There is presented a model of human gait based on power developed by muscle joints in this paper. The mathematical model was presented for healthy subjects, patients with gonarthrosis and patients with Anterior Cruciate Ligaments (ACL). The regression function and the stochastic approximation function were used to the modeling. The differences between the values of model’s coefficients of people with these two pathologies have been compared to the model’s coefficients of healthy subjects. Proposed model is very accurate. The modeling could help to define the level of gait pathology and treatment in a large number of patient

J. Pauk

English

Pauk, J.

JVE International

 © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 APRIL/JUNE,   VOLUME 9,  ISSUE 2,  ISSN 1392-8716  108 283. Mathematical modelling – tool to diagnosis of human apparatus    Jolanta PAUK Bialystok University of Technology Automatics & Robotics Chair Wiejska 45C, 15-351 Bialystok PHONE: +48 85 7469222 E-mail: jpauk@pb.bialystok.pl  (Received 14 April 2007,  accepted 15 June 2007)             Abstract. There is presented a model of human gait based on power developed by muscle joints in this paper. The mathematical model was presented for healthy subjects, patients with gonarthrosis and patients with Anterior Cruciate Ligaments (ACL). The regression function and the stochastic approximation function were used to the modeling. The differences between the values of model’s coefficients of people with these two pathologies have been compared to the model’s coefficients of healthy subjects. Proposed model is very accurate. The modeling could help to define the level of gait pathology and treatment in a large number of patients.   Keywords: modeling, regression function, stochastic approximation function, human gait.  1. Introduction  Human walking is an example of a well-learned fundamental movement pattern that, in normal situations is performed with a great deal of coefficiency and consistency. Gait analysis and modeling has long been used as a research tool to develop complete understanding of the mechanics control of human movement. A standard physical examination can’t provide a complete description of pathology of abnormal human gait. Gait analysis and modeling can. The primary purpose of gait analysis and modeling in clinical application is to provide objective documentation of gait and increase understanding of the mechanism of pathological gait. Mathematical modeling is an important tool for quantifying normal and pathological patterns of locomotion. With this information, more informed decision making about treatment to improve gait function is possible. In this study will be presented the parametric methods for identification of human gait model. There can be many pages explaining what parametric identification method is in depth, but it can be explained as a technique that helps one to structure a mathematical model based on measured data by adjusting parameters within a given model until the simulated output coincides as closely as possible with the measured output. Using parametric identification methods, the dynamic properties of human gait can be identified. Recently, many interactive computer software and hardware packages have been developed that allowed further development of these methods. However, many researchers have attempted the simulation of gait by limiting their model with constraints [10,11,12]. Many of these take an analytical approach to the synthesis of motion, using trial and error, dynamic optimization, or other control schemes to develop the proper force time histories necessary to drive the motion. Onyshko and Winter presented an important model of locomotion in 1980 [1]. In this work, the issue of responsible modeling was stressed in regard to pure synthesis (forward dynamics). One of the few dynamic models of the knee joint was presented in 1988 [2]. The approach involved the development of a 3D model that was later reduced to a 2D model for analysis and simulation of motion. The skeletal model and the origins and insertions were taken from the work of Crownenshield and co-workers [3]. The synthesis of motion was described as a reduction of a second order non-linear system of ordinary differential equations to an algebraic system and then linearized. The tibia was subjected to several dynamic loading situations as different mathematical function shapes, and the resulting dynamic forces were computed. The contact points, joint orientation, and joint and ligament loads were reported, and showed that knee extension  283.  