Computer-aided motion synthesis with movement controlled in respect to kinematic status of biomechanical system Текст научной статьи по специальности «Медицинские технологии»

COMPUTER AIDED MOTION SYNTHESIS WITH MOVEMENT CONTROLLED IN RESPECT TO KINEMATIC STATUS OF BIOMECHANICAL SYSTEM

V.I. Zagrevsky, professor, Dr.Hab. O.I. Zagrevsky, professor, Dr.Hab. Tomsk state university, Tomsk

Key words: computer aided motion synthesis, biomechanical system, control.

Introduction. In literature of last decades, related to training process in the sport, it has become conventional to use the term "exercise technique" in the context of the athlete's way of solving the motor task. In this sense, we agree with Yu.K. Gaverdovski [2, P. 94], who believes that "The exercise technique is a biomechanical way of individual solution of the motor task."

Since within the conceptual framework solution of a motor task presupposes division of the exercise into logically relevant blocks, the use of the structural-parametric approach in the biomechanical analysis of the technology of sports exercises is consistent and vital. The structural-parametric approach supposes dividing an exercise into constituents (periods, stages, phases) [1, 2] and justification of the capacity of increase or decrease of the key parameters of biomechanical characteristics, intended to design of the rational technology of the gymnastic exercise being learnt.

In connection with this agenda, the problem of predicting the outcomes of preliminary structural and parametric adjustment of the exercise technique with the use of mathematical modeling of the athlete's movements is put forward. In general, this problem is transformed into the task of computer aided motion synthesis with transition of the simulated biosystem from a given initial biomechanical state to the desired final state.

The purpose of the study was to make a computer aided synthesis of athlete's motions based on the consecutive transition of the model-based system in single phases of the exercise from the given initial biomechanical state to the final required state.

Objectives of the study.

1. Determine the extent of the problem elaboration of computer aided synthesis of gymnast's motor actions with the given kinematic properties of the biosystem.

2. Develop the mathematical support for the computer aided motion synthesis with the realization of the idea of transition of the simulated biosystem from a given initial biomechanical state to a desired final state.

3. Perform a computer synthesis of a gymnastic exercise "Tkachev catch" on the crossbar.

Materials and methods. The research methods involved to obtain the purpose of the study were: computer modeling by a user of boundary positions of transition from one phase of the exercise to another; computer aided synthesis of athlete's motions in specific phases of the exercise and along the whole trajectory of the biomechanical system.

The first of the methods was used to construct the boundary positions in separate phases of Tkachev catch,

based on a subjective visual presentation of the technique of simulated exercise.

Realization of the second method was carried out based on the automated construction of a mathematical model of the motion synthesis by means of a computer. Results and discussion.

1. Conceptual foundations of a motor program of the motion control. Theoretical analysis of the literature shows that to solve the motor task, the system must have certain biomechanical properties reflecting the biomechanical state of the athlete, manifested as on the entire trajectory and at the individual positions. In this case, an indication of the required values of biomechanical characteristics of motor actions defined for the end time or the end position (and possibly the entire trajectory of the biosystem), describing the quality of a movement - is a goal of a movement. [3]

The transition from one biomechanical state of the biosystem to another by means of the control functions or the programmed functions is, in the synthesis of the human movements on a computer, the programmed control. Thus, the motion program includes at least two components:

- Goal of the movement;

- Programmed control.

Defined goal of the movement is achieved by the control forces (internal and external). The analysis of the equations, describing the purposeful athlete movements in the conditions of the support [3], shows that the external forces act as moments of gravity force for individual links of the model, and their resultant - as overall center of masses of the biomechanical system. Hence, the result of the external forces manifestation is a function of:

- the position of the links of biosystem in the Cartesian coordinate system;

- the biomechanical system configuration.

Since the magnitude of the moments of gravity force of athlete's body parts is derived from the trajectorial position of the links of biomechanical system, we can control the magnitude of the moment of gravity force purposefully only by changing the configuration of the biosystem.

The analysis of other structural components of the equations of purposeful movements of the person shows that their left-hand member includes, as the unknown functions of time, the generalized coordinates, the generalized velocities and the generalized accelerations of the links of biomechanical system, and their right-hand member - the control moments of muscle forces. These functions describe all the possible system properties.

