ISSN 1812-5123. Russian Journal of Biomechanics. 2012. Vol. 16, No. 1 (55): 80–91
ABOUT TWO THE LEAST-SQUARES METHOD MODIFICATIONS FOR LOST DATA RECOVERY IN VIDEOANALYSIS SYSTEM BASED ON ACCELEROMETER DATA A.N. Bobilev1, Yu.V. Bolotin1, A.V. Voronov2, P.A. Kruchinin1 1
Faculty of Mechanics and Mathematics of Moscow State Lomonosov University, 1, Leninsky Gory, 119991, Moscow, Russia, e-mail:
[email protected] 2 Russian State Institute for Medical and Biological Problems of the Russian Academy of Sciences, 76 A, Khoroshevskoe Shosse, 123007, Moscow, Russia
Abstract. In this study, we discuss the lost data recovery in videoanalysis of human movement based on measures of two-component accelerometer. The algorithm uses mathematical model of movement and consists of two stages: identification of unknown parameters and proper stage of recovery of lost information. Both problems are reduced to solution of overdetermined systems of linear equations. Two approaches to recovery problem solution are compared. The first one uses traditional non-recurrent procedure of the least-squares method The second one uses the procedure of suboptimal algorithm of smoothing based on Kalman filtering. These algorithms are applied to recovery of lost values of the angle between the shin of a person and the horizontal.
Key words: mathematical model, videoanalysis sistem, accelerometer, information recovery, least-squares method, Kalman smoothing.
INTRODUCTION Human movement analysis based on videoanalysis systems is popular modern method [12, 15]. It is used in sports medicine, examination of patients with disorders of the locomotor system and in others fields. It is caused by relative availability of equipment and possibility to study rather pure movement (unlike contact tools which can have a pronounced effect on an diagnostic findings). However, at use of similar systems, there can be a series of technical difficulties. For example, during some time interval one of surface markers can be closed from a video camera objective when moving. Accordingly, on this interval, it is impossible to obtain values of one or several limb segment orientation angles. Earlier for the solution of this problem, spline interpolation or correction by the information on length of a limb segment [9] were usually used. The first way uses only the information on trajectories smoothness, and the second one has the big error by change of distance between markers. To eliminate the given videoanalysis system disadvantage, it is supposed to use data of various sensors which measurements are based on other principles [7]. In [6–8], the gap filling (lost data recovery) problem was discussed at the movement videoanalysis system by force plate measurements. For the problem solution, authors write down the complicated dynamic equations, considering many unknown parameters. It is essentially to limit a class of admissible movements. At the present time, availability to application of various measuring devices has made possible to set the same problem with use of other sensors. © Bobilev A.N., Bolotin Yu.V., Voronov A.V., Kruchinin P.A., 2012 Alexey N. Bobilev, Postgraduate Student of Department of Applied Mechanics and Control, Moscow Yuriy V. Bolotin, PhD, Professor of Department of Applied Mechanics and Control, Moscow Andrey V. Voronov, PhD, Leading Researcher of the Russian Academy of Sciences, Moscow Pavel A. Kruchinin, PhD, Associated Professor of Department of Applied Mechanics and Control, Moscow
About two the least-squares method modifications for lost data recovery in videoanalysis system based …
In [2], the lost data recovery method based on additional measurements of the twoaxial accelerometer was proposed. The present article is devoted to detailed description of algorithms of this problem solution for a plane-parallel movements. Movement of one of segments of a person, for example, the hip or the lower leg is surveyed. The main measurement is the angle between a segment and a horizontal, what is calculated under videoanalysis data. We simulate the situation when on some time interval the angle values are lost. SENSORS DESCRIPTION Procedure of the videoanalysis system of human movements is surveyed. Movements of a person are registered by means of video cameras. Markers under which image coordinates of points of a body are defined are put on characteristic points of the body of the person. All measurements of coordinates are recorded by computer and the researcher analyses the obtained trajectories or calculated segment orientation angles by these coordinates. Movements of the person are not complicated by any massive sensors or cables which often prevent to conduct full value researches, for example, at patients with musculoskeletal system disorders