, nd its movement ws mesured by mens of the potentiometers, P1 nd P2. T1 ~ T8 re the LED trgets. The fulcrum of the mnipulndum ws set t the F1 for the free lrge movement shown in Fig. 4, wheres it ws set t F2 for ll the other movements potentiometers, nd then the subject's hnd position ws computed from the potentiometer voltge signls. Visul informtion bout the rm loction ws eliminted by drkening the room or by covering the plexiglss pnel with n opque blck pper. Results of our experiments under the four prdigms discussed in the previous section re shown in Figs. 3B, 4B, 5B, 6B. First, for the free movements between two trgets locted pproximtely in front of the body, the subjects usully generted roughly stright s with singlepeked nd bell-shped profiles s shown in Fig. 3B. These trjectories were in good greement with the experimentl dt reported by Morsso (1981), Abend et l. (1982) nd Flsh nd Hogn (1985), nd further coincided with predictions of the minimum torque-chnge model s seen from Fig. 3A nd B. Second, the lrge horizontl free movements between two trgets were exmined. Sixteen subjects prticipted in this experiment. When the strting posture ws stretching n rm in the side direction nd the end point ws in front of the body, lmost ll hnd s were gently curved nd tngentil velocity profiles were bell-shped. Figure4B shows seven subject's trjectories mong them. Although few subjects sometimes generted rther stright s, these s were lwys slightly convex nd there ws no concve. In this prdigm, the rm movements were ffected by the complicted kinemtics nd dynmics of the rel musculoskeletl system, becuse the strt point ws on the boundry of the work spce. As result of such complicted dynmics, the shpes of the hnd s shown in Fig. 4B were different from those of the hnd s shown in Fig. 3B. From the comprison of Fig. 3B with Fig. 4B, it is cler tht the shpe of the hnd depended on the region of the work-spce where the movement ws executed. The minimum torque-chnge model cn predict such different trjectory shpes ccording to the region of the work-spce where the hnds moved. This is becuse the minimum torque-chnge model ws formulted on the bsis of the dynmics. Third, free movements pssing through vi-point were exmined. When subjects were instructed to move their hnds between two trgets pssing through vi-point, they usully produced gently curved s with single-peked or double-peked profiles; if the vi-point ws locted ner to the line connecting the strt nd the end points, the profiles of subject's hnds were single peked; if the vi-point ws locted further wy from the line connecting the end points, their profiles were double peked. Furthermore, when the subjects produced the highly curved movements with double-peked profiles,
97 the vlley corresponded temporlly to the curvture pek. These qulittive fetures were in consistent with the experimentl dt reported by Abend et l. (1982), lthough the vi-point ws not specified in their experiments while it ws specified in our experiment. The bove experimentl fetures were consistent with the predictions of the minimum jerk model nd those of the minimum torque-chnge model. However, when the strt, the end nd the vi points were locted t the sme positions s in the simultion shown in Fig. 5A, the experimentl results did not support the minimum jerk model but support the minimum torque-chnge model s follows. Specifying vi-points PI nd P2 s shown in Fig. 5B. We mesured vi-point movements, T3~P1 T5 nd T3 ~P2~ T5. The convex movements (T3~PI~T5) were quite different from the concve movements (T3~P2~T5). In prticulr, the profiles of the former were single-peked s shown in Fig. 5B-b, while those of the ltter were double-peked s shown in Fig. 5B-c. These experimentl results were consistent with the predictions of the minimum torque-chnge model. Fourth, we mesured constrined rm movements in which spring forces cted on the hnds. Here the spring constnt ws 70 N/m, which ws the sme vlue tht ws specified in the predictions of the model. As shown in Fig. 6B, one end of the spring ws ttched to the subject's hnd nd the other end ws fixed to the sme position s in the simultion shown in Fig. 6A. The subject ws sked to move his hnd from initil trget T4 to finl trget T6 while resisting ginst