Control forces, necessary for achievement of the goals of the movement, are in this case the programmed forces. As already stated, the control forces may be both external and internal. In particular, it is appropriate to include into the internal programmed forces the control moments of muscle forces in the joints of the athlete.

Programmed motion control, realized at the kinematic level, should limit the changes of the generalized coordinates and their time derivatives in accordance with the goal of the movement and solving the motor task.

Since the kinematics of purposeful movement is formed via equations of programmed control, the equations of the goal of the movement defined on the kinematic level also determine the motion program. 1. Boundary moments of the transition from one phase to another can be considered as the boundary postures characterizing the movement coordination [1]. In this case, within the computer modeling of the

exercise techniques the most informative criteria for determining the phase composition of the exercises may be the extremes of control functions (the programmed control), and the initial and final working postures of the studied exercises [4].

2. The mathematical support for the computer-aided motion synthesis included the consideration of constructing the mathematical models of the motion of biomechanical system and the methods of programming the motion control of a person at the kinematic level.

Model of the musculoskeletal system of gymnast's body shown in Figure 1.

Here: hands are the first link, the body with the head - the second link, feet - the third link. During the exercise an athlete doesn't lose contact with the support, for example, with the crossbar. So, we put athlete's hands at the beginning of the fixed coordinate system Oxy (Fig. 1, A), and combine this system with the end of the crossbar.

For accepted model, let's introduce the following notation (Fig. 1, B): Lt - length of i-th link; Si - the distance from the axis of rotation of the i-th link to its center of mass; pt - inclination of the i-th link to the axis Ox (generalized coordinates of the i-th link); pp, - the angular velocity of the i-th link; p, - angular acceleration of the i -th link; i - letter index used to identify the number of the link (i=1,2,..., N); N -quantity of links of the model. Due to the fact that we took pi for the generalized coordinates of the adopted biomechanical system, pp i and p i will respectively denote the generalized velocity and generalized acceleration of the i-th link. U1, U2 denote the kinematic programmed control in the shoulder and hip joints (Fig. 1, C), which at any moment of time (t) is determined by the difference of the generalized coordinates of biomechanical system.

U1 =(2 -P^, U2 =(3 -(2. (1)

To denote the mass-inertial characteristics of the three-linked model of the musculoskeletal system of athlete's body, let's introduce the following identifiers: Pi - the weight of the i-th link; mt - mass of the i-th link; J] - the central moment of inertia of the i-th link.

Calculational model of the unbranched multilink biomechanical system with rigid ties in the conditions of the support. In compact notation the formular expression of the motion equations of the unbranched multilink biomechanical system, presented in the form of Lagrange equations of the second kind, is given by [5]:

f j cos(0,-fr)-f j/sinl;) + Y cos 4 = Mi - M+1, j=1,..,N, i=1,..,N. (2)

j=1 ,=1

Here Aj]. - dynamic coefficients for the individual anthropometric and weight characteristics of athletes,

determined by the kinematic and mass-inertia characteristics of the model links and calculated by the formula [5]

Ay = 8 ij (Ji + mi Si2) + mj Li Sj (1 - 8 j) + ^ mkLiLj, j>i; если i>j, то AiJ- = А/,

k=j+i

i = l, 2, 3, ... , N; j = 1, 2, 3, ... , N. (3)

In equation ( 3) an 8ij entry indicates the Kronecker symbol. Kronecker symbol equals to

fl, если i = j, 8i i = i

[0, если i Ф j.

The meaning of the Yh coefficients , contained in the left side of the equations is that they are expressions for the generalized forces in Lagrange's equations and in compact notation are given by:

Y = ps +

k=i+1

(4)

Purposeful human movements described by the introduction in the right side of equations (2) of the control moments of muscle forces in the joints (M), recorded for the i-th equation of the system (2) as the algebraic sum of the items M - M+i, where

Mi+i ± 0, if i<N, and Mi+1 = 0, if i = N; M1 - the moment of friction force. (5)

Basic mathematical model of the motion of the biomechanical system constructed in the form (2), may be used in two ways. The first way is to calculate the control moments of muscle forces in the athlete's joints based on optical detection of movements. As for the second way, the basic model (2) can be used in computer-aided athlete's motion synthesis.