or any neurological diseases. In this article, the videoanalysis system using a video stream from one video camera is discussed. For gap filling problem solution, it is offered to use additional accelerometer data. The choice of the given device is caused by several factors. First, to date accelerometer is accessible to carrying out laboratory researches. Secondly, this sensor executed with use of technologies MEMS and Bluetooth, almost does not limit movement. Thirdly, use of accelerometer data does not demand introduction in model of the dynamic equations of human body movement and allows us limiting the kinematical equations for single segment Two-axial accelerometer measurements are projections of the specific force operating on accelerometer sensitive mass onto two orthogonal axes. The measuring device represents the easy small-sized sensor in the rigid case. Accelerometer fastening is made immediately on the body surface in a zone of the least mobility of the skin integument (Fig. 1). SEGMENT MOVMENTS EQUATION We consider problem of segment movement in plane and use model of single segment of a human body limb (the forearm, the upper arm, the lower leg or the hip). We simulate this segment by the inflexible body moving plane-parallel. We will consider plane-parallel movement of interval, connecting two fixed points O1 and O2 on the body. Interval position unequivocally sets segment position (Fig. 2). O1 and O2 are the fixing points of link markers near segment ends. The fastening place of sensor and its orientation are generally unknown. A is the position of sensitive mass of the accelerometer. We describe A position by distance l to O1, and angle β between O1O2 and O1A. Now, we take into account one of two sensitivity axes of the accelerometer. Axis orientation is set by angle α of its deviation from a perpendicular to O1O2. Length l and angles α and β are unknown. Expression for a projection of point A acceleration onto a sensitivity axis of the sensor will be written down from the theorem of addition of accelerations at plane-parallel motion [14]. sin ( aa l sin( ) 2 l cos( ) x1 cos y1 sin ) cos ( x1 sin y1 cos ) . (1)
ISSN 1812-5123. Russian Journal of Biomechanics. 2012. Vol. 16, No. 1 (55): 80–91
81
A.N. Bobilev, Yu.V. Bolotin, A.V. Voronov, P.A. Kruchinin
Fig. 1. Accelerometer fastening
Fig. 2. Limb segment model
Accelerometer indications on the chosen axis are not agreed with accelerations aa of this point. The scheme of work of modern devices also is complicated from the technical point of view. Essentially, their operating is described by “mass on a spring” [5] (Fig. 3). We write the equation of sensitive mass accelerometer motion in a projection onto the sensitivity axis: aa g cos( ) f ,
where f is specific elasticity force which is measured by the device. We consider uncertainties of the device in form of the multiplicative (1 ) and additive a errors. Expressions for accelerometer indications are f (1 ) f a (1 ) aa g cos( ) a .
Multiplicative error (1 ) contains also the components caused by sensor axis orientation errors owing to discrepancy of fixture. Error values are not constant, and change, for example, is depending on temperature. These changes occur slowly enough, and in experiment time intervals further will be neglected. Let us consider also hold-off effect in time of putting off of accelerometer and videoanalysis system indications. We will enter parameter of time delay s : f (t s ) (1 ) aa (t ) g cos( (t )) a . 82
ISSN 1812-5123. Russian Journal of Biomechanics. 2012. Vol. 16, No.1 (55): 80–91
About two the least-squares method modifications for lost data recovery in videoanalysis system based …
Fig. 3. Accelerometer scheme
As a result, the expression for accelerometer indications in terms of videoanalysis data looks like (t ) f (t s ) (1 )l sin( ) 2 (t ) (1 )l cos( )
(1 )sin x1 (t )cos ( y1 (t ) g )sin (1 )cos x1 (t )sin ( y1 (t ) g )cos a .
(2)
IDENTIFICATION PROBLEM The expression (2) contains constants a , , l , , , s , their values are unknown. Also, their determination with use of standard ways of anthropometrical measurements is complicated. It is necessary to use methods of identification of unknown parameters in the manner shown in [6]. Let us consider that s is small in comparison with characteristic time of movement. We will present f (t s ) in the left hand part of (2) in the form of the two first members of Taylor expansion f (t ) f (t ) f (t ) . We write new expression s
s
(t ) X 3 f (t ) X 1 2 (t ) X 2 x1 (t ) cos ( y1 (t ) g )sin X x (t )sin ( y (t ) g ) cos X X f (t ). 4
1
1
5
(3)
6
We use in (3) new designations for combination of unknown parameters a , , l , , , s : X1 (1 )l sin( ) ; X 2 (1 )l cos( ) ; X 3 (1 ) sin ; X 4 (1 ) cos ; X 5 a ; X 6 s .
