A GENERAL COMPUTING ANALYSIS OF HUMAN

Istituto

di Fisiologia

METHOD FOR THE LOCOMOTION*P

Umana dell*Universith di Roma. I Cattedra di Fisiologia P. zale dells Scienzc. WlS5 Rome. Italy

Gmana.

and TOMMGO LEO Centro di Studio dsi Sistemi di Controllo e Calcolo Automatici del CNR. Istituto di Automatica dell’l;niversitA di Roma. Via Eudossiana. 1%0015-i Rome. Ital).

and PEDOTTI

rbTON10

Centro

di Studio per la Teoria dei Sistemi del CSR.

Istituto

dl Elettrotecnica

cd Elettronica

dsl Polirscnico di Ylilano. P. zza Leonardo da Vinci . 32 20141 Nilano. Italy Abstract-This is a study of a method for obtaining a mathematical description of the most significant variables concerning kinematics and dynamics of human locomotion. The method is characterizsd by a typical approach of system theory. It consists of an anal) tlcnl procedure for the processing of data obtained by the most common experimental techniques in this area. i.e. photograph! for recording the movement. and the force platform for measuring the ground reactions. The proposed mathematical algorithms have been designed to obtain the time functions of the variables cited above in an analytical form. and a measurement of the relative indefiniteness induced

by e.xperimental errors. The results of the latter allows a critical evaluation of the experimental procedure. In particular. a maximum limit of rhe number of experimental data to be taken. can be evaluated. The subsystem of the locomotor apparatus which the present work deals with. is the lower limb. The method presented can be applied successfullv fo all of the gaits having a constant msan direction .. of progression and periodic time patterns. I. IUTRODLCTIOU

The stud! of human locomotion bepdn with scientific quantitative methods after the 1860’s. when photographic technique was developed. Muybridge was the first to publish photographic sequences of human locomotor movements taken in different conditions. applying a systematic method suitable for measuring. The result seems. however, to be more artistic than scientific. Mare]; (1885). Braune and Fisher (1898) and. later on, Bernstein (193). improved this technique and obtained very significant results on the mechanism of human locomotion, in particular level walking. concerning its kinematics and very partial data of its dynamics. These latter data Lvere obtained from the former through calculus. .4 fundamental improvement in the analysis on the forces involved in locomotion came from the force platform of Elftman (193-t). with which the resultant of ground reactions and the position of the relative point of application was first measured. Subsequent fundamental works are by Murray rt nl. (196-l) and by Breslcr and Franks1 (1950) which provide a numerical description of temporal. kinema* Recrwd +This Sazionale

16 Jltl_~ 1971. ivork was partially dclle Ricerche.

supported

by

Consiglio 307

tic and dynamical factors of level Lvaiking. These data are relative to only a few subjects. in particular that concerning dynamics. The procedure used to gather the data is very costly in terms of man:‘time. The problem of describing the phenomena involved in locomotion has also been approached from a theoretical standpoint essentially based on modern system theory (Chow and Jacobson, 1971: Beckett and Chang. 1968; Nubar and Contini. 1961; Vukobratovic and JuriK. 1969; Cappozzo and Pedotti. 197.3. etc.). The aim of these works is. in broad terms. the simulation of the control mechanism of muscular forces and. hence. of movements. by guessing the strategy of performance of the central nerh-ous system. New technology allows for the improvement and the development of different aids for locomotory handicapped. In designing the aids a deeper knowledge of movements and forces involved in locomotion, and of the relative control system. are demanded by both an experimental and theoretical aspect. In addition, as stated by Bernstein (1967). the study of locomotion, because of its special characteristics, provides very extensive and interesting material for investigating the processes involved in movement in general from a physiological standpoint. Regardless the field of investigation and application. an exhaustive biomechanical description of the