the spring force. In severl trils for such movements, the subject usully generted trjectories of vrious shpes. However, s the subject got ccustomed to the movements by repeting similr movements, he cme to generte lmost sme trjectories for every tril. Figure 6B shows the free movement (b) nd the constrined movement (c) in which the hnd ws moved from trget T4 to T6. As seen from Fig. 6A nd B, the mesured hnd trjectories were similr to the trjectory predicted by the minimum torque-chnge model. Our experimentl pprtus cn not be utilized to mesure rm movements within verticl plne. Recently, Atkeson nd Hollerbch (1985) mesured humn rm movements in three dimensionl spce using the "selspot system" nd found tht the hnd s were roughly stright or gently curved ccording to the region of the work-spce. These experimentl results qulittively coincided with the predictions of the minimum torque-chnge model described in the previous section (Fig. 7). From the experimentl dt in the present section nd the computer simultions in the previous section, we concluded tht the minimum torque-chnge model could reproduce nd predict multijoint rm movements under vrious conditions. 5 Itertive Lerning Scheme for Optiml Trjectory Formtion In this section, we describe the method to compute the optiml trjectory bsed on the minimum torquechnge model The trjectories shown in Sect. 3 were computed using the lgorithm described here. For moving the hnd of mnipultor from one position to nother between time to nd ts, we give the method to compute the trjectory which minimizes the criterion function CT defined in Sect. 2. To obtin the control which minimizes CT, we must solve n optimiztion problem subject to the constrint imposed by the dynmics of the controlled system: i.e., mnipultor. The dynmics of n-joint mnipultor is generlly expressed s follows: dx/dt= y, dy/dt = hi(x, y) + h2(x)z. (5.1) Here, x, y, nd z re n-dimensionl vector nd represent the position, the velocity nd the torque, respectively, hi(x, y) nd h(x) re nonliner functions. It is cler tht (3.1) cn be trnsformed into (5.1) by setting x = 0, y = 0. For simplicity of nottion, we define stte vrible X nd control vrible u: X r = (x r, yr, zt), U = dz/dt. (5.2) In this section, T denotes the trnspose of vector or mtrix. Note tht x, y, z, nd u re n-dimensionl vectors, nd X is 3n-dimensionl vector. Combining equtions (5.1) nd (5.2), nd then representing the right-hnd sides of them by nonliner function f(x, u), we hve dx/dt = f(x, u). (5.3) Let X 0 nd Xy stnd for vlues of the stte vrible X t the strt point nd the end point, respectively. Tht is, the boundry conditions re given s follows: X(to) = Xo, X(ty) = Xy. (5.4) Furthermore, the criterion function CT defined s (2.3) is rewritten with the control vrible u: 1 tf T CT = -- ~ U udt. (5.5) 2 to We cn summrize the optimiztion problem considered s follows; our problem is to find u(t) which minimizes CT given s (5.5) under the conditions (5.3) nd (5.4). Using the method of vritionl clculus nd dynmic optimiztion theory, we cn get set of
98 nonliner differentil equtions which is necessry condition for minimum to exist. dx/dt = f(x, u), d~0/dt = --(df/~x)t~p, (5.6) U----/pz, where ~0 denotes the Lgrnge-multiplier vector with 3n-components, nd ~Pz represents its n-dimensionl prt which corresponds to z. If the third eqution for u is substituted into the first eqution, (5.6) becomes n utonomous nonliner differentil eqution with respect to X nd ~p. In this wy, our optimiztion problem results in two-point boundry-vlue problem. Tht is, the set of nonliner ordinry differentil Eq. (5.6) must be solved for the two-point boundry conditions (5.4). Although it is in generl very difficult to solve the multipoint boundry-vlue problems for the nonliner differentil equtions, Ojik nd Ksue (1979) nd Mitsui (1981) showed tht kind of qusilineriztion technique is pplicble to these problems. Their technique nmed "initil-vlue djusting method" is regrded s Newton-like method in function spce. We develop n itertive scheme to solve our two-point boundry-vlue problem, which is bsed on Newtonlike method. Although n initil vlue of X is specified (i.e. X(to) = Xo), n initil vlue oflp is unknown. Therefore, when we ssume certin initil vlue of ~0 nd solve