Mathematical model of motion synthesis of the unbranched multilink biomechanical system with rigid ties in the conditions of the support with programmed control at the kinematic level. Let's construct a mathematical model of the motion synthesis of biomechanical systems using the basic model in the form (2). Since the programmed control at kinematic level is given by the difference of the generalized coordinates (1), for N -link model the programmed control with its derivatives can be represented in formula Uz = (p2+i - (p2, Uz = <p2+i - <p2, U2 = &&z+i - (pz. z = ^A..^N - L (6)

Equation (6) can be transformed to

Pz+1 = P, + Uz, Pz+1 = Vz + Uz, Pz+1 = & + Uz. z = U^...N - i. (7)

Superimposed kinematic relationship (7) uniquely determines any of the generalized coordinates and its

first and second derivatives through unknown p1, Ppx,(& and programmed controls with their derivatives

p-1 p-1 p-1

Pp = P +Z Uz, Pp = (&1 +Z Uz, Pp = &1 +Z Uz. P = 2,3,..., N. (8)

z=1 z=1 z=1 V '

Let's replace the second derivatives of the generalized coordinates by time in the system of equations (2) with their relations in form (8). We will look for the solution with respect to the second derivative of the

first generalized coordinate by time. Then we perform further transformations, denoting the number of generalized coordinate as letter index j, and the number of the equation in (2) as letter index i. We sum all the equations (2) and obtain the equation equivalent to the original system of equations, from which we will find p\

N N N N N k-1

M1 - Z Yicos v + EE A Vj2 sin( v j- v) - ZZZ UUzAikcos( (pk- v)

i=1 i=1 j=1 i=1 k=2 z=1 /z-vx

(Pi =-—-. (9)

ZZAcos(v j-Pi)

i=1 j=1

Obtained model has no analytical solution and in the athlete motion simulation on a computer it is necessary to use numerical methods for the integration of equation (9). In our studies there was used numerical method of integration: Runge-Kutta fourth-order accuracy. Integration step was set equal to 0.0001 s.

Programmed control. For dynamic modeling of the individual phases of the movements we must solve the task of transition of the biomechanical system from a given biomechanical state to the final state for a period of time. For this purpose we construct the programmed control in the form of a polynomial function based on the approach outlined in [4].

As boundary positions, separating individual phases of the exercise, we consider those working postures, in which the extremum of the flexion-extension movements in the joints of the gymnast is attained. In these places, the articular flexion switches to extension (or vice versa), and the control has a maximum or minimum of the function and the value of zero of the first derivative. Since the right end of the trajectory of programmed control of considered exercise phase is initial for the left end of the control trajectory of the subsequent exercise phase, the initial data for the left end of the control trajectory of the subsequent exercise phase is accepted as the calculated values of the programmed control and its derivatives of the previous exercise phase.

Thus, the programmed control of the biomechanical system at the individual exercise phases can be described by a polynomial of fourth degree with 3 boundary conditions at the left end the trajectory of the programmed control and 2 - at the right end of the control trajectory. In this case, the equation of the programmed control ( p(t)), first (pP( t)) and second ( pP( t)) derivative with respect to time (t) from t0 = 0 to

t1 = h is given by:

P(t) = b0 + b1t + b2t2 + b3t3 + b4t4; pP(t) = b + 2 b2t + 3b3t2 + 4 b4t3; p(t) = 2 b2 + 6 b3t +12 b4t2. (10)

With 5-boundary conditions to programmed control, first and second derivative, we obtain five equations with five unknowns

P(i0) = K + ht0 + b2t^ + bj^ + bAt*; P(ii) = b0 + b1t1 + b2t12 + b3t13 + b4t14;

(pw = K + 2b2t0 + 3b3t02 + 4b4t03; pP(t1) = ¿1 + ba2t1 + 3b^ + 4b J^. (11)

P>(t0) = 2b2 + 6b3t0 + 12b4t02;

Solving system of equations by analytical method (11), we will find formular expressions for the 5 unknown coefficients bi

b0 = p0; b = &oi b2 = b3 = ^P^ _ _b,3-1^ + i2^. + % 12)

0 0 1 0 2 2 3 h3 h2 h 4 h4 h 2h

Here p0, p&0, &&0 - programmed control, its first and second derivatives at the initial moment of time t = t0,

and p , p - programmed control and its first derivative at the end moment of time t = t,. Since at the

right end of the control trajectory the restrictions on the second derivative are not imposed, then its value in the motion synthesis will be found from (10).

