290

IEEE TRANSACTIONS AUTOMATIC CONTROL, VOL. ON

AC-29,NO. 4, APRIL 1984

The Control of a Prosthetic Arm by EMG Pattern Recognition

Abstract -An electromyographic (EMG) signal pattern recognition system is constructed for real-lime control a prostheticarm through precise of identificationof motion and speed command. A probabiistic modelofthe EMG patterns is f i i formulatedinthe featurespace of integralabsolute d u e (IAV) to describetherelation betweena command, represented by motion and speed variables, and location and shape of the corresponding pattern. The model provides the sample probability density function of pattern classes in the decision space relations between L4V, of variance zero and crossings based on the variance, and zero crossings established in this paper. Pattern classification is carried out through a multiclass sequential decision procedure designed with an emphasis on computational simplicity. The upper bound of probability of error and the average number of sample observationsareinvestigated. Speed andmotionpredictionsareusedin conjunctionwiththe decision proceduretoenhance decision speed and reliability. A decomposition ruleis famulated for the direct assignment of speed to each primitive motion involved ina combined motion..4 learning procedure i also designedforthedecisionprocessorto s adaptlong-termpattern variation. Experimental results are discussed in the Appendix.

I. INTRODUCTION

HE electromyographc (EMG) signals are the electric manifestation of neuromuscular activation associated with contracting muscles. They can be conveniently and safely measured at the surface of the skin near the appropriate musclesbythe surface electrode. It has beenproposed that the EMG signals canbeused to identify motion commandfor the controlof an externally powered prosthesis, based on the factthat synergistic signals can be obtained fromthe intact musculature of an amputee.Butthe extreme compIexities involved in the EMG signals make it difficult to have a precise structural or mathematical model which relates the measured signals with a motion command [l], [2]. Three major approaches been have suggested to solve the motion command identification problem. The first, based on the works of Graupe et al. [3], [4], models the EMG signals as a stationary time series ( A R model), and the model parameters for each of the prespecilied motion classes are identified to form a reference parameter set. The measured EMG signals are classified into one of several prespecified motion classes, either by parameter vector space methods or parallel filtering methods. The second, based on the works of Wirta et ai. [ 5 ] , uses many electrode sites to form a pattern which represents the spatial distribution of the time integrated value of the EMG signals. A linear discrimiManuscriptreceivedMarch18,1982; raised March4.1983.Paper recommendedby A. K. Bejczy,PastChairman of the Automation and Robotics Committee. ih The authorsare w t theDepartment of Electrical.Computer.and Systems Engineering, Rensselaer Polytechnic Institute. Troy. NY 12181.

T

nant function is designed for the classification by proper assignment of weighting coefficients and summationthreshold. The third, based on the works of Saridis et a / . [6], [7], forms patterns of prespecifiedmotionclasses in thefeaturespace of variance and zero crossings, wherethe EMG signal variance and zero crossing are selected as thebestfeaturesubset in thesense of class separability by thefeaturesubset selection procedure. A learning linear classifier is designed to investigate the upper bound of misclassification for each pairwise classification of27 motion classes. Some other important works on prosthetic arm control are the reflexive and trajectory control by Lyman et a/. and Freedy et ai. [8]. [9], thetask classification byLawrence et ai. [lo], and the shoulder torque analysis by Jacobsen et a/. [ll]. Also, many of theaspectsrequired for the practical success of theprosthetic arm research are well described by Jacobsen et a/. [21]. Although previous work has brought some sort of theoretical and practical achievements for the control of a prosthetic arm, further advancement, such as the accurate identification of motion and speed command from the EMG signals and the design of a faster and more reliable command identification procedure, enough to be processed within the limitation of time and error rate, is required to achieve an ultimate goal, the anthropomorphic movement of a prosthetic arm with minimum mental effort. This paper presents a definite contribution to the attainment of the above goal by developing further the latter approach. At first, a probabilistic model of the EMG pattern is formulated based on the pattern trajectory with respect to the command variation. The modelprovides not onlythesample probabilitydensityfunction of pattern classes in thedecision space but also the decomposition of a motion into its primitive motions. Then, amulticlasssequentialboundeddecisionprocedure is designed for pattern classification. The decision speed and reliability areenhancedbyspeed and motionpredictionswhich adjust pattern locations and shape, and specify prior probability distribution of pattern classes before each classification. A learning procedure is also designed for the automatic update of stored pattern information so that the decision procedure is able to adapt the pattern variations. Finally, the decomposition scheme is provided to assign speed to each primitive motion involved in a combined motion, so as to establish direct control of the a r m .
11. PROBABILISTIC MODEL EMG PA-ITERNS OF