the initil-vlue problem for the differentil Eq. (5.6), the finl vlue X(ty) does not lwys rech to the trget vlue Xf. We define residul error t the termintion s follows: E = Xf-- X(ty). (5.7) Let be n initil vlue of ~p: ~(to) = ~. (5.8) Since X(tl) depends on the initil vlue of W, the residul error E is regrded s function of. Finding the solution which stisfies the boundry conditions is equivlent to obtining * such tht E(*) = 0. It cn be solved by the Newton method: o~k + 1 = o~k -- (SE()/OcO - 1 E(~k), (5.9) where k is the initil vlue of lp t the k-th itertion. However, this scheme cn not be relized becuse it is impossible to compute 8E/Bet nlyticlly. Hence, we modify the itertive scheme (5.9) s follows. We first define positive perturbtion prmeter e. By solving differentil eqution (5.6) for ~p(to) = +eei nd X(to)=Xo, we cn obtin the residul error E(~+eej). Here, e; is unit vector whose j-th component is 1, but the others re ll 0. We crry out the bove procedure for j = 1, 2,..., 3n, respectively, nd then compute the following 3n x 3n mtrix, S(, Q: S(~, 5) = [{E( + eel)- E(~)}/e, {E(~ + ee2) -- E(cO}/e,... {E( + 8e3, )- E(c0}/e ]. (5.10) It should be noticed tht S(, t) is difference pproximtion of 8E/S, if e is smll enough. Finlly, we cn get the following itertive scheme regrded s Newton-like method insted of the scheme (5.9): k + 1 = k -- S(o~k, I~) - 1 E(o~k). (5. l 1) In this itertive scheme, the solution cn be obtined numericlly becuse the two-point boundryvlue problem for the nonliner differentil eqution ws trnsformed into n initil-vlue problem of the sme eqution. It must be emphsized tht this itertive scheme cn be used irrespectively of whether the trget position is expressed in the tsk-oriented coordintes or in the joint-ngle coordintes. There is no gurntee for the itertive scheme (5.11) to converge for n rbitrry strting point l. In generl, s the nonlinerity of the controlled object is stronger, itertion tends to diverge. For exmple, when the mnipultor shown in Fig. 2 ws chosen s the controlled object with the physicl prmeters of Tble 1, the dynmics of the system is strongly nonliner nd the itertion diverged for the strting point l = 0. However, in tht cse, it ws possible to modify the scheme so tht itertion converges by decresing of chnging ; tht is, the modifiction term (the second term in (5.11)) is multiplied by slowdown fctor 7, (0<7 < 1). Figure 9 shows the results of the computer simultion for the bove itertive scheme,locm 1 + /"- + 2 ] Y r! 500msec 0 X (time) Fig. 9. Simultion results of itertive lerning control, which cn be regrded s Newton-like method. In order to clculte the minimum torque-chnge trjectory for the free point-to-point movement (T3---}T6), ten itertions were crried out. The left figure shows the ten hnd s nd the right figure shows the corresponding profiles. The number ttched to these curves represents the itertion number
99 setting 7 = 0.4. In this cse, the optiml trjectory ws lmost perfectly relized by the tenth itertion. The bove itertive scheme is pplicble to the trjectory plnning nd control of industril mnipultors. In this ppliction, it is n essentil problem how to solve the complex nonliner differentil Eq. (5.6). As shown in ppendix, its pproximte solution cn be obtined only by moving the mnipultor nd mesuring the trjectories repetitively. In this itertive lerning control, the optimum trjectory is directly clculted while neither explicity clcultions of inverse kinemtics (i.e. coordintes trnsformtion) nor inverse dynmics is necessry. In other words, this lgorithm corresponds to step 5 in Fig. 1, in which the three computtionl problems (trjectory determintion, coordintes trnsformtion nd genertion of motor commnd) re simultneously solved by this lgorithm. Consequently, this method is very ppeling from n engineering point of view. 6 Discussion We proposed the minimum torque-chnge model, in some sense, by expnding the minimum jerk model from the viewpoint emphsizing the dynmics of the controlled object. Recently, Flsh (1987) proposed combintion of the minimum jerk model with the "equilibrium trjectory hypothesis" which ws bsed on the spring-like behvior of the humn rm. According to the equilibrium trjectory hypothesis, the CNS defines the time history of