Expression (3) is the equation for unknown components of vector X ( X1, X 2 , X 3 , X 4 , X 5 , X 6 )T . Derivatives of the measured values in this case are calculated by finite difference relations. The expression (3) has been written down for all moments of time t j ( j 1, , N 0 ) at which measured values do not contain failures. We combine these equations to overdetermined equations system AX B , ISSN 1812-5123. Russian Journal of Biomechanics. 2012. Vol. 16, No. 1 (55): 80–91
(4) 83
A.N. Bobilev, Yu.V. Bolotin, A.V. Voronov, P.A. Kruchinin
where B ( f (t1 ),..., f (t N 0 ))T , and components Aij of matrix A are equal to: A1 j 2 (t j ), (t j ), A2 j A3 j x1 (t j ) cos (t j ) ( y1 (t j ) g )sin (t j ), A4 j x1 (t j )sin (t j ) ( y1 (t j ) g ) cos (t j ), A5 j 1, A6 j f (t j ).
Here, (t j ) , f (t j ) are measurements attributed by the software to time t j . Vector X calculation by (4) is solved by modified least-squares methods [10, 11]. Effective procedure of identification in similar problem in detail is analyzed in [6]. PROBLEM OF LOST DATA RECOVERY For lost angle data estimation, also we use the equation (3). Now, we use this equation for time intervals with the incomplete data on marker O2 coordinates, and consequently also an angle is unknown. Values of combinations of parameters X i were obtained at a previous identification stage. We consider two modifications of procedure of lost information recovery for videoanalysis system. The first of them uses the non-recurrent procedure of the ordinary leastsquares method [6, 8]. The second one uses Kalman filtering procedure [11]. Ordinary least-squares method use Reduction of lost data recovery problem with unknown angles to linear equations system Ax B solution was analyzed in details in [6, 8]. We repeat here in brief the basic stages of this procedure. Let us consider that the data obtained by system during the moments of time ti (i 1, , N ) are lost. We denote , . We present the values , , in each time moment as the sum of the a priori estimation designated by an index “a”, and the small a . We linearize (3) relative , , . The deviation a , a , result is c1 (t ) c2 (t ) c3 (t ) b(t ) ,
(5)
where c1 (t ) X 3 ( x1 (t )sin a ( y1 (t ) g )cos a ) X 4 ( x1 (t )cos a ( y1(t ) g )sin a ); c2 (t ) 2a X 1; c3 (t ) X 2 ; b(t ) f (t ) 2a X 1 a X 2 X 3 ( x1 (t ) cos a ( y1 (t ) g )sin a ) X 4 ( x1 (t )sin a ( y1 (t ) g )cos a ) X 5 .
These values are the known functions of measured values x1 (t ), y1 (t ) , identified parameters X i and a priori estimated functions a (t ) . The equation (5) is correct in moments of sensor measurement ti (i 1, , N ) . We use finite difference to derivatives calculation 84
ISSN 1812-5123. Russian Journal of Biomechanics. 2012. Vol. 16, No.1 (55): 80–91
About two the least-squares method modifications for lost data recovery in videoanalysis system based …
(ti ) ( (ti 1 ) (ti )) / , (ti ) ( (ti 1 ) 2 (ti ) (ti 1 )) / 2 .
Eq. (5) can be written in form q1i (ti1 ) q2i (ti ) q3i (ti1 ) b(ti ), i 1,, N ,
(6)
where q1i c3 (ti ) / 2 ; q2i c1(ti ) c2 (ti ) / 2c3 (ti ) / 2 ; q3i c2 (ti ) / c3 (ti ) / 2 .
Here, values (t0 ), (t N 1 ) can be obtained, using the correct videoanalysis system data before and after a recovery interval. By analogy, we write similar equation for the second sensitivity axis: r1i (ti1 ) r2 i (ti ) r3i (ti1 ) d (ti ), i 1,, N .