Acommandwhichgeneratesashort-time arm movement is specified by motion and speed. To develop a probabilistic model of the EMG patterns in cooperationwithexperimentaldata, motion and speed are properly defined as two variables in their domain sets. Themodel is formulated in thefeaturespace of integral absolure oalue (IAV) and describes the relation between a

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LEE AND SARIDIS: CONTROL OF A PROSTHETIC A R M
Domain S e t [m}x{s]

291

b

Feature Space

VY

pronation

Fig. 1. The graphical representation of a motion set.

Domain S e t [m]x{s]

Feature Space

command the and location and shape of the corresponding pattern, where IAV is defined as the time integration of absolute value of the EMG signal for the specified time interval.
Motion and Speed Variables

The EMG samples collectedfor a command( m ,s) E { m } x { s } form a pattern in an appropriate feature space. The dimension of the feature space is determined by the number of features multipliedby the number of measurement sites, so that a pattern containsboth thetimeinformation (selected features) andthe spatial information (measurement sites) of the corresponding command. A pattern formedbyaspeculativedeterministiccommand ( m ,s) E { m } x { s } canberepresentedby a random vector x( m , s), the randomness of which is considered mainly due to various measurement noises x( m , s) is called pattern trajectory in the sense that it represents the pattern-moving with respectto the variation of a command ( m ,s), as shown in Fig. 2(a). Sincesamples for acommand ( m , can onlybecollected s) For convenience, speed s'.J or s i , * * of a combined motion m l - J or m i .J , k is defined in terms of the speeds s i and s J or S I , sJ, and from the commands randomly distributed around ( m ,s) due to s k of its primitivemotions m' and mJ or mi, J , and m k as the fact that amputee's muscle control which is highly random in m its nature generates the command, a complete description of a follows: pattern with respect to a command requires the randomization of motion and speed variables in pattern trajectory. An expression x ( m ,s; m , s), called a probabilistic model o the EMG patterns, is f nowused to represent probabilistically the relation between a where command and the location and shape of the corresponding pattern, as shown in Fig. 2@). An explicit mathematical expression of the model can be and is commutative associative. and obtained in the feature space of IAV in cooperation with sample A speed set { s } is now defined as a collection of al possible data collected from two electrode sites located on the biceps and l speeds. triceps, as a specific application of the above argument.

Motions of the three-degrees-of-freedom arm which has humeral, elbow, and wrist axes can be conveniently reconstructed by six primitive motions and their velocities, where the six primitive motions defined here are humeral rotation in and rotation out, elbow flexion and extension, and wrist supination and pronation. Let mi, i = 1 . - , 6 represent the six primitive motions and let ; vi, i =1 . .,6, represent their corresponding velocities. A com; bination of two or three primitive motions mi and m J or m', m J , and m k with a particular velocity ratio v l / v J or c'/vJ or v ' / v J / v k , which donot conflict, determinesadouble-combinedmotion mi,' or triple-combined motion mi*J,k. double-combined or a A triple-combinedmotionsubset {mi,'} or { m r 9 J * k } thendeis fined as the collection of all possible different velocity ratios in the combination of two or three primitive motions mi and m J or m i , m J , and mk. A motion set { m } is nowdefined as the collection of six primitive motions, twelve double-combined motion subsets, eight triple-combined motion subsets, and no motion. l t is conveniently represented by a graph illustrated in Fig. 1, where six nodes represent six primitive motions, twelve edges represent twelve double-combined motion subsets, and eight triangular areas surrounded by threeedgesrepresenteight triplecombined motion subsets. A complete description of arm movement requires the specification of absolute velocity or speed of a motion, since a motion has an infinite number of velocity realizations of its primitive motions which maintain the same velocityratio. The speed si of a primitivemotion m' is first defined as the ratio betweenthe actual velocity u' and theprespecifiedreferencevelocity as follows:

(b)

Fig. 2. Pattern as a mapping command. Mapping from (a) from a deterministic command ( m , to a pattern x ( m , s). (b) Mapping from s) a random command ( m , m, s) to a pattern x ( m , s ; m,s). s;