the hnd equilibrium positions determined by the neuromusculr ctivity; hence, the sprink-like force is exerted on the rm ccording to the difference between the ctul nd equilibrium hnd positions. Flsh reported tht the equilibrium trjectory hypothesis successfully cptured both the qulittive fetures nd the quntittive kinemtic detils of the mesured movements. Flsh further discussed tht the minimum jerk description might fit the hnd equilibrium trjectories better thn the ctul trjectories. We now reconsider these mthemticl models nd hypothesis from the viewpoint of the computtionl theory shown in Fig. 1. If one ccepts the minimum jerk model or the equilibrium trjectory hypothesis, it lso implies tht the three computtionl problems shown in Fig. 1 re solved step by step; tht is, the desired trjectory is first plnned in terms of the motion of the hnd in extrcorporl spce (step 1), the hnd motion is second trnsformed into the joint motion of the musculoskeletl system of the rm (step 2), nd finlly, corresponding torque nd force re generted so s to relize the desired trjectory (step 3). On the other hnd, in the minimum torquechnge model, the three problems re simultneously solved, nd hence plnning nd execution processes cnnot be explicitly seprted; tht is, the CNS clcultes the optiml torque directly s indicted by step 5 in Fig. 1. In conclusion, the difference between these models is summrized s follows. The minimum jerk model nd the equilibrium trjectory hypothesis imply tht the motor system is divided between higher levels (e.g., the CNS) nd lower levels (e.g., neuromusculr system); in the higher levels, the desired trjectory is plnned independently of the musculoskeletl dynmics; in the lower levels, the ssocited torque nd force re generted. The minimum torque-chnge model implies tht the optiml motor commnds (torques nd forces) re directly obtined from the dynmics of the musculoskeletl system. As seen from Fig. 3B, trjectories mesured in the plnr point-to-point movements were pproximtely stright but they were not completely stright. Further, hnd trjectories were evidently curved for lrge plnr point-to-point movements s shown in Fig. 4B. Two contrry interprettions re considered for these experimentl results. One interprettion is tht the plnned trjectory is stright but the ctul trjectory is not stright becuse of the incomplete control. Another interprettion is tht the motion is plnned nd controlled in ccordnce with the dynmics of the musculoskeletl system; in other words, the plnned trjectory is not stright originlly. The equilibrium trjectory hypothesis is comptible with the former interprettion. Our minimum torque chnge model tkes the stndpoint of the ltter interprettion. The fct tht the minimum torque-chnge model depends hevily on the dynmics of the musculoskeletl system plys n importnt role to predict the trjectory for the movement ffected by the externl force (e.g., spring force). According to the minimum torque-chnge model, the plnned trjectory which is the sme s the relized trjectory ws quite different from tht of no lod movement s shown in Fig. 6A. It my be possible to explin the constrined movement shown in Fig. 6B by mens of the equilibrium trjectory hypothesis. However, it is doubtful whether the equilibrium trjectory hypothesis could reproduce the experimentl fetures in the vi-point movement shown in Fig. 5B. As described in previous sections, the minimum torque-chnge model succeeded in predicting nd reproducing very skilled rm movements. However, we do not intend to totlly deny the possibility tht the CNS performs the step-by=step process (i.e. step 1 ~2 3) for some kinds of voluntry rm movements. It is possible to suppose tht the CNS performs severl kinds of computtionl schemes (1 ~2~3, 1 ~4, 5 in Fig. 1) ccording to different types of voluntry movements. When certin unskilled movement is intended