MATHEMATICAL MODELLING – TOOL TO DIAGNOSIS OF HUMAN APPARATUS.  J. PAUK    © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 APRIL/JUNE,   VOLUME 9,  ISSUE 2,  ISSN 1392-8716  109caused elongation of the medial and lateral collateral and anterior cruciate ligaments (ACL), but not the posterior cruciate ligament (PCL). It was also shown that the ACL carried the highest load. This study did not consider the effects of muscles or parametric changes such as the loss of a ligament or the manipulation of the geometry. A later study presented 3D, 7 segment models with revolute joints [4]. This model was driven by inverse dynamic forces and moments obtained from a gait analysis laboratory, and a forward solution were obtained using ADAMS dynamic modeling software. Ground contact was modeled as spring-damper forces on an ellipsoid foot to reflect the progression of the center of pressure. Torsional spring-dampers were placed at each joint to regulate the motion. The model was applied to the swing phase of gait, and joint moments were altered in order to obtain kinematic patterns similar to the original measured motion [4].  The goal of this work is to investigate the prospects of dynamic model as a tool for understanding and treating pathological musculoskeletal motion.   2. Material and method  In this study 45 healthy subjects (the average age 46 yr.), 37 patients with gonarthrosis (the average age 67 yr.), and 36 subjects with Anterior Cruciate Ligaments (ranged between 20 and 37 yr.) were evaluated. Subjects were recruited into Centre of Bioengineeria in Milan. The optoelectronic system Elite-3D was used for the measurements. The system is based on an on-line data processing of signals from a number of TV cameras. It is possible to recognize in the field of view of each TV camera those bright areas which are of interest for motion analysis. Quantization of biomechanical variables and spatio-temporal parameters of walking was performed by means of a computerized system for automatic acquisition of kinematics and ground reaction forces. A working volume 3 m long, 2.5 m high, and 1.2 m wide was calibrated by a precision grid, which was displaced in three different parallel planes. The resultant accuracy was assessed by measuring the movement of a special stick with three retro reflective markers placed on it. Pre-processing of raw data involved a tracking procedure, three-dimensional reconstruction of the marker’s coordinates, correction for optoelectronic distortion, and filtering. The frequency of acquisition was set at 50 Hz. All the subjects were analyzed with the S.A.F.Lo protocol. The location of the markers on each subject was the following: two on the Posterior Superior Iliac Spines, one on the Sacrum Bone, two on the Lateral Femoral Condyles, two on the Lateral Malleoli, and one on the Wrist. The inertial parameters were also derived using the measurement and some kinematics data from optimization, according to the adjustments of the Zatsiorsky-Seluyanov’s parameters [7]. In this work ground reaction data were collected with two KISTLER platforms placed in these labs. Three forces and three moments relative to each force plates were recorded. Force plates were also calibrated by leaving the special eight markers device on each force plate at the time. The mean distance computed between the two spheres on the stick differed in general from the actual distance (400 mm) by less than 0.3 mm. Using data from the ground reaction platform, the kinematic data (trajectories, joint angles, acceleration etc.) has been combined with ground reaction forces and interia parameters in order to compute the joint moments and powers.   2.1 The power’s model of human gait using the regression function   The two parametric identification methods were used to the modeling of human gait: the regression function and the stochastic approximation function.  The computation of power joint is a relatively straightforward application of Newtonian mechanics. The mechanical power associated with joint rotation is computed from the combination of the joint moment and the joint angular velocity (the rotational velocity of one segment relative to another) [8,9]. The power was normalized to the body weight of each subject. The formula of power joint is facilitated through the use of Eqn.1: iii MP→→⋅= ω ,   (1)  where P - joint power, M - joint moment, ω - angular velocity.  The power’s model proposed by the author is based on instantaneous power developed by muscle joints. The human gait model using the regression function is determined by Eqn. 2 [8,13,14]:  auY nn ⋅=Λ, n=1,2,…,N, (2)  where: ΛY  – model’s output (power joints in the n instant), un – model’s input (power joints in the n instants before), a – unknown parameters of human gait, N – sample size.  