From (12) it follows that the numerical values of the b coefficients are determined not only by the restrictions on kinematics of the programmed control ( p0, Pp0, p0, p , p ), but also by the duration (h) of the considered exercise phase

h = t1 _ t0. (13)

The definition of h at the individual exercise phases is carried out in an iterative process, constructed on a gradual approach to its value, which meets the conditions of the restrictions imposed on kinematics of the programmed control on the left and right ends of its trajectory.

Technological aspects of the implementation of the reduction task. Let's introduce the notation. Let there be W allocated phases of the flexion-extension athlete actions in the joints within the analyzed exercise. Suppose k - a number of the image, where the image is considered as a kinetogram frame, delimiting exercise phases. Total amount of the boundary positions will be - P (k=1, 2, ... , P; P=1, 2, ... , W+l). Denote t - current time to the range from Tz to Tz+1 (z=0, 1, ..., W). Let also hz - the duration of the exercise phase between the boundary positions determined by the expression hz = Tz+1 _Tz, where z=0, 1, ..., W. In the computer implementation the value of Tz (the start of a separate exercise phase) was set equal to zero everytime.

Each kinetogram frame, delimiting exercise phases, corresponds the generalized coordinates of the biomechanical system, recorded in the form of two-dimensional array Qkii ( Fig. 2 - C), where k - a number of the image , i - a number of the link of the biomechanical system (k=1, 2, ... , P; i=1, 2, ... , N). Figure 2 -A shows the athlete's boundary positions, separating the support period of the gymnastic exercise "Tkachev catch" on the crossbar into individual phases, which correspond to specific numerical values of the array

Fig. 2. Synthesized trajectory of the biomechanical system (B) on the initial data of athlete's working postures (C) at the boundaries of the exercise phases (A).

Let's formulate the problem of athlete's motion synthesis from the given values of boundary positions on some parts of the trajectory of the biomechanical system. The trajectory of athlete's motion consists of several parts and their boundaries are given by the sequence of generalized coordinates (Q^) of the biomechanical system. Here k - a number of a point, splitting the exercises into separate phases (k=1, 2, ..., P), i - a number of the link of the biomechanical system, corresponding the number of a generalized coordinate (i=1, 2, ..., N), P - the amount of positions of the modeled biosystem. Let the k points correspond to time points T1, T2, . . . , Tp. Suppose also that we know the initial biomechanical state of the simulated system (Qki; Qk i; Qki) defined by the generalized coordinates (Qk t), generalized velocities

(Qki), generalized accelerations (Qki) of the links (i) of the athlete's body, at the points (k) of trajectory

partition, regarding the exercise phases (k=1, 2, ..., P). Here k - a number of an image of boundary positions in the area of separation of the exercise into phases. It requires a period of time hk to transit the biomechanical system from the initial state with biomechanical kinematic parameters (Qki; Qk i; Qki) to

the final state (Qk+u; Qk+1i; Qk+1i) on the parts of the trajectory (k, k+1), corresponding the individual

phases of the exercise. Our task is interpreted as a task of reducing, related to the controlling the final kinematic state of the biomechanical system by generalized coordinates, generalized velocities and generalized accelerations.

Computer implementation of the formulated task requires the construction of a mathematical model of the motion synthesis of the biomechanical systems in the form (9), the formation of the programmed control in the form of (10-12) and the iterative search of the h at the individual exercise phases in accordance with (13).

A point in time h of the synthesis completion of the k-th exercise phase was set on the basis of the conformity of the synthesized trajectory position of the model links by the generalized coordinates (j and the specified for this exercise phase sequence of generalized coordinates (Qk+1i). In mathematical

form, this condition can be written as

8i=abs| Qk+1,i - щЛ (14)

Here &i - the permissible deviation of the synthesized j from the given Qk+1., abs - absolute value. In our

studies, there was the hard condition for the convergence of the iterative process - si was equal to 0.1 degrees. The iterative process of determining h at the individual exercise phases is carried out in several stages.