It is noted that the representation of arm movement in terms of motion and speed, as described previously, is quite compatible with the way an amputee tries to make arm an movement. Furthermore, the introduction of tlie reference primitive velocities of a motion providesconvenient a way to connect the amputee'smuscletensioncontrol to theactual arm speed by selecting the reference primitive velocities of a motion to correspond to the amputee's most comfortable muscle tension of the motion. In a summary, a command which generates a short-time arm movement is completely described by a pair { m , s} where m represents a motion variable m E { m }, and s represents a speed variable s E { s }, as defined previously.
Formulation of a Probabilistic Model

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The selection of feature IAV for model construction brings mathematical and practical convenience due to its proportionality to the speed of a motion at a constant load, or vice versa, in many muscle activities [12]. Pattern trajectory x( m , s) can then be expressed in the two, dimensional feature space of IAV as follows: x,(m,s)=f(m,s)+w t
'a

(4)

where the additive measurement noise w is assumed to have zero mean. The expected value of x , ( m , s), f ( m ,s), is called the pattern trajectory function and represents the movement of the mean point of pattern with respect to the variation of command. A typical form off(m,s), ( m , s ) E { m ' . ; } x { s } ,is illustrated in Fig. 3 on the basis of experimental data, where either motion or speed is considered as aparameter. f(m, s) can nowbe expressed as follows,based on the linear relationshipbetween IAV and speed:

0

b

*

Fig. 3. Pattern trajectory function.

Then, x , ( m , s: m , s) can be approximated as follows: where K( m ) represents the pattern trajectory function at the unit speed. K ( m ) is then approximated by a iecewise linear function in eachsubdomain { m'vj} or { mi3J*'] withexperimentally specified reference coordinates. For example, with the following reference coordinates:
K(m')=

x , ( m , s : m , s ) = f ( m , s :m , s ) .

(11)

[21,

K ( r n J )=

[:],

and K ( r n > j ) =

[21

It is noted that thevariances of m and s depend on the amputee's degree of training acquired by experience. An explicit probabilistic model is obtained in the subdomain { m ' . J } x { s } from (9) and (11) as follows:

x,(m,s:m,s)=gl(a)sL,+g,(B)sLg

(12)

(6)

where m>J= { m'*Jls'/sJ = l/l}, and with new parameters OL and j , 0 < a,j < 1, K ( m ) , m E { m i , ' } can be approximated by the 3 3 following equation:

where parameters a and fl are randomized. The formulated model will be the basis of the derivation of sampleprobabilitydensityfunction and theconstruction of a decomposition scheme as w be discussed later. i l 111. SAMPLE PROBABILITY DENSITY FUNCTION (SPDF) Command identification bypattern recognitionrequiresthe specification of a suitable number of referencecommands as pattern classes.Here,27referencemotions and onereference speed are selected to form 27 reference commands. It is noted that the reference speed is specified at each classification by the speed prediction so as to adjust the location and shape of reference patterns accordingly. A set { m , } of reference motions is formally defied in the follo~g:
{m~}.={~,{,m1,z=l;..?6},{rn~',i,~; 12possiblecombinabons of I andj}, { m > J * ki, j , k ; 8 possible combina, tions of i , j , and k } } , (13) where I#I = no motion m: = m1 m'.J = { m . l ' s = 1/1} , ''s/J
'

(7)

where
1:
L a [ Lp] =

forallmE { m ' . ' l s ' = z , O < s j < l } {m'.'10J,s)}, in generatingacommand is highlyrandom and, furthermore, learning scheme can remove other critical long-term variations. can be derived from (12) by specifymg the probabhty dlstribu-

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293

tions of a, $, and s. The distributions of a,$,and s are approximately specified as follows, based on the assumption that motion and speed of thereference command s) are Gaussian distributed around ( m $ J s: ,)

(rn:’,

E{

X z } =-

6’

Kz

a = $Ia’I,a’- N ( O ,a) :;

$ = $l$’I, B’- N ( O , $ ) ;
S-

N(s,u ) :