30s

A. CAPPOZZO.T. LEO and A. PED~TTI

mechanical system. of the forces and movements involved in locomotion. are of fundamental importance. Differsnt locomotor acts performed by subjects having ditfirsnt factors (sex. age, height and weight. etc.) must be analysed bvith rigourous scientific methods and described in mathematical terms. Particular attention must also be paid to pathological gaits. A large number of measurements have to be taken. The experimental data must enter an analytical procedure to obtain the values of variables not directly measurable. Both experimental and analytical procedures have to take a reasonable amount of time and must result in numbers lvhose reliability can be controlled. In this context. the aim of this work is to provide a method for obtaining an analytical description of the most significant variables concerning kinematics and the dynamics of human locomotory acts which have a constant mean direction of progression and periodic time patterns. The method has been designed to meet the requirements cited above, namely economy from the point of view of man/time and of the possibility of controlling the reliability of the results. The method consists of an analytical procedure for the processing of the data obtained by the most common esperimental techniques in this area, i.e. photography for recording the movement and force platform for measuring the ground reactions. The proposed mathematical algorithm allows the analytical representation of the kinematic and dynamical variables concerning the movement and a measure of the relative uncertainty induced by experimental errors. This latter result permits a critical evaluation of the experimental procedure. In particular, a maximum limit of the number of the experimental data to be taken. can be evaluated. Beyond this limit, in fact. no useful information can be added. The sensitivity of the method to variations of each experimental variable and of each parameter of the mechanical system, is also discussed. The subsystem of the locomotor apparatus which the present work deals with, is the lower limbs. As will be discussed later, the analysis is concerned with one single lower limb; in the case of asymmetric gait (typically, pathological gait) the analysis must be carried out for both the lower limbs independently.

In reality. the angular displacement of the body segments have a direct relationship with the resultant moments of the muscular forces regarding each joint. Such a relationship describes. in simple form. the more complicated relationship between the movements of the structure and the single muscular forces; it constitutes a suitable mathematical model of the system lower limb. which can be applied in studies relating to the coordination and control of mok-ement mechanism, and to various practical problems of functional rehabilitation. By focusing our attention on the system theory aspects of the problem the following considerations can be emphasized: (1) The system under study (Fig. 1) has as input variables the external forces and the muscular moments. and as output variables the ones concerning the movement. The moments at the articulations cannot be directly measured. all the other variables are measurable and are affected by measurement errors. (2) The mathematical model of the mechanical system is a set of non-linear differential equations, the parameters are known only in a statistical manner: that is. they have to be defined by means of a nominal value and a range of variability about this nominal value. According to these considerations. rhe problem of applied mathematics which we are facing is the reconstruction of the non-measurable input variables on the base of statistical knowledge of the remaining input variables, the output variables. and the parameters of the mathematical model of the system. This is a quite classical problem of mathematical estimation. by which various suitable methods of solution are available. Given the specific factors involved in our

H

xH.z,,

thigh

2. PRELI>IIN.-\RY

COSSIDERATIOSS

The present analysis deals with the lower limb during a locomotor act. The lower limb is an articulated system of links, in dynamical equilibrium under the action of external forces performed by the rest of the body through the hip joint. by the ground reactions, inertia. gravity forces. and the action of internal forces performed by the muscular contraction. Under the action of these forces the links undergo angular displacements about the joint axes. determining the overall translatory displacement of the body. which characterizes locomotion.

A bOf

Fig. I. See text.