The unknown vector a is determined by Eqn.3:  ( ) ,1 YUUUa TT ⋅⋅⋅= −                  (3)  whereU - the matrix of input data, Y - the vector of output data, a – unknown parameters of human gait.  2.1 The power’s model of human gait using the stochastic approximation function   To identify the value of model’s coefficients in time, the stochastic approximation function has been proposed in  283.  MATHEMATICAL MODELLING – TOOL TO DIAGNOSIS OF HUMAN APPARATUS.  J. PAUK    © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 APRIL/JUNE,   VOLUME 9,  ISSUE 2,  ISSN 1392-8716  110 this paper. The equation of human motion using the stochastic approximation function can be expressed by Eqn.4: ... ,1 2 1a a an n n n k n kΛ= + + +− − −Y Y Y Y  (4)  where the vector a is expressed as below: ( )11 −− −+= iiiTiiii aUyUPaa ,  (5) ( )TiiiiiTiiii UPUPUUPPP1111 1−−−− +−= .  (6)  Where Yn– model’s output (power developed by muscle joints in n-instant), U n– model’s input (power developed by muscle joints in n-instants before), a– unknown parameters of human gait, P  - matrix of covariance. The initial value of vector a0 and of matrix 0P  must be given by Eqn. 8-9:      [ ] ,0000Ta …=       (7) ,0Λ= rP  (r≈102 ÷ 103),  (8) where Λ  - is a unitary matrix.  3. Results  Table 1 shows power’s model coefficients of healthy subjects into seven phases of human gait: the initial contact (IC), the loading response (LR), the midstance (MSt), the terminal stance (TSt), the initial swing (ISw), the midswing (MSw), and the terminal swing (TSw). The standard deviation value of the model’s coefficients is also given in the same table.   Table 1. The power’s model coefficients of healthy subjects using the regression function (±SD)  Phase a1 a2 a3 IC 0,636 ±0,102 -0,068 ±0,003 0,046 ±0,008 LR 1,232 ±0,117 -0,350 ±0,045 -0,076 ±0,007 MSt 1,039 ±0,154 -0,138 ±0,035 -0,091 ±0,009 TSt 0,576  ±0,122 -0,408 ±0,101 -0,057 ±0,005 ISw 0,641  ±0,107 -0,187 ±0,050 -0,097 ±0,021 MSw 0,610 ±0,151 -0,061 ±0,005 -0,051 ±0,008 TSw 1,226 ±0,221 -0,263 ±0,051 -0,151 ±0,080 There are presented model’s coefficients of patients with Anterior Cruciate Ligaments and with gonarthrosis in Tab.2-3.   Table 2. The power’s model coefficients of patients with gonarthrosis using the regression function (±SD)  Phase a2 LR -0,075±0,050 TSt -0,800±1,150 Isw -0,096±1,101    The model’s coefficients of patients with gonarthrosis were compared to the model’s coefficients of healthy subjects. In the initial contact, the midstance, the midswing and in the terminalswing no difference between model’s coefficients. The value of coefficient a2 is different in the three phases of human gait: the loading response, the terminal stance, and the initial swing. There are presented model’s coefficients for patients with Anterior Cruciate Ligaments in Table 3.    Table 3. The power’s model coefficients of patients with Anterior Cruciate Ligaments using the regression function (±SD)  Phase a2 IC -0,193±0,120 LR -0,146±0,040 TSt 0,533±0,306 ISw 0,418±0,025    There is no difference of model’s coefficients in the midstance, the midswing, and in the terminal swing compared to the healthy subjects. The significant difference of coefficient a2 is in: the initial contact, the loading response, the terminal stance, and in the initial swing.  With the help of a program in MATLAB by the stochastic approximation function the model’s coefficients with the best fit to the data were easily computed. Table 4 shows power’s model coefficients for the hip joint, the knee joint, and for the ankle joint of patients with gonarthrosis and patients with Anterior Cruciate Ligaments using the stochastic approximation function. The standard deviation is also given in the same table.    Power’s model coefficients of these two pathological groups were compared to the power’s model coefficients of healthy subjects [10]. Analysis of model’s coefficients shows, that there is no difference of coefficient a1 and a2 for all joints of human body. The significant difference is noted for the coefficient a3.  283.  MATHEMATICAL MODELLING – TOOL TO DIAGNOSIS OF HUMAN APPARATUS.  