Human Walking Analysis, Evaluation and Classification Based on Motion Capture System

Gait analysis is the systematic study of human walking. It is helpful in the medical management of those diseases which affect the locomotion systems. Recently, the gait motion capture systems are becoming widely used by doctors and physical therapists for kinematics analysis and biomechanics and motion capture research, sports medicine and physical therapy, including human gait analysis and injury rehabilitation. This chapter describes some new progress on human walking analysis that our group made in the past few years based on motion capture system. Generally, ageing causes many changes to neuromuscular system of a human being, for an example, his walking capabilities degenerate by ageing. Because these changes sometimes result in an increase the number of falls during daily walking, especially after the age of 75, it is very important to study the age related changes in the walking gait of elderly subjects. Many researchers studied stability of human walking gait and it was quoted that human walking gait stability decreases with age increasing the risk of falls in elderly people. Many studies have been reported about the change in the kinematics parameters with age (Arif et al., 2004). This paper only focuses on the progress of walking modeling and walking stability. Especially, in order to simplify the method of data acquisition, this paper suggests process of reduction on dynamic stability features through feature selection. That will help us analyze stability in a more clear way.


Introduction
Gait analysis is the systematic study of human walking. It is helpful in the medical management of those diseases which affect the locomotion systems. Recently, the gait motion capture systems are becoming widely used by doctors and physical therapists for kinematics analysis and biomechanics and motion capture research, sports medicine and physical therapy, including human gait analysis and injury rehabilitation. This chapter describes some new progress on human walking analysis that our group made in the past few years based on motion capture system. Generally, ageing causes many changes to neuromuscular system of a human being, for an example, his walking capabilities degenerate by ageing. Because these changes sometimes result in an increase the number of falls during daily walking, especially after the age of 75, it is very important to study the age related changes in the walking gait of elderly subjects. Many researchers studied stability of human walking gait and it was quoted that human walking gait stability decreases with age increasing the risk of falls in elderly people. Many studies have been reported about the change in the kinematics parameters with age (Arif et al., 2004). This paper only focuses on the progress of walking modeling and walking stability. Especially, in order to simplify the method of data acquisition, this paper suggests process of reduction on dynamic stability features through feature selection. That will help us analyze stability in a more clear way.

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The maximum Floquet multipliers (FM) are used to measure orbital stability of upper body in difference walking speed (Marin et al., 2006). Orbital stability changed very little with speed. The general purpose of (Kavanagh, 2006) is to examine factors that may influence acceleration features of the upper body during walking. Balance in quiet upright stance, which was studied by (Stirling & Zakynthinaki, 2004), does not imply motionless stability, in fact a ML and AP body sway occurs. Almost 95% of the anterior-posterior sway happens around the ankle and the hip axis. Sutherland et al investigated the development of mature walking in children of age from one to seven years old (Sutherland et al., 1980). The objective of literature (PH. Chou, 2003) was to investigate the gait maturation of Taiwan children. Elderly subjects exhibits gait pattern characterized by reduced velocity, shorter step length and increased step timing variability (Hylton et al., 2003). It is mentioned that elderly people reduce their walking speed to improve their walking stability in the literature. While many walking stability indices have been proposed, there is still no commonly accepted way to define, much less quantify, locomotors stability.