;\ general computing method

309

In a steady state gait the functions q, (r). i = 12.3. case. we have approached the problem in two steps: (a) research of the best fitting functions of the kine- are periodic with period Tequal to the duration of matic variables by applying optimization methods of a double step. The coordinate of the hip s&r) and. for gaits on least squares kind: (b) reconstruction of the input variables by using sloping terrains or on stairs. also zH (t). can be usefully the preceding results. This problem is simplified by divided into two terms: the fact that the equations of the mathematical model. S”(f) = K,.t + SH(f) (I) when resolved with respect to the moments. are algeIn(f) = K,.t + T”(C) (2) braic and not differential equations. The method is then a quasi-optimum method. -H(T) -H(O) and ,S, = Another important problem related to the bioT mechanics of locomotion, concerns the forces transmitted by the surface of joints. In reality these forces the functions ,Yri(r)and Y-H(t)are periodic. By considering the functions ~,.t~~.r7~.sH and ZH have two components: one due to external forces and the other due to muscular forces. While the former periodic. we can assume that the time patterns in can be evaluated with sufficient accuracy, by using locomotion are reproducible. These time patterns. in the results obtained as briefly illustrated above and fact. are very likely to be affected by ‘>erky” comby applying simple algebraic equations. the latter can ponents due to a continuous regulation of the very only be estimated through large assumptions on the sensitive balance of the whole body. Nevertheless. the range of these components is below the limits of sensistructure of the muscular system and through the LW of direct measurements of the state of contraction of tivity of the experimental system adopted. and thus entirely concealed by the measurement errors. each muscle (Paul. 1967; Morrison, 1969). The lower limb can be described mechanically by projecting it into two orthogonal planes (the sagittal Experimental techniques based on photography. and frontal planes) and then carrying on the analysis widely applied in biomechanics, allow a measurement independently of the two projections. of the values assumed by the functions a,(t). ski(t) and Both displacements and forces have their prevalent Z”(I) in m known equally spaced instants of time durcomponents, on the sagittal plane (Bresler and Frankel. 1950). The method. which is the subject of this ing the double step period. The displacement functions are then. sampled at paper. is presented with reference to these prevalent a sampling rate equal to m/T, which is the number components. Obvious transformations of coordinates make the algorithms directly referrable to the com- of frames per second of the cinecamera. The actual value of the sampling rate to be chosen ponents lying on the frontal plane. A three-link mechanical model of the projection of the lower limb is worthy of a separate discussion. It is. in fact. possible to find a value of it which is optimal from a on the sagittal plane is used (Fig. I). practical point of view. This will be done when a The lengths of the links are assumed to be constant. deeper knowledge on the characteristics of the disdespite the fact that the distances between the projection on the sagittal plane of the joints centers vary placement functions and their measures. is available. These sample values are naturally influenced by induring the motion. This assumption will be discussed later and its effect on the reliability of the results will accuracies of different origin: the markers borne by be evaluated. the subject indicate only approximately the position The mechanical model which would have eight of the rotation axes of the joints and undergo displacedegrees of freedom, as a consequence of the above ments of the skin in relation to the bones during assumption. is assumed to have only five degrees of motion; the optical system of the cinecamera can infreedom. troduce errors: the inaccuracies caused by the reading of the film must also be included. The measurement of the generic sample of the generic displacement function ci(.) [r~,(.).r~~(.),t~~(.)..x~.).:J.)]can be written For the description of kinematics five free co- as: ordinates must be associated to the five degrees of 1i: = d, + 6’k k = I. . . 1f1. freedom of the mechanical model. where The chosen coordinates are shown in Fig. 1. With dl, = d&T,) actual value of the sample of the funcreference to these coordinates, kinematics will be detion ci(t) at the time kT, (T, = samscribed by the following vectors of functions of time pling interval) d: = d*(kT,) measure of the sample d(kT,) displacements vector error on the measure of d(kT,). ek = e&T,) velocities vector accelerations vector

The error e. induced by multiple and independent causes. can be hypothesized to be random with a normal probability distribution. In addition. one can assume it undergoes independent increments; in other

A. C.~PPOZZO,T. LEO and A. PEDOTTI

310

words the error on a sample does not depend on the errors on all the other samples. .Lfathenmrical procrdw

It is imperative to design a method for obtaining the best interpretation of the measured data. Statistical measurements theory provides mathematical tools which allow the estimation of an analytical formulation of the function d(.) from the measured samples d*,, minimizing the effect of the experimental errors (best fitting). A measure of the accuracy of such an estimation is also provided. The theory cited above suggests that the estimation of the function d(.), Eld(.)(. has to be sought within a set of functions Rz belonging to a more general set R characterized by being a series of elementary functions whose number varies within the set itself. Since the function d(.) (which, for the purposes of this section will represent one of the functions q i(.).~2(.),~J(.)..&(.).Zn(.))is periodic and must have all the properties of wellbehaved functions, the more convenient set of functions R is the one whose elementary function is a trigonometric one. Hence:

Assume that the sampled data dz are subject to a random error whose statistical characteristics do not change with time. Each sample d; can be seen as an observation from a normal population having a mean value r,(kT,) and variance G:. These assumptions are justified by the measurement errors characteristics cited above. Time is supposed to be known precisely: if an error in it is made. its effect will be taken into account in the error on the sampled data measures. Under these assumptions a method of least square can be used for the estimation of the parameters ro,rjSj and of the variance ~5 of r,(r). The variances and covariances of these parameters are also estimated (Carnahan rt al., 1969). The procedure consists in the minimization of the mean square error with respect to the parameters zO,rj./Ij That is min F = mi; It [r,(kTJ - d*(kT,)]‘. %.q.8, I”.X).I

The parameters that verify equation (5) are found by setting SF -= 1 sx, 0

”

d( .)eR =

r,(r) = ~0 + xj (xj sin Wjt i

+~jcoswjt);O~r~T;n=l,...,N,

1

I

ii r,(r) = z,, + z

[

0

El,=0

qo=O.

