Experimental and Theoretical Investigations on the Control of Limbs by the Functional Electrical Stimulation of Muscles

Copyright © IFAC Control Science and Technology (8th Triennial World Congress) Ky oto. Japan . 1981

EXPERIMENTAL AND THEORETICAL INVESTIGATIONS ON THE CONTROL OF LIMBS BY THE FUNCTIONAL ELECTRICAL STIMULATION OF MUSCLES T. V. Nguyen and G. Vossius Institut fur Biokybernetik und Biomedizinische Technik, UniverSliilt Karlsruhe, Federal Republic of Germany Abstract. Problems occurino durin a the development of control systems within orthoses for p aralyzed p atients are analysed. Fxperimental results obtained from a s i~ ? le closed-loop p osition control systeM ~howed that ~ore sophisticated control strateoies are necessary to remove the deficiencies of actual control sys tems.With reaard to the implication of appronriate control strateqies a Model for the "electri c ally sLim ulated ~uscles-li~b­ load" systeM i s d~rived. ~n adaptive control straten y f o r the Functional Electrical StiMulatio n is then described and, in vieH of the im p l urn entation on a di~ital computer, a feasible p erformance of this strateoy is presented. The effects of different modulation techni~ues on the feasibilit y of control systeMs were also investioated. Pulse frenu e nc y Modulation system~ are found ~ore robust than p uls e am o litude modulation systems. Keywords. Adaptive control; biomedical; direct dioital control; hierarchical systems; nonlinear control s v ~tems; time varyino systems. ~ ulse control sys tems;

These effects severelv liMit the further development of control systems. To choose a suitabl e control stratery The first successful application of the Functional Electrical ftimulation (FESl, for the FES is indeed not an easy task. Re cently some pronres was made ~ainl y however without control terms, Has made by Liberson and others (1961l.since then on the probleM of muscle fatinue (Peckseveral studies have heen reported on ha m, ~o rtimer, 19 77l. The variations of muscle responses du e to electrical stithe field of FES. By no w there are tw o major areas of a pp lication: the p r o vimulation could also b e Minimized h y sion of functional Movements in the usinr. intramuscular electrodes (Mortimer, Peckhal'l, 19 73l. All these encouhand of ~uadripleni c p atients (Peckham, ra qe new attem p ts to develon contr o l ~lortimer, 1977l and the improvement of the nait of hemipleoic p atients (Kral j strate~ies, Hhich ray effect smooth, proportionall y renulated moveMents of and others, 1971l. ['o r all these aPDlilimbs via FES. This paD er o utlines cations an accurate, reaulated control some experimental and theoretical asof electricall y elicited Muscle conp ects resultina fr o m the deve lo pm ent tractions is essential. of control svsteMs by FES. The hitherto p u h lishe d o aoers oive an insiaht in the main control n roblems of P ~I~PLE CLOSED-LOOP CONTROL the FES at the p resent tiMe (Crano and OF THE LI~B POSITION others, 19 80 ; Stanic, Trnkoczy, 1'J74l: INTRODUCTION

- The tiMe variance of the c ontrolled system, i.e. th e "muscles-limb-load" system ("'LLl, du e to the channes in efficacy of sLimulation arisinn from shifts of stimulating electrodes durin g movements and to the fatinue of muscle s .

In order to oet an insiaht in the methods of FES and to check how far a control system based on simplifyinn assumptions can be employed in some simple ap p lications, a closed-loop control system for liMb p osition was investinated. Fio. 1 shows the principal ? rocedures of this s ys tem.

- The nonlinear input-output relationship of the MLL system.

3747

T. V. Nguyen and G. Vossius

37 48

u:

w K: T I c:

output of the controller difference between the command wand the position feedback x proDortional r:ain factor : time co~stant of the intenrator constant

Results

Fin.

The block sche~e of the simple closed-loop control systen.

1.

i .ssum!Jtions

The

followinn assumotions were rade:

-

The a l,tanonistic muscle pair has a inear i zed i IIE'ut-output re la ~ionship. ';'he stim1l1us is the input, the elicited force is the output. Its transfer ~unction is qiven by: 1

v

( 1)

(1

where s is the complex frequenc y , V is the nain factor and tk the contraction time of

the nuscle.