100 for the first time, the CNS determines the desired trjectory in the visul coordintes, trnsforms the coordintes nd clcultes the ssocited motor commnd; tht is, the CNS performs the step-by-step process (1~2~3 in Fig. 1) t first. However, we suppose tht in the course while the movements is repetedly executed, scheme bsed on the minimum torque-chnge criterion is grdully cquired. We will propose neurl network model, which lerns the energy to be minimized, for this process in our next pper. This scheme corresponds to step 5 in Fig. 1. Hence, for skilled movements such s simple motion between two trgets, the optiml trjectory nd the ssocited torque re utomticlly clculted without step 1, 2, nd 3, using the scheme cquired by the lerning process. The minimum torque-chnge model successfully reproduces the observed trjectories under the vrious conditions (e.g. plnr free movement, vi-point movement nd constrined movement under the externl force) from the single criterion function Cr. We now consider physiologicl or physicl dvntges of minimizing C~.. One possible nswer might be mechnicl reson. Since the control which minimizes chnge of torque genertes the smooth torque nd trjectory, such control reduces wer nd ter on the musculoskeletl system. Furthermore, in tht control, the consumption of energy is reltively low (though it is not minimum) becuse unnecessry force is voided. Another explntion might be found in inherent dynmics of neurl networks in the CNS. This explntion is closely relted to our neurl network model which produces the minimum torque-chnge trjectory, nd it will be explined briefly. Though our itertive scheme is useful to clculte the optiml trjectory with computer, the CNS does not seem to dopt such n itertive lerning scheme bsed on Newton-like method. We know tht some neurl networks cn solve computtionlly difficult problems such s the trveling slesmn problem or erly visions, which cn be regrded s nonliner optimiztion problems with some constrints, by minimizing some cost function (energy) (Hopfield 1982; Hopfield nd Tnk 1985; Poggio et l. 1985; Koch et l. 1986). Becuse of the success of the minimum torque-chnge model, the problem of trjectory formtion for prticulr types of movements cn lso be regrded s nonliner optimiztion problem with constrint given s nonliner dynmics of the controlled object. We recently found multi-lyer neurl network model which cn generte the minimum torquechnge trjectory. This network performs two kinds of prllel informtion processing: lerning process nd optimiztion process. In the lerning process, n internl dynmics model of the controlled object (e.g. rm or mnipultor) is cquired by djusting weights of synptic connections in the network. During this phse, in some sense, the network lerns the energy to be minimized. In the optimiztion process, using the cquired internl dynmics model, the motor commnd which minimizes the cost function (energy) is clculted s result of endogenous dynmics of the neurl network. In this wy, the minimum torquechnge model, if it is reconsidered t the hrdwre level of Mrr, might not so severely contrdicts with the neurl network model proposed by Bullock nd Grossberg (1988), which does not explicitly compute ny criterion function. The minimum torque-chnge model presented in this pper succeeded in predicting the trjectories of plnr two-joint rm movements. It is our future work to investigte whether our model is pplicble to other types of movements. In prticulr, redundnt multijoint movements nd obstcle-voidnce movements re interesting. Fortuntely, our itertive lerning scheme is pplicble to these movements. Acknowledgements. We would like to thnk to Mr. Nobuo Fukud in our lbortory for his technicl ssistnce in the experiments. Appendix In this ppendix, we give the method to solve the Euler-Lgrnge Eq. (5.6) indirectly by moving the mnipultor nd mesuring its trjectory repetitively. Let us rewrite (5.6): dx/dt = f(x, u), (A. 1) d~0/dt = -(~f/x)r~, u=~pz. (A.2) (A.3) The solution X(t) of differentil Eq. (A.1) cn be obtined from the mesured trjectory by moving the mnipultor under the control u(t). Furthermore, differentil Eq. (A.2) cn lso be solved indirectly s follows. If we set 6u(t)- 0 (tht is, u(t) is fixed), the vritionl eqution of (A.1) is expressed s d(rx)/dt = (Of/dX)~SX. (A.4) Since (A.2) nd (A.4) re djoint with ech other, it follows tht doprrx)/dt=o. Consequently, ~prbs = const. (A.5) (A.6) Let gxj(t) be the solution of vritionl Eq. (A.4) for JXJ(to) = ej where ej is the j-th unit vector. 6xJ(t) is pproximtely equl to the vrition obtined by mesuring the trjectory when only j-th component of X(to) is perturbed. Hence, ~)XJ(t) (j = 1, 2...,3n) re obtined s follows. (Note tht X nd lp re 3n dimensionl vectors, respectively). We first define positive perturbtion prmeters e i, (0<ej< 1). X(t) is the trjectory for given control u(t). Next, we chnge the initil vlue of X into X(to) + ejej,