J. PAUK    © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 APRIL/JUNE,   VOLUME 9,  ISSUE 2,  ISSN 1392-8716  111 Table 4. The power’s model coefficients of patients with gonarthrosis and LCA using the stochastic approximation function (±SD)  Joint Coeff. Gonarthrosis LCA Hip a3 -0,063 ±0,047 -0,045 ±0,015 Knee a3 0,143 ±0,121 -0,113 ±0,101 Ankle a3 -0,052 ±0,031 -   Statistical analysis was performed on the whole population of healthy subjects and patients with gonarthrosis and those with Anterior Cruciate Ligaments. A characterization of the difference was obtained by computing the following parameters: the standard deviation, correlation, variance, and confidence intervals.   4. Conclusion  Gait Analysis is a good tool for the documentation of human movement.  In this study the regression function and the stochastic approximation function were used to identify dynamic properties of human gait of healthy subjects and patients with gonarthrosis and those with Anterior Cruciate Ligaments. The methods showed the most accurate results. The regression function and the stochastic approximation function complete each other. The stochastic approximation function allows to analysis model’s coefficients currently. Model’s coefficients using the stochastic approximation function have lower values than model’s coefficients using the regression function. There is shown that some model’s coefficients should not be analysed during research process in this paper.  The reason of rejected model’s coefficient is no difference of model’s coefficients between healthy subjects and presented groups of pathology such as the gonarthrosis and Anterior Cruciate Ligaments. Through the numerical analysis and experiments, it is concluded that these methods are the most suitable for accurately monitoring of the dynamic properties of human gait. It is very likely that applying kinetics data, especially power joints, helps to define gait pathology and treatment in a large number of patients. With the help of advanced computing software, parametric identification method has proved to be convenient and powerful method for monitoring human gait. The advantage of the proposed technique is possibilities to classify human gait to different groups of pathology. It also provides the information from which a treatment recommendation may be made to enhance the subnormal gait patterns evident in these patients. These methods of identification represent human movement in a very accurate way during walking  a man in sagittal plane.    References  [1] Onyshko S, Winter D. A. (1980) A mathematical model for the dynamics of the human locomotion. Journal of Biomechanics; 13: 361-368. [2] Moeinzadeh M. H., Engin A. E. (1988) Dynamic modeling of the human knee joint. Computational Methods in Bioengineering. ASME Winter Annual Meeting,       145-156. [3] Crownenshield R., Pope M. H., Johnson R. J. (1976) An analytical model of the knee.  Journal of Biomechanics: 397-405. [4] Ishac, M.G., Winter D.A. (1990) Internal validity of link-segment models of gait analysis. Sixth East Coast Clinical Gait Conference. Michigan State University: 75-78. [5] Frigo C. (1998). Functionally oriented and clinically feasible quantitative gait analysis method. Medical & Biological Engineering & Computing; 36 (2): 179-185 [6] Pedotti A., Ferrigno G. (1995) Optoelectronic-based systems, three-dimensional instrumentation, three-dimensional analysis of human movement. Human Kinetics Publ.: 57-77. [7] Pedotti A., Frigo C. (1992) Quantitative analysis of locomotion of basic research and clinical applications in Functional Neurology; 7, (4): 47-56. [8] Pauk J., Jaworek K. (2002) Parametric identification of lower limbs during walking of a man. Design a. Nature. WET Press: 361-366. [9] Pauk J. (2005) Identification of human gait model, Ph.D. thesis. Polish Academy of Science. [10] Amirouche F. M. L., Ider S. K., Trimble J. (1990) Analytical method for the analysis and simulation of human locomotion. Journal of Biomechanical Engineering; 112: 379-386. [11] Koopman H. F. J. M., Grootenboer H. J., de Jongh H.J. (1989) Mechanical model to simmulate and predict human walking patterns. XII International Congress of Biomechanics. Los Angeles. [12] Chow C. K., Jacobson D. H. (1972) Further studies in human locomotion: Postural stability and control. Mathematical Biosciences; 15: 93-108. [13] Mendel J. (1973) Discrete techniques of parameter estimation. M. Dekker. New York. [14] Manerowski J. (1999) The model’s identification of flying objects. ASKON. Warsaw.            The author is a fellowship-holder from Polish Foundation for Polish Science in 2007. The work has been approved by the appropriate ethical committees related to the Center of Bioengineeria in Milan. I thank you for prof. C. Frigo for help in the collection, analysis and interpretation of data. This paper is supported by grant W/WM/11/05 from Bialystok Technical University, Poland.  