(7)
Thus, the system of 2n linear equations (6), (7) with N unknown members (t1 ),..., (t N ) is obtained. Its matrix form is Ae x Be ,
(8)
where 0 0 ... 0 q21 q31 0 0 ... 0 q12 q22 q32 0 0 q 0 ... 0 13 q23 q33 ... ... 0 0 q1N 1 q2 N 1 q3N 1 0 0 ... 0 0 0 q1N q2 N Ae , 0 0 0 ... 0 r21 r31 r12 r22 r32 0 0 ... 0 r13 r23 r33 0 ... 0 ... ... 0 0 r1N 1 r2 N 1 r3N 1 0 0 ... 0 0 0 r1N r2 N
(t1 ) x ... , (t ) N
ISSN 1812-5123. Russian Journal of Biomechanics. 2012. Vol. 16, No. 1 (55): 80–91
85
A.N. Bobilev, Yu.V. Bolotin, A.V. Voronov, P.A. Kruchinin
b(t1 ) q11 (t0 ) b(t2 ) ... b(tN 1 ) b(t N ) q3 N (t N 1 ) Be . d ( t ) r ( t ) 1 11 0 d (t2 ) ... d (tN 1 ) d (t ) r (t ) N 3 N N 1
We solve system (8) by least-squares method [10, 11]: ~ x ( AeT Ae )1 AeT Be .
~ We have estimated values in time moments ti (i 1, , N )
(ti ) a (ti ) (ti ) . Iterative procedure is applied to accuracy rising. At the first step as a priori estimation on a restoration interval [t i , t N ] , value of linear function of time is used: a (t ) ((t N 1 ) (t0 ))(t t0 ) / (t N 1 t0 ) (t0 ). On each following interval as “a priori” value, the value obtained at the previous step is accepted. There is a need notice that this algorithm can be applied in a case when one of sensitivity accelerometer axes is taken into account only. For this purpose in matrixes A and B, it is necessary to leave only the first or last N lines. Kalman filtering use Other algorithm of an lost angle φ recovery by accelerometer data uses procedure of smoothing by means of R. Kalman filtering [11]. For use of this procedure, it is necessary to reduce a restoration problem to estimation in dynamic system xn 1 n xn Bn , zn H n xn rn. Here, xn is a discrete state vector, n and rn are dynamic disturbances and measurement errors, modeled by discrete white noises, zn are measurements, n and H n are known matrices. In this case, the problem solution probably is possible by means of the smoothing algorithm based on the Kalman filtering [1, 11]. There is a need make the assumption of solution smoothness, similar to that are accepted at signal smoothing for search of a casual field [3, 13], n1 n n , where n is discrete white noise. Noise intensity 2 is a parameter of the problem. We use finite difference expressions to obtain the equations where value of angles, angular rates and accelerations in n-th and n+1-th moments of time are interrelated by:
86
ISSN 1812-5123. Russian Journal of Biomechanics. 2012. Vol. 16, No.1 (55): 80–91
About two the least-squares method modifications for lost data recovery in videoanalysis system based …
(tn1 ) (tn ) (tn ) (tn )2 / 2, (tn1 ) (tn ) (tn ), (tn1 ) (tn ) n .
Here, is a discretization interval. We denote x(t ) ((t ), (t ), (t ))T . Here, xn ((tn ), (tn ), (tn ))T is value of x vector at point of time tn . Then, xn 1 n xn Bn , where 1 2 / 2 0 n 0 1 , B 0 . 1 0 0 1
We write formalized measurement zn H n xn rn with the aid of (3) for first sensitivity accelerometer axis as f1 f1 ( x) . Approximate value of function f1 in x xn is equal to its Tailor expansion in point x xna :
f1( xn ) f1( xna ) H n1( xn xna ) , f where H n1 1 x
T
a1n1, an21, an31, an41 , x xna
a1n1 X 3 x1 (tn )sin((tn )) ( y1 (tn ) g ) cos((tn ))) X 4 x1 (tn ) cos((tn )) ( y1 (tn ) g )sin((tn ))) , an21 2 X 1(tn ), an31 X 2 , an41 1. “Measurements” are equal to zn1 H n1xna f1 ( xna ) H n1xn rn1 and similarly for the second axis zn 2 H n 2 xna f 2 ( xna ) H n 2 xn rn 2 . White noise rn1 , rn 2 covariance will be called r . A priori point xna is Kalman filtering predictor result [1, 11]. Summarizing all above, we obtain the following formal problem: xn1 n xn Bn , M [n m ] 2nm ;
2r zn H n xn rn , M [r r ] 0 T n m
0 nm ; 2r
1 2 / 2 0 n 0 1 , B 0 ; 1 0 0 1
ISSN 1812-5123. Russian Journal of Biomechanics. 2012. Vol. 16, No. 1 (55): 80–91
87
A.N. Bobilev, Yu.V. Bolotin, A.V. Voronov, P.A. Kruchinin
a1 H n 1n1 a n2
an21 an22
an31 an32
T
an41 ; an42