(14)

where K, is a constant, the value of which isdependent upon the EMG signal sampling interval. Fig. 4 provides an experimental illustration of lemmas, while the theoretical derivation is givenin AppendixA. The following theorem describes the result of transformation. Theorem 2: The SPDF P { ( x , , x z ) T l ( m ,s)} i approximately s given by the following equations: p { ( x , , x z ) 7 = (:x,x,: x,:,:x)
T~

where N ( m , 0 ’ ) represents a Gaussian distribution with mean m and variance 0 2 , and the conditions a ua -=ZS 1, and a, s are , implicitly assumed. This assumption has been verified directly by the matching test between the derived pattern distribution and the actual pattern distribution and indirectly by the effectiveness of proposed decision procedure. The following theorem summarizes the derivation. Theorem I: The SPDF of a reference command (mi;’, s), P{ xol(mi;’,s)} is given by the following equation:

(m,s))

1
2aNJ\/r,r,(l- rb)(l- rr)

(x: - Nr,) -exP{ - 2Nrb(1 - rb) + 2Nr,(1- r,) (x,”- N r b )

[

2

where

where a= x : x :

- ”: x , .

x:(X:-Xf)-X:,(x,b-X;)’ (17)

Ja=Ix:(x~-X:)-x,”(Xf-xX:)IS,,

s,

=

x:(x:-x:>-x:(xp-x,”) x ;. :
- x:x;

and N is the number of sample digitization. Proof: See Appendix B. The approximated SPDF derived in the feature space of variance and zero crossings will be used in the pattern classification procedure to take advantage of the better class separability imposed by those feature subsets [6], [7].

8, ss, and Jp are obtained from a,sa, and Ja by replacing i by j :
L a ,[ L p ]
=

L

N. MULTICLASS SEQUENTIAL DECISION PROCEDURE
Pattern classification is carried out through a multiclass sequential decision procedure designed for the effective control of the probability of error. The decision rule and the stopping rule of the procedure use simple mathematical formulas defined as the likelihood probability and the decision measure, respectively. Definition 1: Likelihood probability H ( q , x k ) k 1:

if thereexists a mappingfrom (a, to sa)
( x , b , x : ) , [ ( P P B ) to
(:X) .?:]

otherwise, (19)

, x:, x,”, and x are defined in (6) and satisfythe : x : conditions x:/x,” # ./; and x:/x,“ # $/x,”. :x Proof: Direct results from the application of rule of random vector transformation to (12) and (14). x:, x:, SPDF in the Feature Space of Variance and Zero Crossings

The following lemmas provide the relations between IAV, variance, and zerocrossings so that the SPDF of the feature space of variance and zero crossings is obtained from the SPDF of the feature space of IAV. Lemma 1: Let x, and x, represent IAV and variance of the EMG signal, respectively. Then the following relationship holds: x, = yx;.
B

where x k { xl; . ’ , x k } represents a sequence of k observed samples, wi, i = 1; . .,m represents m pattern classes, and rn i=l

DecisionRule: If

Lemma 2: Let the random variable x represent zero crossings , of the EMG signal. Then the following relationship holds:

then x k E w,.

Definition 2: Decision measures D ( x k )

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Integral Absolute Value (x 2 x lo-" v o l t )

I

xa

Fig. 4. The relations between IAV,variance, and zero crossings.

o ( x k ) &s lmax j ...
,

,m.

{H ( ~ , X , ) ) .

Then, the following sequence of inequalities is obtained:

Stopping Rule:

In case that k < Nmx,

if D(x,) 2 a , , then stop sampling and apply the decision rule for decision if o(xk) a,, then take one more sample. <
In case that k 2 N,,.
Q)

~(l-ct,)

k=l

I

P ( x k ) d x k = (1- 0,)

(28)

sk

stop sampling without regard to the value of D ( x k ) and apply the decision rule for decision. The threshold a,, $ < a, < 1, is called stopping rule threshold. of sampleswhich Nma istheprespecifiedmaximumnumber

where S, represents the entire space of x,. The above result is formalized in the following theorem. Theorem 3: The stopping rule threshold a, has the following relationship with the upper bound of the probability of error PC:

bounds the decision interval for the stability and smoothness of arm control. It is noted that the design emphasizes the overall procedural simplicity to achieve the faster computation required where for the on-line decision making.
Stopping Rule Threshold Versus Probability of Error

PC6 1 - a , ,

(29)

f s,). h e mapping should satisfy the following condition:
J ,

J,

s ,

= s,@s,.