Mathematical modelling – tool to diagnosis of human apparatus

Journal of Mechatronics and Artificial Intelligence in Engineering

Journal of Vibroengineering

 © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 APRIL/JUNE,   VOLUME 9,  ISSUE 2,  ISSN 1392-8716 
 
108 
283. Mathematical modelling – tool to diagnosis of human apparatus 
 
 
 
Jolanta PAUK 
Bialystok University of Technology 
Automatics & Robotics Chair 
Wiejska 45C, 15-351 Bialystok 
PHONE: +48 85 7469222 
E-mail: jpauk@pb.bialystok.pl 
 
(Received 14 April 2007,  accepted 15 June 2007) 
 
 
 
 
 
 
 
 
 
 
 
 
Abstract. There is presented a model of human gait based on power developed by muscle joints in this paper. The 
mathematical model was presented for healthy subjects, patients with gonarthrosis and patients with Anterior Cruciate 
Ligaments (ACL). The regression function and the stochastic approximation function were used to the modeling. The 
differences between the values of model’s coefficients of people with these two pathologies have been compared to the 
model’s coefficients of healthy subjects. Proposed model is very accurate. The modeling could help to define the level of 
gait pathology and treatment in a large number of patients.  
 
Keywords: modeling, regression function, stochastic approximation function, human gait. 
 
1. Introduction 
 
Human walking is an example of a well-learned 
fundamental movement pattern that, in normal situations is 
performed with a great deal of coefficiency and 
consistency. Gait analysis and modeling has long been 
used as a research tool to develop complete understanding 
of the mechanics control of human movement. A standard 
physical examination can’t provide a complete description 
of pathology of abnormal human gait. Gait analysis and 
modeling can. The primary purpose of gait analysis and 
modeling in clinical application is to provide objective 
documentation of gait and increase understanding of the 
mechanism of pathological gait. Mathematical modeling is 
an important tool for quantifying normal and pathological 
patterns of locomotion. With this information, more 
informed decision making about treatment to improve gait 
function is possible. In this study will be presented the 
parametric methods for identification of human gait model. 
There can be many pages explaining what parametric 
identification method is in depth, but it can be explained as 
a technique that helps one to structure a mathematical 
model based on measured data by adjusting parameters 
within a given model until the simulated output coincides 
as closely as possible with the measured output. Using 
parametric identification methods, the dynamic properties 
of human gait can be identified. Recently, many interactive 
computer software and hardware packages have been 
developed that allowed further development of these 
methods. However, many researchers have attempted the 
simulation of gait by limiting their model with constraints 
[10,11,12]. Many of these take an analytical approach to 
the synthesis of motion, using trial and error, dynamic 
optimization, or other control schemes to develop the 
proper force time histories necessary to drive the motion. 
Onyshko and Winter presented an important model of 
locomotion in 1980 [1]. In this work, the issue of 
responsible modeling was stressed in regard to pure 
synthesis (forward dynamics). One of the few dynamic 
models of the knee joint was presented in 1988 [2]. The 
approach involved the development of a 3D model that 
was later reduced to a 2D model for analysis and 
simulation of motion. The skeletal model and the origins 
and insertions were taken from the work of Crownenshield 
and co-workers [3]. The synthesis of motion was described 
as a reduction of a second order non-linear system of 
ordinary differential equations to an algebraic system and 
then linearized. The tibia was subjected to several dynamic 
loading situations as different mathematical function 
shapes, and the resulting dynamic forces were computed. 
The contact points, joint orientation, and joint and ligament 
loads were reported, and showed that knee extension 
 283.  MATHEMATICAL MODELLING – TOOL TO DIAGNOSIS OF HUMAN APPARATUS.  J. PAUK 
 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 APRIL/JUNE,   VOLUME 9,  ISSUE 2,  ISSN 1392-8716 
 
109
caused elongation of the medial and lateral collateral and 
anterior cruciate ligaments (ACL), but not the posterior 
cruciate ligament (PCL). It was also shown that the ACL 
carried the highest load. This study did not consider the 
effects of muscles or parametric changes such as the loss 
of a ligament or the manipulation of the geometry. A later 
study presented 3D, 7 segment models with revolute joints 
[4]. This model was driven by inverse dynamic forces and 
moments obtained from a gait analysis laboratory, and a 
forward solution were obtained using ADAMS dynamic 
modeling software. Ground contact was modeled as 
spring-damper forces on an ellipsoid foot to reflect the 
progression of the center of pressure. Torsional spring-
dampers were placed at each joint to regulate the motion. 
The model was applied to the swing phase of gait, and 
joint moments were altered in order to obtain kinematic 
patterns similar to the original measured motion [4].  
The goal of this work is to investigate the prospects of 
dynamic model as a tool for understanding and treating 
pathological musculoskeletal motion.  
 