101 nd then we mesure the trjectory XJ(t) for the sme control u(t). Finlly, we get 6XJ(t) from X(t) nd XJ(t) s follows; 6X~(t)_~(XJ(t)- X(t))/ej. (A.7) Chnging the perturbed component of X(to) nd performing the bove procedure, we get 3n different solutions of (A.4): 6X l, (~X 2..., fix n. Now, let us define 3n 3n mtrix, Dx s follows: DX = (~X 1, 6X 2... (~xn). (A.8) Dx is regrded s fundmentl solution mtrix of (A.4). From eqution (A.6) nd definition (A.8), we cn esily see tht lp(t)rdx(t) = ~P(to)rDx(to), (t o < t < tl). (A.9) Furthermore, it is cler tht Dx(to) = I, where! represents unit mtrix. Therefore, (A.9) turns out to be Dx(tff~(t) = ~(to). (A. 10) As seen from the bove discussion, when ~(to) is given, the solution of differentil Eq. (A.2) cn be obtined from (A.10). It should be noticed tht we must move the mnipultor nd mesure its trjectory 3n+ 1 times repetitively until we get the solution ~p(t) of (A.2). References Abend W, Bizzi E, Morsso P (1982) Humn rm trjectory formtion. Brin 105:331-348 Atkeson CG, Hollerbch JM (1985) Kinemtic fetures of unrestrined verticl rm movements. J Neurosci 5:2318-2330 Bizzi E, Accornero N, Chpple W, Hogn N (1984) Posture control nd trjectory formtion during rm movement. J Neurosci 4:2738-2744 Bryson AE, Ho YC (1975) Applied optiml control, Wiley, New York Bullock D, Grossberg S (1988) Neurl dynmics of plnned rm movements: Emergent invrints nd -ccurcy properties during trjectory formtion. Psychol Rev 95:49-90 Cnnon SC, Zhlk GI (1982) The mechnicl behvior of ctive humn skeletl muscle in smll oscilltions. J Biomech 15:111-121 Flsh T (1987) The control of hnd equilibrium trjectories in multi-joint rm movements. Biol Cybern 57:257-274 Flsh T, Hogn N (1985) The coordintion of rm movements: An experimentlly confirmed mthemticl model. J. Neurosci 5:1688-1703 Hsn Z (1986) Optimized movement trjectories nd joint stiffness in unperturbed, inertilly loded movements. Biol Cybern 53:373-382 Hogn N (1984) An orgnizing principle for clss of voluntry movements. J Neurosci 4:2745-2754 Hollerbch JM, Flsh T (1982) Dynmic interctions between limb segments during plnr rm movement. Biol Cybern 44:66-77 Hopfield JJ (1982) Neurl networks nd physicl systems with emergent collective computtionl bilities. Proe Ntl Acd Sci USA 79:2554 2558 Hopfield J J, Tnk DW (1985) "Neurl" computtion of decisions in optimiztion problems. Biol Cybern 52:141-152 Kwto M, Furukw K, Suzuki R (1987) A hierrchicl neurlnetwork model for control nd lerning of voluntry movement. Biol Cybern 57:169-185 Kwto M, Uno Y, Isobe M, Suzuki R (1988) A hierrchicl neurl network model for voluntry movement with ppliction to robotics. IEEE Control Sys Mg 8:8-16 Kwto M, Isobe M, Med Y, Suzuki R (1988b) Coordintes trnsformtion nd lerning control for visully-guided voluntry movement with itertion: A Newton-like method in function spce. Biol Cybern 59:161-177 Koch C, Mrroquin J, Yuille A (1986) Anlog "neuronl" networks in erly vision. Proc Ntl Acd Sci USA 83:4263-4267 Mrr D (1982) Vision. Freemn, New York Mitsui T (1981) Newton method for the boundry-vlue problems of the differentil equtions (in Jpnese). Mth Sci 218:41-46 Morsso P (1981) Sptil control of rm movements. Exp Brin Res 42:223-227 Nelson WL (1983) Physicl principles for economies of skilled movements. Biol Cybern 46:135-147 Ojik T, Ksue Y (1979) Initil-vlue djusting method for the solution of nonliner multipoint boundry-vlue problems. J Mth Anl Appl 69:359-371 Poggio T, Torre V, Koch C (1985) Computtionl vision nd regulriztion theory. Nture 317:314-319 Polite A, Bizzi E (1979) Chrcteristics of the motor progrms underlying rm movements in monkeys. J Neurophysiol 42:183-194 Uno Y, Kwto M, Suzuki R (1987) Formtion of optimum trjectory in control of rm movement minimum torquechnge model Jpn IEICE Technicl Report, MBE86-79, 9-16 Received: December 22, 1988 Dr. Yoji Uno Deprtment of Mthemticl Engineering nd Informtion Physics Fculty of Engineering University of Tokyo Hongo, Bunkyo-ku Tokyo, 113 Jpn