(41)

O

for qi = q j , d(O,-)Pl, d(O,+)pl, d(+,6)AO

d(+,-)=3,
(38)

Since x, is in area 1 where s, > sJ, from (3) and (41), we have that Observing that sJ yaries from zero at the point A to s, at the point G along the h e A G while s, is kept constant at s, s is approximatelycalculated by thefollowingequationswiththe assumption of a uniform rate of increase of sj along A G :
s.=-sAC
J

for6E{+,-,O,+}.

The predicted reference motion Q p , Q p = ( q i , q,P,q!), can be specified by the following motion predicbon rule.
Motion Prediction Rule

AG

=-

OB OE‘

(43)

The decomposition rule is now derived from (42) and (43) as follows. where the constants r,, re, and r, are prespecified decision thresholds and s g n represents the signum function. The following prior probabilityassignment rule is nowproposed as an implementation of the principle that higher probability is assigned to the reference motion of shorter distance from the predicted reference motion.
Prior Probability Assignment Rule Decomposition Rule

A double-combinedmotion m i . / whichisrepresentedbya sample point x,, x, = (! x, isdecomposed into thespeeds, si and s j , of its primitive motions by the following rule: Case 1:

T>+, x , x ,

Xf

xr

If the similarity distance d(Q,, Q j ) from Q p to Q, of a pattern class wJ is d j , j = l ; . . , m , then the priorprobabllity P ( w , ) is assigned by the following equation:

Case 2:

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A

PROSTHITIC ARhi

297

si and s are obtained from (44)and (45) by replacing i by j , , where xf/x,b # x:/x,b. See Appendix C for the detailed derivation. The decomposition of a triple-combined motion can be accomplished by decomposing it into one double-combined motion and one primitive motion, with arbitrary selection of a double-combined motion. Thedecompositionprovidesprecise anddirect control of an arm.
ViI. LEARNING PROCEDURE
An on-line learning procedure is provided to update the parameter values of the SPDF such as reference coordinates and variances, so as toadapt the long-term variation of pattern location and shape arising from variation of an amputee's physical condition, degree of training, and properties of recording equipment. At first, learning samples are collected for eachpattern class by the learning sample selection rule which uses a properly defined window region, as described in the following. Definition 4: A window region w,(x,) for a class or with reference coordinate x, is defined as follows:

0

Fig. 6. Learning of attern location. Windowregion Wr(x,) is updated wth $ update of reference coordinate x,. e

VIII. OVERALLPROCEDURE
The block diagram presented.jn Fig. 7 explains how individual components of various functions work together toward the accomplishment of desired control goals. The flowcharts given in Figs. 8 and 9 explain the developed decision and learning procedures in more detail. The procedures developed here actually work as the organizer and, partially, coordinator of the hierarchically intelligent control system [15]. A different neuromuscular channel is also provided for the off-line learning and the correction of severe error. The final values obtained through thedeveloped procedures wl be fed into the lower level of the coordinator which reconil struct the motion on a per-interval basis and t r on the motors un of the prosthetic arm to activate it for this particular interval. The subject should continue generating those synergistic EMG signals for the next interval until the motion is completed.

where wr is a window region threshold and x, is a sample vector.
Learning Sample Selection Rule

A set L,(x,) of learning samples for a class w, is constructed by the following rule:

m. EXPERIMENTAL. VERIFICATION
The effectiveness of the theories and the algorithms developed are experimentally verified by the analysis of collected data and the computer simulation of decision and learning procedures. Due to the large volume of complete experimental results, only parts of the important results which support the developed theories and algorithms are briefly described in the Appendix D. For more details, see [13].

If x, E o, and x , x, 4 LAX,).

E

W,(x,), then

x,

E &, () x,

otherwise

The useof thewindowregionfor on-line samplecollection reducestheeffect of classification error on the evaluation of pattern location andshape. Now, the variation of pattern location, detected bythe difference between the mean of the learning samples %(, A ,x) E{ x,Ix, E L,(x,)} g d the exisfing mean x , automatically calls , for updating the procedure described in the following algorithm. Algorithm I -Reference Coordinate Learning Algorithm: Input: Initial reference coordinate x,"; threshold constant a t; acceleration constant a; a sequence of positivenumbers q,, i = 0,l;. ., Output: New reference coordinate xyw. Method: Step 1) Set i = 0 Step 2) Calculate X ( x ) from &x) , ; (: Step 3) If l l ~ ( x i ) - x:1I2 G t go to.Step 7) Step 4) x; q i [ x , ( x ; ) - xila Step 5)Selectthenewwindowregion W , ( X ~and )con+' struct L,( X i + l ) Step 6) i = i +1and go to Step 2) Step 7) x = x. " : : This algorithm defines,a stochastic appoximation procedure. Its convergence with probability one is guaranteed provided that the sequence of positive numbers qi, i =1,2;. ., and E { ~ ~< 03 satisfied Dvoretsky's conditions [14]. Once the reference coordinates are updated, the variances are estimated from the newly formed learning sample [13]. The procedure of learning the pattern location is illustrated in Fig. 6. It is assumed that the pattern variation is slow enough for the algorithm to keep up with it in real time.