- The systen "limh-external load" a ssumed to be an undamped mass m:

Sinusoidal command inout frer,uencies up to 1.2 Hz were applied to assess the E'erformance of the control systems The results can be summarized as follows (Fin. 2): - The control syste~s were statle in most cases. Some date exhihited rather larne deviations between x and w. - The position reversals were sometimes unsatisfactory. - Too hinh a proportional nain caused oszillations in the systems. The results, similar to those of other investiqations (Crano and others 1980; Stanic, Trnkoczy, 1974) are in fact not suprisinq because of the reasons mentioned in the introductory section. Moreover multielektrode stimulation in a muscle is necessary for a larqer control ranne.

is ':

1/ (ms

2

':: ·. ~ ~ : L :"' ED

~ ··J -:- ;;;C' L i...~ L)

'J Ar.

VA:'IAOL..E

1

(2 )

:;"~LE

x

X

.1ethods Experiments were carried out on the Lhe nastrocnemius and tibialis anterior muscles in the hind len of an~esthetized adult cats. The hind len o f the c at was fixed b y a mechanical f rame so that onlv the foot remained ~ovable in one ~irection. The annle o f the ankle joint was measured ty an electr o noniometer. 'T'he outE'ut of the contr o ller (a di n ital computer) v as multiplexe u and di n ital-analon co nverted, and drove the ~ultichan­ l, el stimulat0r. Th e p ulse an ;:; litude "" odulati0n ( Pl\~) an d ~ ulse fre n uency modulation (PFM) ,,' ere simultaneousl~' : erformed in all ex ~ erirents. Intra~ uskular electrodes were used. ~ee . auyen (19 80 ) for further details. \oJith p ro ,-, orClosed-loo E' svst e ms tional ~ lus intenral controllers ";ere i r, vesti r: ate d :

(3)

v 1 s ec (A)

Fin.

L.

TI"'E

VV TI~

1 s ec

E

(8 )

Results from the simDle cQntrol s!,ster( A) satisfactorv results (B) unsatisfactor y p osition reversals.

Co n s equently ~ore c o mnlex c o ntrol strateqies, which take into account the above mentioned effects, should be dev elop. Intuitively adaQtive control is u est suited for diminishin a the influe nces of the time variance and nonli~ earities in the syste~. Uoreover for clinical ap p lications it is necessar y to automaticall y adjust and readjust the controller p arameter s durin~ o n eration.

3749

Experimental and The ore ti cal I nvest i gat i ons o n the Co ntr o l of Li mbs

A MODEL FOR THE MLL S YSTEM

(e)

The model is assumed to have a nonliIle ar static pa rt and a linear dvnamic p art with a dead time. A model must b e ~ recise enouch to yield sufficient control effects, cut need not too complex with recard to evaluation durinG operat ion. own experiments h~ve shown, that the n o nli nea r pa rt consists of a dead zo ne, a nearl y p roportional zone and ~ sa turation zone for both modul~tion te c hniques.

~ ur

The linear pa rt can he described ty a mechanical s ys t em as show n i n Fig. 3.

an0

o

further c

e

= 0 ;

d

e

=

0

it becomes

3rd order diff erential eouation:

~~b~+ci+dx= V F - V F - v i -bY F (9) Pip 2 F A F A The assumptions 7 and 8 lead t o the fact th a t muscle s of the same tyoe have the same contraction time, thounh their c ains can b e different fr o m each o th er. . he p resented rodel (Fin. 4) is an extension of the linear rodel n r o nosed by '1 a q d ale n 0 (1 968) fo r s mall p er t u r can c e s . Ou r own experimental results a re co nsistent I-l ith the here n resente d model (Vossius, 19 8n ) . rHAN!") H .:J-t "' UIJ C' TICH.

F •

~

F

i9.

3.

Mechanical model the MLL system.

1

F,

for

Fin. antaoonistic pair and th e li.mb are c haracterized by their mass m, their s erial and p arallel elastic components c, c , (i= 1,2) and their dampinG

4.