2. Material and method 
 
In this study 45 healthy subjects (the average age 46 yr.), 
37 patients with gonarthrosis (the average age 67 yr.), and 
36 subjects with Anterior Cruciate Ligaments (ranged 
between 20 and 37 yr.) were evaluated. Subjects were 
recruited into Centre of Bioengineeria in Milan. The 
optoelectronic system Elite-3D was used for the 
measurements. The system is based on an on-line data 
processing of signals from a number of TV cameras. It is 
possible to recognize in the field of view of each TV 
camera those bright areas which are of interest for motion 
analysis. Quantization of biomechanical variables and 
spatio-temporal parameters of walking was performed by 
means of a computerized system for automatic acquisition 
of kinematics and ground reaction forces. A working 
volume 3 m long, 2.5 m high, and 1.2 m wide was 
calibrated by a precision grid, which was displaced in three 
different parallel planes. The resultant accuracy was 
assessed by measuring the movement of a special stick 
with three retro reflective markers placed on it. Pre-
processing of raw data involved a tracking procedure, 
three-dimensional reconstruction of the marker’s 
coordinates, correction for optoelectronic distortion, and 
filtering. The frequency of acquisition was set at 50 Hz. 
All the subjects were analyzed with the S.A.F.Lo protocol. 
The location of the markers on each subject was the 
following: two on the Posterior Superior Iliac Spines, one 
on the Sacrum Bone, two on the Lateral Femoral Condyles, 
two on the Lateral Malleoli, and one on the Wrist. The 
inertial parameters were also derived using the 
measurement and some kinematics data from optimization, 
according to the adjustments of the Zatsiorsky-Seluyanov’s 
parameters [7]. In this work ground reaction data were 
collected with two KISTLER platforms placed in these 
labs. Three forces and three moments relative to each force 
plates were recorded. Force plates were also calibrated by 
leaving the special eight markers device on each force 
plate at the time. The mean distance computed between the 
two spheres on the stick differed in general from the actual 
distance (400 mm) by less than 0.3 mm. Using data from 
the ground reaction platform, the kinematic data 
(trajectories, joint angles, acceleration etc.) has been 
combined with ground reaction forces and interia 
parameters in order to compute the joint moments and 
powers.  
 
2.1 The power’s model of human gait using the regression 
function  
 
The two parametric identification methods were used to 
the modeling of human gait: the regression function and 
the stochastic approximation function.  
The computation of power joint is a relatively 
straightforward application of Newtonian mechanics. The 
mechanical power associated with joint rotation is 
computed from the combination of the joint moment and 
the joint angular velocity (the rotational velocity of one 
segment relative to another) [8,9]. The power was 
normalized to the body weight of each subject. The 
formula of power joint is facilitated through the use of 
Eqn.1: 
iii MP
→→ ⋅= ω ,   (1) 
 
where P - joint power, M - joint moment, ω - angular 
velocity. 
 
The power’s model proposed by the author is based on 
instantaneous power developed by muscle joints. The 
human gait model using the regression function is 
determined by Eqn. 2 [8,13,14]: 
 
auY nn ⋅=
Λ
, n=1,2,…,N, (2) 
 
where: 
Λ
Y
 – model’s output (power joints in the n instant), 
un – model’s input (power joints in the n instants before), 
a – unknown parameters of human gait, 
N – sample size. 
 
The unknown vector a is determined by Eqn.3: 
 ( ) ,1 YUUUa TT ⋅⋅⋅= −
                 (3) 
 
whereU - the matrix of input data, Y - the vector of output 
data, a – unknown parameters of human gait. 
 
2.1 The power’s model of human gait using the stochastic 
approximation function  
 To identify the value of model’s coefficients in time, 
the stochastic approximation function has been proposed in 
 283.  MATHEMATICAL MODELLING – TOOL TO DIAGNOSIS OF HUMAN APPARATUS.  J. PAUK 
 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 APRIL/JUNE,   VOLUME 9,  ISSUE 2,  ISSN 1392-8716 
 
110 
this paper. The equation of human motion using the 
stochastic approximation function can be expressed by 
Eqn.4: 
... ,1 2 1a a an n n n k n k
Λ = + + +− − −Y Y Y Y  (4) 
 
where the vector a is expressed as below: ( )11 −− −+= iiiTiiii aUyUPaa ,  (5) 
( )Tiii i
i
T
ii
ii UPU
PUUP
PP
1
11
1 1 −
−−
− +−= .  (6) 
 
Where Yn– model’s output (power developed by muscle 
joints in n-instant), U n– model’s input (power developed 
by muscle joints in n-instants before), a– unknown 
parameters of human gait, P  - matrix of covariance. 
The initial value of vector a0 and of matrix 0P  must be 
given by Eqn. 8-9: 
     [ ] ,0000 Ta …=       (7) 
,
0
Λ= rP  (r≈102 ÷ 103),  (8) 
where Λ  - is a unitary matrix. 
 