X. CONCLUSIONS This research focuses on the construction of an EMG pattern recognition system which provides precise identification of a motion and speed command fromthesynergisticallygenerated EMG signals. The system developed such functional capabilihas ties as high level of decision making, learning, and interaction with control environment through motion and speed prediction. The various schemes devised in this paper work together toward the enhancement of control precision, decision speed, and reliability and adaptation to environment which are vital to the success of prosthetic arm control. Also, this approach to EMG control of a prosthetic ann aimed at the effortless control and little training of the amputee seems advantageous over other techniques that require considerable training and effort to generate mostly uncoordinated motion of the arm. x Further research is recommended to complete the design of the ~ ~ ~ } coordinator in relation with the subsystem control and the consideration of a real amputee as a subject to investigate any differences in pattern formation or any other physiological and psycological treatment required for the practical success [21]. It would also be valuable to investigate the effect of training on the pattern formation.

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1984

Hierarchically Intelligent Control System

t
S e q u e n t i a l Bounded Neuromuscular Speed and Motion F'redictor D i r e c t Command Generator

I

I Subsystem1 ISubsysteml

lSubsy8

u

I

i

t

i

I

Fig. 7. Block diagram of overall procedure.

3

Coordinator

Bank of Subsystems

4 ISDecify system p a r a m e t e r s : s t o p p i n g r u l e 1 t-hreshbld- a t , m i x i m u m number of samples ,N , window r e g i o n t h r e s h o l d m t , e t c . be 0
STAR

crocomputer system four with microprocessors processing implementation of the algorithms.
APPENDIX A

for parallel

-

DERIVATIONLEMMAS 1AND 2 OF

S e t up t h e p a r a m k e r v a l u e s i n SPDFs: referncecoordinates xr. Variances 02, and dg a t u n i t SDeed.

The features IAV, variance, and zero crossings, represented by x , , x , , and x , , respectively,areobtainedfrom the measured EMG signal x ( ? ) by the following equations:

I

S e t k = 1. Take a sample x a n ds p e c i f y it i n t h e f e a t u r e s p a c e o$ I A V 2nd i n t h e f e a t u r e space o f v a r i a n c e a n d z e r o c r o s s i n g s . CalculEte likelihood probability H(m , x 1, r = l , . . , m , by u s i n g SPDPs.

I

. i=O N-1

x z = N - 1 q [ x ( i A t ) x ( ( i + l ) A r } ] , q ( x ) 6 ( 1; i f x < o 0; otherwise.

t

( W

t
[ C a l c u l aetc i smo n s u r e d e i ea
D(xk).

. .
.Y==

I

l A p p l yd e c i s i o nr u l ef o rc l a s s i f i c a t i o n . 1 Averagethesamples in the feature space o f IAV.

1

The fact that the amplitude of the EMG signal at time t , x ( t ) , is considered to have approximately Gaussian distribution with zero mean and variance u2cl, and theassumptions that the [l] sampling interval NAr is short enough to consider approximately same for all t , t E [0, N A t ] , and that the ergodlclty conditions are satisfied for IAV and variance result in the following equation:

Apply d e c o m p o s i t i o n r u l e t o s p e c i f y t h e speed of each primitive motion. A , B t Learning Loop

A

X , =E{

x 2 ( i a t ) }=u ; ,

(A51

Fig.

8.

Flowchart o decision procedure f

where the deterministic values x , and x , represent the features of a particular EMG signal x ( t ) . The Lemma 1 is then derived directly from (A4) and ( 5 . A) The investigation of the Lemma 2 starts with the definition of the success probability r,, i = 0,. . .,N - 1, such that
rI~P,[x(iAt)x{(i+l)Ar}