The MLL mo del.

c",le

Sl

p1

com j"l onents d

(i= I, 2) .F.lasticit y and i dampinq are assumed t o depe nd on the o ctu nl neuronal mu scle innervation (~ aodal ~ no and o thers, 1968 ) The input i ' i(i= I, 2) is the internal force oener a t ed 0 y the contractil elements. The e xternal load is characterized by its r ass M and its elastic and d ampin~ come ponents c and de FA is an additional e external force. Dy assu~inn a ri g id c 0 nnection between limb and load the same displacement x is valid for both: (4 ) (5 )

c s lxl+cs2x2-(CS1+Cs2+ce)x -d

L'

x

- (m +m)x e

F

A

Olle can see from Eo . 9 that the pOSition o f the lim b (i.e. of the load) can be characterized hy two relationships: that h etween electrical stimulation and limb p osition (Gp ) ' and that between external

forc e a nd

li mh j"lo sition

This consideration is ther implications.

im no rtant

(G

F

for

) fur-

Now consider the external load beinn replaced by the correspondinn interactin o force FS. Inste ad of FA' FS is now in Eq. 9 , whose coefficients are then d et ermined only by the parameters of ~uscles and limb. Thus F.~. 9 retains its validity. Under certain circumstances this re p resentation has some advantaaes. In this way the rodel p arameters depend only on the "internal state" of ~uscles and limb. Even by l o ad variations the identification need not b e re pe ated.

(6 ) ~ DAPTIVF.

CONTROL OF LIMB

~OVEYEN TS

~~ .

4-6 yield a 4th order differential eq uation in x, where th e Generator force f and the external force F are the i A excitation input functions. By assumption of the symmetry conditions: (7 )

Based on the above r en ti oned Model the ~ rinci p al features of an ada p tive control strateov which ray compensate the influences of time variance in the system can be stated. Fin. 5 show s the olock sche~e of this hierarchically orqanized control straten y.

T. V. Nguyen and G. Vossius

3750

Fig.

5.

The block scheme of the hierarchical control strateqy.

-

The lowest level of thL hierarch y is closed control loo p where the limb osition and the force applied to the limb are inputs in the controller. This level should auarantee the stability of the limb movements and compensate s~all disturbances. The control alnorithms ill t. ; is level should he able to actuate o: ultich : nncl stimulators and easy to modify. ~

- The next level d e als with the adapt ive features o f the system: s y stem i dentificati o n, s y nthesis and modificatio n of the c o ntrol alnorithms for thc lowest level.

c alculatinn time and its ease of mathemanipulation.The nonlinearities in the system can also be determined, if they are described by nol y noms. For diaital com p utation a recursive LS alr orithm should he used (Isermann,1977).

~atical

System: ( 10)

G

(

p

-1 -n B z _b..,;l:..._ _+.,-._._+_h..:.n,-Z_ _ Z - d Z = -1 -m 11 + •. +amz l+a z 1 )

(z

)

(11) (z -1 )

C (z

T he hinhest level deals with planninn ~ nd initiation of intended move ~ ents Jn d additionall y it is involved with Lhe supervision of the entire system.

-1

-

11 (

Z

-1 - 1

)

( 12) )

Estimatinr. model: The

linear p art of the submodel "sti~UIUS-Dositi o n" and the s ~ hmodel "force- p OSition" c orres p ond r e spectively w ith the relationship Gp and Gp

.Ar.

N(u)=

in the nrevious section. The first o ne re p resents the formal relation between ,.tir.ulus ( s timulus ~arameter: nulse " m,:>litude and p ulse frer..uency) and re';1l1tinq dis),lacement of the limb ,,'ithGut l o ad. It is nonlinear. The second o ne characterizes the interaction bet~ een the limb and the ex t ernal load.

"

PRACTICIIL PERFORMANCE FOR LOW IIND I~~ERMEnIATE LEVELS

TilE

rdent. ification can br' ? erformed L" 'eans of methods for ~ arameter estimat ion. IImonn s t these the least sr..uare - ethod (LS ) has been us e d in many anlicatio n s because of j LS reliabilit y in converqence, its relativel u small

A

n

(13)

le:

A

C (z

~iable

a~d

-1

)

( 15)

,,-1 .... -m 1 + ., 1 z + • • + a mZ

The S y stem i ,,,ntification

2

u + . . . +'1 u

( 14)

where ~ is " earitv, z !\

....

'1 1 u+TlU

"A (z -1 )

the p ol y non for the nonliis the discrete com n lex vad the aead time.

ele~ents

of the vector:

A

e ( 16) A..

b .. 1 J ~re

unknown

an~

should b e

esti~ated.