3. Results 
 
Table 1 shows power’s model coefficients of healthy 
subjects into seven phases of human gait: the initial contact 
(IC), the loading response (LR), the midstance (MSt), the 
terminal stance (TSt), the initial swing (ISw), the 
midswing (MSw), and the terminal swing (TSw). The 
standard deviation value of the model’s coefficients is also 
given in the same table.  
 
Table 1. The power’s model coefficients of healthy subjects 
using the regression function (±SD) 
 
Phase a1 a2 a3 
IC
 
0,636 
±0,102 
-0,068 
±0,003 
0,046 
±0,008 
LR
 
1,232 
±0,117 
-0,350 
±0,045 
-0,076 
±0,007 
MSt 1,039 
±0,154 
-0,138 
±0,035 
-0,091 
±0,009 
TSt 0,576  
±0,122 
-0,408 
±0,101 
-0,057 
±0,005 
ISw 0,641 
 ±0,107 
-0,187 
±0,050 
-0,097 
±0,021 
MSw 0,610 
±0,151 
-0,061 
±0,005 
-0,051 
±0,008 
TSw 1,226 
±0,221 
-0,263 
±0,051 
-0,151 
±0,080 
There are presented model’s coefficients of patients 
with Anterior Cruciate Ligaments and with gonarthrosis in 
Tab.2-3. 
 
 
Table 2. The power’s model coefficients of patients with 
gonarthrosis using the regression function (±SD) 
 
Phase a2 
LR
 
-0,075±0,050 
TSt
 
-0,800±1,150 
Isw 
-0,096±1,101 
 
 
 
The model’s coefficients of patients with gonarthrosis 
were compared to the model’s coefficients of healthy 
subjects. In the initial contact, the midstance, the midswing 
and in the terminalswing no difference between model’s 
coefficients. The value of coefficient a2 is different in the 
three phases of human gait: the loading response, the 
terminal stance, and the initial swing. 
There are presented model’s coefficients for patients 
with Anterior Cruciate Ligaments in Table 3.  
 
 
Table 3. The power’s model coefficients of patients with 
Anterior Cruciate Ligaments using the regression function (±SD) 
 
Phase a2 
IC 
-0,193±0,120 
LR
 
-0,146±0,040 
TSt 0,533±0,306 
ISw 0,418±0,025 
 
 
 
There is no difference of model’s coefficients in the 
midstance, the midswing, and in the terminal swing 
compared to the healthy subjects. The significant 
difference of coefficient a2 is in: the initial contact, the 
loading response, the terminal stance, and in the initial 
swing.  
With the help of a program in MATLAB by the 
stochastic approximation function the model’s coefficients 
with the best fit to the data were easily computed. Table 4 
shows power’s model coefficients for the hip joint, the 
knee joint, and for the ankle joint of patients with 
gonarthrosis and patients with Anterior Cruciate Ligaments 
using the stochastic approximation function. The standard 
deviation is also given in the same table.    
Power’s model coefficients of these two pathological 
groups were compared to the power’s model coefficients 
of healthy subjects [10]. Analysis of model’s coefficients 
shows, that there is no difference of coefficient a1 and a2 for 
all joints of human body. The significant difference is 
noted for the coefficient a3. 
 283.  MATHEMATICAL MODELLING – TOOL TO DIAGNOSIS OF HUMAN APPARATUS.  J. PAUK 
 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 APRIL/JUNE,   VOLUME 9,  ISSUE 2,  ISSN 1392-8716 
 
111
 
Table 4. The power’s model coefficients of patients with 
gonarthrosis and LCA using the stochastic approximation 
function (±SD) 
 
Joint Coeff. Gonarthrosis LCA 
Hip a3 -0,063 
±0,047 
-0,045 
±0,015 
Knee
 
a3 0,143 
±0,121 
-0,113 
±0,101 
Ankle
 
a3 -0,052 
±0,031 
- 
 
 
Statistical analysis was performed on the whole population 
of healthy subjects and patients with gonarthrosis and 
those with Anterior Cruciate Ligaments. A characterization 
of the difference was obtained by computing the following 
parameters: the standard deviation, correlation, variance, 
and confidence intervals.  
 