(6)

where j=

(xj sinwjt + /Ijcoswjt)

1 .

1. . . ..I1 2F

dF

(3)

where !V is a finite number. The determination of E[d(.)], estimation of d(.). develops along two levels: (a) Estimation of the more convenient set R,$R, that is determination of the number of harmonics Ii in the Fourier series expansion r,(t). R, =

(3

x0 I = -2- I l”=u” I,=U, L’,=h,.

By solving system (6) of 2n+ 1 equations in the 2n + 1 unknown parameters ao.aj.bj. the following solutions are obtained:

(4)

(b) Determination of the function t-,&which fits the data better. That is. estimation of the parameters ro,rj.flj which will be called a,,aj,bj Several facts pertaining to the choice of Rx must be underlined. The number of harmonics of the function d(.) can not be stated a priori. The measurement errors. superimposed on the sample data, make it impossible to evaluate the small amplitude (high order) harmonics with sufficient accuracy. The measurement errors entity is not known a priori. Disregarding the mathematical method used, a maximum of (m-1)/2(m = number of sampled data) harmonics can be estimated as stated by the sampling theorem. In these cases, the measurement theory suggests, the estimation of the parameters of each function r,(.), with n = l,...,m-l/2. and its inaccuracy. In this manner a pattern of results is obtained from which, as will be seen later, Tican be chosen by inspection.

(7) bj = i$sin[jc(k

- 1)l.d:

where c=-.

2Tr nl

The variance ~5 is estimated by calculating F in equation (5) for x0 = ao, Xj = aj and Bj = bj

sf =

$

I

(8)

{cosbc(k - 1)l.d:)’

+ i I

[sin Cjc(k - l)] . d$) ’ 11

211

A general computing method The variances of the parameters. according formulas given by Davtes (1957). are:

to the

var (‘Ii?) = 5’0 = ‘5 ItI 7 > var ( calculating the mean vltluc of R,: if it is equal to zero. the mo\zment is in a stead! stat2. In Table 1. where the indiv-idual characteristics of the IWO subjects are summarized. the assumed variances on the body parameters referred to the nominal values of the parameters are also shown. In Tables 2 and 3 and Figs. J-7 the input variables and the main results of the analytical method are collected with reference to both gaits considsrsd. In Table 2 the results concerning the best fitting procedure applied to the displacement experimental data are shown. Parameters of the best fitting functions. chosen according to the criterion illustrated in Section 3. are also indicated. by framing. The quality of the lit can be judged by referring to the variance estimation. framed for each function in Table 2. and to the plots vs time of best fitting functions and relative experimental data shown in Fig. 3. In Fig. 5 the time patterns of the ground reactions sagittal plane components and of the position of the relative point of applktion are shown.

Table 2. Best fitting paramctcrs

I

T

I

i_ L

/

315

316

A. CAPPOZZO.T. LEO and .A. PEDO~

‘srnNcE

STANCE

(a)

PHASE

SWING PHASE

(b)

Fig. 4. Plots vs time of the five measured kinematic variables: level walking (a), walking up stairs (b). Continuous line refers to the results of the best fitting; circles represent the measured values.

STANCE (a)

PHASE

STANCE

PHASE‘

(b)

Fig. 5. Plots vs time of vertical component (R,). horizontal component (R,)acting on the sagittal plane and abscissa of the point of application (s,) of the ground reaction; level walking (a). walking up stairs (b).

A general computing method

SWING

PHASE

ia)

STANCE

PtlASE

SWING

P+tASE

(b)

Fig. 6. Muscular moments (Nm) acting at the hip (&I,), knee (&I,) and ankle (AI,) joints. with their relative indeterminate assumed equal to Is,,.