zn ( zn1 z n 2 )T , rn (rn1 rn 2 )T . The Kalman filtering algorithm is applied to construction of estimation with the minimum dispersion of an error in direct and reverse time. Initial and final values of state vector x and covariation matrix P are set proceeding from value before and after a restoration interval. The measuring systems obtained under sensors data are:
x0 (0 , 0 , 0 )T , xN ( N , N , N ), p1 P0 PN 0 0
0 p2 0
0 0 . p3
Estimated values ~ xN and N are the result of optimal smoothing algorithm on the fixed interval as in [1, 11]. EXPERIMENTAL TEST OF THE RECOVERY PROBLEM SOLUTION The developed algorithms are tested on results of the exploration spent in Russian State Institute for Medical and Biological Problems of the Russian Academy of Sciences. The light-reflecting markers (reflectors) have been pasted on the lateral projections of joints rotation centres of the right leg. Patient made movements (knee-bend type). The videoanalysis system “Videoanaliz–2D” registered movements by a video camera with synchronous of twoaxis accelerometer indications. We use accelerometer from biomechanical equipment MuscleLab 4000e, firm Ergotest (Norway). It is placed on the lower leg as shown in Fig. 1. Movement recording frequency is 50 Hz. Accelerometer recovering frequency is 100 Hz. We use even accelerometer indications only. It is spent three records on six–seven knee-bends for one patient. By results of measurements, the coordinates of markers are calculated. Videoanalysis data accelerometer data for one of records are presented in Fig. 4. The analysis of structure of a accelerometer signal shows an appreciable oscillatory component at ~10 Hz frequency. It is most likely a consequence of a muscular tremor. To minimize an error caused by influence of the tremor and an error of discretization of the data, we use preliminary smoothing of data by Hann window with width 1 sec [4]. The width of a window is caused by characteristic frequencies of patient movements from 0.2 to 0.5 Hz. Identification of unknown parameters is made for each of accelerometer sensitivity axis according to the algorithm stated earlier. The analysis of dependence of identification results on length of identification time interval has shown that the time interval 5 sec, comparable with time of full knee-bend, allows us to identify well enough unknown components of the vector X. The length augmentation of this interval leads to change of component X estimations on 5%. At augmentation of length of an interval over 15 sec, the condition number for the solved problem does not change.
88
ISSN 1812-5123. Russian Journal of Biomechanics. 2012. Vol. 16, No.1 (55): 80–91
About two the least-squares method modifications for lost data recovery in videoanalysis system based …
Fig. 4. Example of videoanalysis data and accelerometer data
The obtained identified parameters are used further in recovering problem. In Fig. 5, we show the results of the algorithm use and errors for a characteristic interval. Parameters of Kalman filtering as a result of selection are accepted equal to 100 rad sec 2 , r 10 m sec 2 . In procedure of with use of least-squares method, it was made two iterations. From the second iteration, the standard deviation is invariable and equals 10–6 rad. Therefore, we execute only two iterations. The dependence of standard deviation of recovered data estimation on length of an time interval at the fixed left end are presented in Fig. 6 Apparently, the order of an estimation error remains at augmentation of an interval about several seconds. On the average, standard deviation of estimation error for time interval in 1 second for least-squares method is 1.06·10–3 rad, and for procedure of Kalman filtering it is 1.27·10–3 rad. Changes of standard deviation of an error from record to record are not observed. The result of Kalman smoothing not always appears more precise algorithm comparing with least-squares method. Nevertheless, the order of an error for both methods coincides, and accuracy of estimation has appeared reasonable, since for videoanalysis an instrument error of determination of an angle is about 1°.