4. Conclusion 
 
Gait Analysis is a good tool for the documentation of 
human movement.  In this study the regression function 
and the stochastic approximation function were used to 
identify dynamic properties of human gait of healthy 
subjects and patients with gonarthrosis and those with 
Anterior Cruciate Ligaments. The methods showed the 
most accurate results. The regression function and the 
stochastic approximation function complete each other. 
The stochastic approximation function allows to analysis 
model’s coefficients currently. Model’s coefficients using 
the stochastic approximation function have lower values 
than model’s coefficients using the regression function. 
There is shown that some model’s coefficients should not 
be analysed during research process in this paper.  The 
reason of rejected model’s coefficient is no difference of 
model’s coefficients between healthy subjects and 
presented groups of pathology such as the gonarthrosis and 
Anterior Cruciate Ligaments. Through the numerical 
analysis and experiments, it is concluded that these 
methods are the most suitable for accurately monitoring of 
the dynamic properties of human gait. 
It is very likely that applying kinetics data, especially 
power joints, helps to define gait pathology and treatment 
in a large number of patients. With the help of advanced 
computing software, parametric identification method has 
proved to be convenient and powerful method for 
monitoring human gait. The advantage of the proposed 
technique is possibilities to classify human gait to different 
groups of pathology. It also provides the information from 
which a treatment recommendation may be made to 
enhance the subnormal gait patterns evident in these 
patients. These methods of identification represent human 
movement in a very accurate way during walking  
a man in sagittal plane.  
 
 
References 
 
[1] Onyshko S, Winter D. A. (1980) A mathematical model 
for the dynamics of the human locomotion. Journal of 
Biomechanics; 13: 361-368. 
[2] Moeinzadeh M. H., Engin A. E. (1988) Dynamic 
modeling of the human knee joint. Computational Methods 
in Bioengineering. ASME Winter Annual Meeting,       
145-156. 
[3] Crownenshield R., Pope M. H., Johnson R. J. (1976) An 
analytical model of the knee.  Journal of Biomechanics: 
397-405. 
[4] Ishac, M.G., Winter D.A. (1990) Internal validity of link-
segment models of gait analysis. Sixth East Coast Clinical 
Gait Conference. Michigan State University: 75-78. 
[5] Frigo C. (1998). Functionally oriented and clinically 
feasible quantitative gait analysis method. Medical & 
Biological Engineering & Computing; 36 (2): 179-185 
[6] Pedotti A., Ferrigno G. (1995) Optoelectronic-based 
systems, three-dimensional instrumentation, three-
dimensional analysis of human movement. Human Kinetics 
Publ.: 57-77. 
[7] Pedotti A., Frigo C. (1992) Quantitative analysis of 
locomotion of basic research and clinical applications in 
Functional Neurology; 7, (4): 47-56. 
[8] Pauk J., Jaworek K. (2002) Parametric identification of 
lower limbs during walking of a man. Design a. Nature. 
WET Press: 361-366. 
[9] Pauk J. (2005) Identification of human gait model, Ph.D. 
thesis. Polish Academy of Science. 
[10] Amirouche F. M. L., Ider S. K., Trimble J. (1990) 
Analytical method for the analysis and simulation of human 
locomotion. Journal of Biomechanical Engineering; 112: 
379-386. 
[11] Koopman H. F. J. M., Grootenboer H. J., de Jongh H.J. 
(1989) Mechanical model to simmulate and predict human 
walking patterns. XII International Congress of 
Biomechanics. Los Angeles. 
[12] Chow C. K., Jacobson D. H. (1972) Further studies in 
human locomotion: Postural stability and control. 
Mathematical Biosciences; 15: 93-108. 
[13] Mendel J. (1973) Discrete techniques of parameter 
estimation. M. Dekker. New York. 
[14] Manerowski J. (1999) The model’s identification of flying 
objects. ASKON. Warsaw. 
 
 
 
 
 
 
 
 
 
 
 
The author is a fellowship-holder from Polish Foundation for Polish 
Science in 2007. 
The work has been approved by the appropriate ethical committees 
related to the Center of Bioengineeria in Milan. I thank you for prof. C. 
Frigo for help in the collection, analysis and interpretation of data. 
This paper is supported by grant W/WM/11/05 from Bialystok 
Technical University, Poland. 
 


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Mathematical modelling – tool to diagnosis of human apparatus

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