In Fig. 6 the moments are graphed as calculated values and indicating the relative range of uncertainty. Such an uncertainty was assumed equal to twice the instantaneous value of s,,, (equation 17). i.e. the true value of the moments lies within the range of uncertainty with a probability equal to 98 per cent. The weight with which each homogeneous group of the input variables contributed to the moments uncertainties. is visualized in Fig. 7. Errors in determining the position of the structure, with respect to the vector of ground reactions, are the major determinants of the errors on the muscular moments. This position depends on the displacement variables. on the segment length and on the abscissa of the point of application of the ground reactions. The contribution of the body segment masses, moments of inertia. and location of the centre of mass uncertainty are negligible. The inaccuracies induced by the uncertainty on the ground reaction components R, and R, are of relatively less importance. On the basis of the results obtained in 13 tests on walking upstairs (Cappozzo and Leo, 1973) and 5

tests on level walking, the further following consideration can be made. When the speed of progression of the gait goes beyond certain values, depending on the locomotory act performed, the inaccuracy on the moments is significantly affected by the acceleration variables errors. COSCLCSION

Even if in literature studies on human locomotion and the determination of some variables which define the gait are available, the method presented in this work and the results obtained take on a special importance. In fact. the problem has been approached from the system theory point of view making use of the biomechanical aspects of locomotion possible in a wider context concerning the analysis of the whole locomotory apparatus. The system analysed (lower limbs) is considered not separated from the measurement apparatus and data processing. In other words. it is not described in a deterministic way. but by taking into account the statistical properties of the variables.

A. CAPPOZZO,T. LEO and A. PEDCTTI LEVEL

‘AM !

WALKINO

Al

Fig. 7. Contribution to the moments uncertainties due ip the vectors displacement d, velociry d, acceleration d, ground reaction R, mechanical parameters p.

As far as the ‘man walking’ system is concerned, the most suitable choice of the model of the system and the variables necessary to describe its behaviour have been made. This choice made a model of the lower limbs possible which satisfied both the needs of exactness and detail, on the one hand, and simplicity, on the other hand, thus minimizing the inevitable lack of precision. The model chosen here takes both the features of the biological system and the phenomenon in which this is involved into account. The system was considered as a system of articulated links defined by parameters which were constant in time. At the same time, we took into consideration the approximation introduced by this assumption introducing the sensitivity of the system to the variations of the parameters and defining the statistical properties for each one of these. The problem has been faced and resolved as a problem of optimal estimation of the input variables of a system where the output, parameters, and their statistical properties are known. Another outstanding outcome of the results obtained was the fact that the variables describing the movement were obtained in analytical form. Especially, when considering the periodic nature of the movements, we used a representation in terms of

Fourier’s series. This enabled us to use a tLpica1 ‘best fitting’ method for the pre-elaboration of measured data. Furthermore, this representation. is important in itself and forms an important base for future study of locomotory apparatus. The patterns of the torques and the forces on the joints obtained by this method as ths other variables which define the movement (free coordinates and their first and second derivatives) take on a special importance because now they are joined to their statistical properties. In fact, for each of these variables, the nominal average pattern and the range of uncertainty was obtained. The range of uncertainty mentioned above includes all the patterns which one can possibly obtain by repeating the test in the same experimental conditions. The assertion is based on the repetitive nature which is typical of the conventional locomotory acts on the hypothesis regarding the statistical properties of errors. The experimental method used has the advantage of offering the possibility of maximum liberty to the subject who can walk freely. that is, without any apparatus attached to him and without using a treadmill. Another important result which is a direct outcome of the analytical method used, consists of the determination of the optimal number of frames necessary and sufficient to completely describe the behaviour of the system. This number, which is clearly inferior to that which is generally used in an analysis of this kind, permits a considerable saving of the time necessary to read the frames and of the time necessary for elaboration. At this point it must be emhasized that such analysis can be extended by simply modifying the deterministic model, to the study of a larger class of movements (all the periodic ones). In fact, the problem of describing the phenomena involved in movement. from a theoretical standpoint, essentially based on system theory. has, till now. been approached by using only the available deterministic data. This is indeed limiting. Qualitatively new results on these topics can be achieved. instead, by using, for the description of the system, the variables in an analytical form and with significant statistical properties. In light of the above considerations, the variables obtained in this work. with their range of uncertainty. can usefully form the base of a study, along mathematical lines, whose object is a deeper description of the locomotory apparatus on different levels. REFERESCES