ISSN 1812-5123. Russian Journal of Biomechanics. 2012. Vol. 16, No. 1 (55): 80–91
89
A.N. Bobilev, Yu.V. Bolotin, A.V. Voronov, P.A. Kruchinin
Fig. 5. Estimated angle values and errors of estimation in interval from 6.5 till 8.5 sec with using of ordinary least-squares (LS) method and Kalman filtering
Fig. 6. Standard deviation of gap filling for different time interval length. For notations see Fig. 5
CONCLUSIONS We analyze the problem of data recovery (gap filling) for videoanalysis system based on the additional measurements of the two-axes accelerometer. It is discussed two algorithms of data recovery: the first one is based on the method of least-squares, the second one uses Kalman filtering. A necessary stage of proposed procedure is model parameters identification. 90
ISSN 1812-5123. Russian Journal of Biomechanics. 2012. Vol. 16, No.1 (55): 80–91
About two the least-squares method modifications for lost data recovery in videoanalysis system based …
In the case of application of the method of least-squares, the use of several iterations allows us to improve accuracy of estimation. Thus, for experimental data, the restoration error has on the average appeared within accuracy of measurements of the videoanalysis system. Discrete Kalman filter use allows us to consider smoothness conditions, simplifies the computer calculations of restoration and will allow using effectively further procedures of stochastic estimation measures [12] at the solution of more difficult practical problems. However, selection of covariance of the model noises is uncommon and the problem solution is complicated. It is necessary to notice that these covariances remain invariable for all tests spent simultaneously. Their selection can be carried out on the test movement spent in the beginning of a series of tests. ACKNOWLEDGEMENT Work is executed with partial support of the Russian Fund for Basic Research (Grant 09–01–00809). REFERENCES 1. Alexandrov V.V., Boltyanskii V.G., Lemak S.S., Parusnikov N.A., Tikhomirov V.M. Optimal control of motion. – Moscow: Fizmathlit Press, 2005. – 367 p. (in Russian). 2. Bobilev A.N., Voronov A.V., Kruchinin P.A. Restoration of lost readings of videoanalysis movement system with using accelerometer measurements // All-Russian Biomechanical Conference “Biomechanics 2010”. Thesis. – Saratov: Saratov State University Press, 2010. – P. 46–47 (in Russian). 3. Bolotin Yu. V., Yurist S. Sh. Suboptimal smoothing filter for the marine gravimeter GT-2M // Gyroscopy and Navigation. – 2011. – Vol. 2, No. 3. – P. 152–155. 4. Hamming R.W. Digital filters – New Jersey: Prentice-Hall, 1983. – 221 p. 5. Ishlinskii A.Yu. Applied problems of mechanics. Vol. 2. – Moscow.: Science, 1986. – 331 p. (in Russian). 6. Kruchinin P.A., Mishanov A.Yu. Estimation measures in problem of data recovery of human movement videoanalysis by support normal reaction measurements // Russian Journal of Biomechanics. – 2008. – Vol. 12, No. 3. – P. 58–73. 7. Kruchinin P.A., Mishanov A.Yu. Using of mathematical models of human movement at treatment measuring data in biomecanics // “Physics and Radio Electronics in Medicine ans Ecology. FREME 2008”: 8-th International Scientific and Technical Conference. – Vladimir, 2008. – Book 1. – P. 259–262 (in Russian). 8. Kruchinin P.A., Mishanov A.Yu., Saenko D.G. About abilities of common treatment of reading of videoanalysis movement system and stabilographical platform // Mathematical modelling of human motions at norm and at some kinds of pathology. – Moscow: Moscow State University Press, 2005. – P. 28–53 (in Russian). 9. Kuo A.D. A Least-Squares Estimation Approach to Improving the Precision of Inverse Dynamics Computations // Journal of Biomechanical Engineering. – 1998. – No. 2. – P. 148–159. 10. Lawson Ch., Hanson R. Solving Least-Squares Problems. – New Jersey: Prentice Hall Inc., 1974. – 232 p. 11. Maybeck P.S. Stochastic models. Estimation and Control. – New York: Acad Press, 1979. – 291 p. 12. Perry J. Gait Analysis: Normal and Pathological Function. – New York: McGraw Hill Inc., 1992. – 432 p. 13. Stepanov O.A., Blazhnov B.A., Koshaev D.A. Investigation of efficiency of using of satellite measurements at gravity determination in a aircraft // Gyroscopic and Navigation. – 2002. – No. 3. – P. 33–45 (in Russian). 14. Targ S.M.. Short Course of Theoretical Mechanics. – Moscow: Higher School Press, 1986. – 416 p. (in Russian) 15. Voronov A.V. History of biomechanical video filming // http://www.biosoftvideo.ru/article/ (in Russian). Received 20 February 2012
ISSN 1812-5123. Russian Journal of Biomechanics. 2012. Vol. 16, No. 1 (55): 80–91
91