Experimental and Theoretical Investigations on the Control of Limbs

l ar adaptive control systems usina LS v arameter estimation and DB renulators nave however indicated that the estimated p arameters converge and the closedlOOpS are stable in many circumstances (Astrom, Ljung, 1975).

Position renulator Since the design a lld modification of a d ead-beat controller (DB) is particularly simple, it will he used for rer: ulating the displacement due to the stimulus (Relationshi p Gp)' The parameters of this renulator (R ) can be r d irectly calculated from the estimated "A

parameters in Gp'

G

F

EFFECTS OF

1977) .

F

I

M€ ASUREME NT

l

OF FOR CE

I

+

1

.., I I

+-

- !. _________ J I

I

I I

t G F

J

I

I I I

I I

:~

Rp

r

1

I I I

I I I

J

I I I

r---------I I ~I

LA

i - - -- -- ------ --- - - ---,

r=;

RF

TECHNIqUES

~ODULATION

The effects of PAM and PFM techniques o n the behaviour of control systems are interestina with respect to the exten-

'"

and N (Isermann,

I

3751

S T IIoU.ATCJl

~ I I

-

,....----,

N f--

X

+

Gp

~--~----------------

t ME A~E .. ENT t

I Fin.

~_ompensation

6.

OF P OSI TION

I

The block scheme of the

f or forc " disturbance

hs discussed E~. 9 allows the phy sical s e pa ration of th e MLL s y stem into two s u brnodels. Thus one can inter p ret the fo rce FS as disturbance. Two separate contr ol ler parts can then be im p licate d . ': he first one is the above mentioned J B re (~ ulator Rp' the second one (R ) is F fo r compensation of the disturbance FS' transfer function of this compensat i n a p art is g iven by:

~' h e

lowest level.

sion and optimization of control effects. Computer simulations were made to investiqate these effects.By neglecting the non linearit y a 4th order syst em was used. Closed-loop control systems with proportional plus integral p lus derivative controllers actinq in PAM technique, and those with intenral c ontrollers acting in connection with intearal pulse frequency modulators (IPFM) (Noges, 1975) and additional velocity and acceleration feedback were sirnulated. ~ion

'11

( 17)

Fiaure 6 shows the block scheme of the lowest level. At the tine of writinn L"e control strateny is pa rtly implere ~.() rt on it rne nted. It is hop ed to "l erformance at an e ,lrly da te. Two major ; , r cb le ms are o ~ interest here: the conve r qe nce of the pa ramet e r estimation a nd the s ~ ability of the closed-loop sy stern. Th , mathematical p roofs are fa r from tri via l be cause of the nonlinearities and ti ~ e variance in the s y stem. Extensive simulations and rre~ise p roofs to these p ro b lem s in simi-

,

'0

n

Time (.ee)

Posit ion

'[ 1

F in.

7.

PAM

" "

rime ( ze: '

simulated step-forced responses.

3752

T . V. Nuguyen and C. Vossius

Fioure 7 shows two ste p -forced respons es of the simulated systens. The si~u­ lations show that with reSDect to settlino time and stationary accuracv PA~ sy stems are su p er i or to PFM s ys te~s o ut they are sianificantly more sensitive to parameter variati c ns in the s ·; stems. CONCLUSION Exp 2 rimental and theoretical aspects res ultinq from the development of control s y stems for ~E S were discussed. It is necessary to apply adaptive control ~ trate qies to compensate the influence u f tim R variance and nonlinearities inLerent in the c ontrolled systems. Bas ed o n a model for the MLL system an ada p tive control strateg y was pr oposed, ~ hich involves th e pa rameter estimation, a self tunin n re a ulator and the co ~p ensation of disturbance. T he investi ' ations on PA~l and PFM techlliques show that PFT~ systems are more robust th all PAM systems. Furth e r investinations on other methods of parameter estimati o n and tynes of controller, and also on combinations of di f ferent modulation t ~ chni q ues (resp ecti v el y switch strateoies b etween them) are n n cessar y to achieve more satisfactory results. HE FERENCES Astr om , K. J., L. Ljuna (1975). Th eory and ap p lications of adaptive regulators based on recursive pa rameter es ~ imation. Proc. of the 6 th IFAC world conClress, pap er no. 50.1. Crago, P. E., J. T. ~l ortimer, and P. H. Peck llar. (19 80 ). Closed-l oop control of force durino electrical stimulati on of muscle. IEEE Trans. Biomed. Eng., BME-27, No. 6, 306-312. Isermann, R. (1977). Diqitale Renelsysteme. Sp rin ge r-Verlag, Berlin. Kralj, A., A. Trnkocz y , and R . Acimovic (1971). Improvement o f locomotion in hemiple r. ic p atients with multichannel electrical s timulation. Proc. of the Conf. o n Human Locomoter EnaineerinCl. University of S ussex. 45-50. L iberson, ~]. T., and ot ll ers (1961). Fun c tional electroth e ra py : stimulati o n of the pe roneal nerve synchronized with the swinq p hase of the aait of hemipleaic p atients. Arch. Ph y:o . Med. Rehab., 10 1-1 05 . --MaCldaleno, R . E., D. T. Hc Rue r, and G . P. Moore (1968). S mall p ertur ba tion d y namics of the neuromuscular s ; stern in trackina tasks. Technical Report NASA CR-1212. Sys tems Technol ogy , Inc., Hawthorn, Cali fornia. Mortimer, J. T., P. H. Peckham (1 9 73). Intramuscular electrical stimulat ion. I n 1'1 . f'. Fie 1 d san d L. A . Leavitt (Eds.), lleural Or a anisation and its Re levance to Prosthetics. Intercontinental Med ical Book Corp., New York. pp . 131-146.