Bcckett, R. and Chang, K. (1965) An evaluation of the kinematics of gait by minimum energy. J. Bimrchanics 1, 147. Bernstein, N. A.. Mogilanskaia. Z. and Popova, T. (1934). Techniques of‘Morior1 Srudirs (in Russian). cited by Bernstein (1967). Bernstein, N. A. (1967) T/w Coordinatiou md Rrgulation of ,\fo~emruts. Pergamon Press. Oxford. Braune, C. W. and Fischer. 0. (1598. 190-t) Der Gang des Menschen (Human Gait). In .-lbhand~wger~ der Sarchs. Gessellschoft

der Wisserlschaftm

21-28.

319

,A central computing method Bresisr. B. and Franked. J. P. (1950) The forces and moments in the leg during level walking. AS,LIE Trans. 71. 27 (Paper So. -%A-62). Cappozzo. .-\. and Leo. T. (1973) Biomechanics of Walking up Stairs. 1st Cl.S41-IFTo.~f.~f Symp. Theory and PracMY of‘ Robots umi .~firnipularors. Udine. Cnppozzo. A. and Pedott). A. (1973) Modelling biological systems by optimization methods. In .-Ldcnnces in C$erwrits umi Spsrems. Gordon & Breach, New York. Carnahan. B.. Luther. H. A. and Wilkes. J. 0. (1969) .-lpplird .Vumericid .Ilrthods. Wiley, New York. Chow. C. K. and Jacobson, D. H. (1971) Studies of Human Locomotion via Optimal Programming. Math. Bioscierlces 10, 239. Drillis. R. and Contini, R. (1966) Body Segment Parameters. Tech. Rept. 116.03. New York University. School of Engineering and Science. New York. Elftman. H. (1931) .A kinematic study of the distribution oi pressure in the human foot. .4naromical Record 59. 181.

Mare?. E. J. ( l8Y51L~I.Lferhode Graphiqur dnus /es Sciences ~.\-prrirnerlrr~l,s. Masson. Paris. Morrison. J. B. ( 1969) Function of the knee joint in various activities. Bio-.Ilrri. Emgng 1. 573. hlurrav. M. P.. Drouaht. A. B. and Korv. R. C. (1964) Walking patterns o?normal men. J. Bol;r Jhr Swg. (.;lj 46. 33% Muvbridge. E. (1901I T/w Hwwr~ Figure ill Marion. Chapman & Hall, London. Nubar. Y. and Contini. R. (1961) A minimal principle in bio-mechanics. 51ril. ,\fat/r. 5ioph.w. 23. 377. Paul. J. P. (1967) Forces transmitted by joints in the human body. Proc. .Cfech. Engng 181, 3. i’ukobratovic. &I. and JuriSic. D. (1970) Contribution to the synthesis of biped gait. IEEE Trcnrs. Bio-Med. Eilqt~g 16. I.

- mgd, sin B1 T i- m&i, cos & - m&1, sin 8: +

+ m$,l,l,coS(Q~

+ e2) -

- rn&l,r, sin(0, + &) - m,Zjzli + + m$,d,l,

C0ge2 + ed -

- m&d&,

sin(e,

-

t+g1,

+

8:)

-

e1 i

sin

+ .I[,, + + R,1, sin e1 - &I, COS e2

MH = Jji, + + m,d,. = s,l, + fl ECU; ,] = s; , = s;, + Sf2+ s;,.

(3)

20

A. CAPPOZZO,T. LEO and A. PEDOTTI

In relation to the forces transmitted at the joints, keeping in mind that mathematical relationships can be written only with reference to their components due to the external forces, the following equations can be obtained:

f,:

= f,:

t tj2d, sin .02 + Qjd, cos 02).

cos B,)

F,%,= R, - m&

+ tl,l, cos 0, - QIi,sin 0, - 8JScos02 . + @l, sin O2 + f3,d, cos f& - @dl sin &)

(1)

F,, = F,,,