Background of walking symmetry
The concept of gait symmetry itself is no acceptable unified definition, more often many people's research assumes that the normal gait is symmetric, which is to simplify data collection and analysis of gait. In fact gait symmetry was only evaluated with a small number of biomechanical studies that used quantitative data of the two lower limbs. In addition, the gait symmetry was not actually carried out enough exploration. It is because there is no participation of a considerable more number of test objects. One hand, the gait parameter of information provided is the effects of the movement not the reasons of the movement. This may also affect explaining the behaviour of the lower limbs. The other hand, the gait symmetry is not clearly defined, and the use only single gait parameters or a simple statistical method to compare which has made study of gait symmetry been more limited. Maybe it stands to reason that a healthy man or woman has left leg and right leg symmetrically, and normal gait seems symmetric with ones right-side and left-side. What about the walking stability or dynamic balance if there is not so symmetric? Can we get some quantity or relationship of symmetry and balance? Gait symmetry is multifaceted. Normal walking seems right-left symmetry because normal people walk with their right foot and left foot. The ability of maintaining balance is essential to keep normal walking. Poor balance is an independent risk factor for falling, so anything that improves balance can have a positive effect on safety and function after stroke. Balance is described as the ability to maintain or move within a weight-bearing posture without falling. We have not found any trends study relating to walking stability and symmetry of human gait. And the trends indicated what meanings. Yang in his doctoral thesis paper (Yang, 2001) proposed the symmetry indicators of step length and stride length, to describe the step in the role of gait symmetry. But phase symmetry and the step length symmetry are not fully reflects the characteristics of gait, especially the joint angle in the role of symmetry.

Modeling assumptions and theoretical framework
Walking is a complex dynamic activity. A good human model for gait analysis should be simple, but extensive enough to capture the dynamics of most walkers.
Almost existing models, which have from 2 segments to more than 15 segments, have two pitfalls. Firstly, they paid more attention to the sagittal plane, and overlook the other two planes, transverse plane and frontal plane. Secondly, it is not enough to describe the particularity of the feet. Most of them regarded the foot as a point. Thus, it is difficult to decide the gait cycle, such as initial heel contact, heel rise, and toe off. So a new walking model, so called Fourteen-Linkage (FL) model, is proposed.  We suggest a walking model, which is a collection of 19 points, 14 segments and 12 joints, used to specify the position and the configuration of a human body, as shown in Fig. 1. Position describes the location of a body segment or joint in space, measured in meters. In FL model, the 19 points can be decided by the 30 markers we measured from motion capture system. The relationships between the model and markers are also shown in Fig. 1(a). Body segments are considered to be rigid bodies for the purposes of describing the motion of the body. 14 segments are composed of these points, such as head (S 1 ), shoulder (S 2 ), left and right upper-arms (S 3 , S 4 ), left and right forearms (S 5 , S 6 ), trunk (S 7 ), pelvis (S 8 ), left and right thighs (S 9 , S 10 ), left and right shanks (S 11 , S 12 ), and left and right feet (S 13 , S 14 ). 12 joints between adjacent segments are composed of these segments, such as head-trunk (θ 1 ), head-shoulder (θ 2 ), shoulder (θ 3 , θ 4 ), elbows (θ 5 , θ 6 ), hips (θ 7 , θ 8 ), knees (θ 9 , θ 10 ), ankles (θ 11 , θ 12 ), see Fig. 1(b).

Definitions of walking stability
In human movement, kinematics is the study of the positions, angles, velocities, and accelerations of body segments and joints during motion. FL model M consists of displacement P, segment angle Φ, joint angle Θ, and their velocity V and acceleration A, angular velocity Ω and angular acceleration Λ at time t, represented as 9-tuple as Equation (1).
Note that it is an absolute measure, meaning that it changes according to the orientation of the body. Joint angle is the angle between the two segments on either side of the joint. It is defined as Equation (3).
where θ s is the joint angle in space, and θ xoy , θ yoz , θ zox are the projections of joint angle in transverse plane, frontal plane and sagittal plane respectively. Since joint angle θ s is relative to the segment angles, it doesn't change with the body orientation. Velocity may be linear (change in displacement) or angular (change in angle). Normally, velocity is derived from displacement or angle data by the process of differentiation. Acceleration is change in velocity. Again, it may be linear (change in linear velocity) or angular (change in angular velocity). Acceleration, too, is usually calculated from the displacement data by differentiating twice. It can also be measured directly by an accelerometer.