In the first stage, the motion synthesis of the simulated system for the next exercise phase is performed with the unchangeable programmed posture of the model, received by the end of the previous exercise phase. Directly at the beginning of the modeling process the athlete's posture corresponded with the starting position, by the generalized coordinate Q1 i. The synthesis is terminated when the value of the

generalized coordinate of the first link, calculated at one of the integration steps, does not exceed the permissible deviation, set to 0.1 degrees, from the generalized coordinate of the first link in accordance with equation 14. The process of motion synthesis closure is taken for h and the bi coefficients, necessary for the formation of the programmed control at the next step of the iterative cycle, are recalculated. At the second step the movement is synthesized with the programmed control calculated at the first iteration step. Motion synthesis ends the same way as at the first stage, when the deviation of the

7

generalized coordinate of the first stage from the end position of the synthesized k-th exercise phase doesn't exceed the specified accuracy, ie s^absQ^ -

At the third stage, the iterative process of convergence was considered for all the programmed controls and it was implemented to meet the conditions as in the (14) for each generalized coordinate. Further, the synthesized generalized coordinates of the simulated biosystem and their first and second derivatives at the moment of time Tk+1=h were taken as the initial data for the synthesis of the subsequent exercise phase. 3. Computer synthesis of the exercise "Tkachev catch" on the crossbar in the computational experiment. Synthesized within the computational experiment trajectory of the biomechanical system, corresponding to the conditions of the task of reduction, is shown in Figure 2 as a sequence diagram (Fig. 2-B) with a time step of 0.04 s. The calculation was made to move with the initial angular velocity of the model links, which equals to:

Q11 = 1,63pad / c, Q12 = 1,56pad / c, Q13 = 5,27pad / c.

Structural and pedagogical analysis of the synthesized exercise shows that at the time of crossing the vertical position under the support by the center of masses (Fig. 2-B, frames 16-18) gymnast is bent in the shoulder and hip joints, which is a mistake of the performance technique of the "Tkachev catch". At this point, the athlete should be slightly bent over backwards in these joints that will be a contribute to more efficient throwing movement of the feet. In this regard, let's construct a new motion model, in which we additionally introduce the appropriate posture (slightly bent over backwards) at the moment when the gymnast body is crossing the vertical position under the crossbar. The rest positions remain unchanged. For the additional postures (Fig. 3 C, k = 4) we will define the generalized coordinates of the model links (in degrees):

Fig. 3. Synthesized within the computational experiment trajectory of the biomechanical system, with the control correction at the moment of crossing the vertical position under the support.

Performed computational experiments showed the possibility of modification of the gymnastic exercises technique with the construction of a variety of options for structural-parametric adjustment of the control actions, that meet the criteria of optimality of the motor tasks. In this case, for the control correction we used the possibility of dividing the exercise according to phase composition. Except the exercise phases, separated on the basis of alternation of athlete's active actions (under the "postures"), there are phases, associated with the frequency and the nature of the influence on the athlete's body of certain external forces or physical effects [1, 2]. Those are the phases of lowering and lifting during the swinging movements, etc. The introduction of additional posture under the number 4 (Fig. 3 C) separates the phases

of lowering and lifting and it is not unnatural in the description of the exercise technique. Conclusions.

1. The forecast of changes of the sport exercises technique, based on structural and parametric adjustment of their phase components, can be obtained on the basis of computer-aided motion synthesis of biomechanical systems.

2. As boundary postures, in area of separation the exercise into the motor phase structures, should be used:

- Initial and final working positions of an athlete;

- Boundary postures of an athlete in the area of the phase separation on the basis of alternation of athlete's active actions (extremes of the programmed control).

- Boundary postures of an athlete in the area of the phase separation on the basis of the frequency and the nature of the influence on athlete's body of certain external forces or physical effects.

3. The method of computer-aided motion synthesis, based on the simulation of the boundary postures at the individual exercise phases, can be very effective in implementing the idea of the controlling component of the orienting basis of action, based on the comparison of the subjective components of the motor program of movement control with its objective biomechanical components.

References

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3. Zagrevsky, V.I. Designing of the optimal technique of sports exercises via a computer experiment / V.I. Zagrevsky, D A. Lavshuk, O.I. Zagrevsky. - Mahilyow: A.A. Kuleshov MSU, 2000. - 190 P. (In Russian)

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