N("! u ye n, 7. V ., G . Vossius (19 8 0). F.xDerim en telle und theoretische Unters uchunaen zur Reae lun ("! der Bewer.unosarossen fO r d ie funktionelle S ti mulat ion. Rio~ed. ~ech ., 25, 316-318. Noges , E., P. r'. Frank (1975). Pulsfre q uenzm od ulierte Rene lun q 5SYSteme. ~ . Oldenhura Verlao, ",inchen. Peckham, P. H ., J. T. ' lo rti me r (1 9 77). Resta urat ion o f hand function in the r. uadriple n ic thr o ur.h el ectrical stimulation. Tn F. T. Hamhrecht and J. R . ".eswic'<: ( Eds .), Function;,l Elect rical St i mu lat ion : Anplicatlon in Neural Protheses. Yarcel ~e kker, Tnc., ; i E'~1 Yo r k. nD . 8 3-95. ~ tanic, U ., fI. Trnkoczv (1974). Closedloo p nos iti o nin n of hemip ler.ic p atient's jo int hy means of functional electri ca l st im ulati o n. IEEE T ran s. B i 0 mf> d. E n Cl., B ~, E - 2 1, N0 ~ 365-370. Voss iu s , G. (1980) Funktionelle Stimulation ael h hmter ~u5keln. Interner Bericht d es Instituts fUr Biok ybe rnetik u nd Biomedizinische Technik de r UniversitAt Karlsruhe. Thi s research is suppo rted by cra nt of Bundesministeriurn fUr Forschunn un d Te c hno l oqi e. Discussion t o Paper 112.5 C. Cobelli (Ital y ) : Could you please comme nt on the purpose of your CVS model? D. Moller (Federal Repub lic of Germany) : Our efforts to derive this verif i ed ou tli ned mathematical model of CVS have been made so that we c ould use this model for the identification of the contractility of the heart, especially in t he pathophysiolog i cal case of h eart d ise ase , in which we predict, for instanc e, the degree of heart insufficiency and also the therapeu tic effects o f glycosides on t he heart muscle . Some further wo rk i s going on in this direction , and we ar e now able to predict Signif i cant l y these relevant heart parameters. Discussion to Paper 112 . 6 D. Moller (Federal Republic of Germany) How do you prove the stability o f the i dentification me th od in your nonlinear sys tem, by simulation or by using a mathe matical method? W. Setz er (Fede ral Republic of Germany) On ly by simulation of the algorithm under different test conditions. Y. Kosugi (Japan): Could you expl a in in more detail the stimulation condit i on , i.e. the current amplitude , pulse wi dth and pulse frequency? W. Setzer (Federa l Republic of Germany) : The surface stimulation is 100-500~s , 0 .. . 100(150)mA ; imp lanted electrodes 100-500~s , 0 .. . lO(l5 ) mA . Impulses Y. Kosugi (Japan ): How did you modula te the st i mul i, AM or FM? W. Setzer (Federal Republic o f Germany ) : Both , AM a n 1 "'11.