Walking data preprocessing
Data preprocessing is an important issue for data analysis, as real-world data tend to be incomplete, noisy, and inconsistent. Data preprocessing includes data cleaning, data integration, data transformation, and data reduction. Although numerous methods of data preprocessing have been developed, data preprocessing remains an active area of research, due to the huge amount inconsistent or dirty data and the complexity of the problem. Before we talk about the methods of walking data preprocessing, let us have a look at walking data measuring.

Walking data measuring
There are many measurements of human gait, such as basic data (spatial and temporal data), kinematics (displacement, velocity and acceleration data), kinetics (force and moment data), electromyography (electrical activity of lower limb muscles), and image and graphics (individual silhouette images, monocular, image sequence, video). We adopt the kinematical approach in modeling the human movements. Two kinds of data, motion data and acceleration data are measured using two different systems. The Motion data are gotten from motion capture system (Vicon MX System by OMG Plc), while acceleration data are obtained by a 3-axis accelerometer. The providers walk along 5m straight line in level plane 3 times at them natural normal walking speeds. 30 markers are attached to the body, as shown in Fig. 3. Fig. 4 shows a snapshot of data acquisition using a motion capture system in the University of Aizu, Fukushima, Japan. The highlights show markers on human body.  The motion capture system can detect the three dimensions displacement data at tree directions: anterior-posterior, left-right, and superior-inferior. The sampling rate of motion data is 120Hz. At the same time, a tri-axial accelerometer unit is mounted with CASI, the same point as marker 18, see Fig. 3. The accelerometer is connected with a mobile phone to save the acceleration data. After that the data can be transferred to computer. Acceleration data are also collected in three dimensions as same as motion data, including the gravity acceleration. But the directions are not same. The sampling rate of acceleration data is 90Hz.
To calibrate the accelerometer, before each testing session, it was placed with each of the orthogonal axes vertically, to estimate the ±1g values. By the way, a movie is taken by a video camera while he/she is walking. 44 normal persons from 20 to 69 year old are measured. These subjects are classified into 5 groups (20+, 30+, 40+, 50+, and 60+) by the age.

Data cleaning
Data cleaning attempts to fill in missing values, smooth out noise, and correct inconsistencies in the data.

Missing data
In the walking data measured by the motion capture systems, there are some missing data because the system can not detect the markers at a moment. Many interpolation methods could be used, such as nearest neighbor interpolation, linear interpolation, cubic spline interpolation, piecewise cubic Hermite interpolation and N-th degree polynomial interpolation. We choose the spline method to interpolate the missing data because the cubic spline interpolation is a piecewise continuous curve, passing through each of the values in the source data. An example is shown in Fig. 5.

Noisy data
In acceleration data, there are many noisy data, as shown in Fig. 6. We should try to identify and cut these noisy data from the source data.
Noisy data Noisy data Useful data And there are some noisy data in motion data caused by vibration. Since the walking signal resides in the low frequency range, it is easily affected by interference from other signal and noise sources. Butterworth low-pass filter is used to reduce noise by passing signal which frequency below twice walking cadence. Some examples are shown in Fig. 7 to Fig. 8.

Inconsistent data
Because of the faults of the measure systems, one marker can be identified as two or more. In the source file, there is more than one column to store them. So these inconsistent data must be processed by the mean methods, see Equation (4).

Data transformation
Data transformation converts the data into appropriate forms for analysis further. The coordinate of acceleration data is not parallel the space coordinate because the accelerometer is set up obliquely, so we need to normalize the coordinate.

Calibration
The aim of calibration is to transform raw data to acceleration of gravity (g). The standard of vibration amplitude is maintained in terms of electrical output of reference accelerometer corresponding to a known value of displacement. Acceleration singles are sampled at 90Hz using purpose-written software and saved on computer for subsequent analysis. To calibrate the accelerations before each testing session, they ware placed with each of the orthogonal axes vertically, first pointing up, then down, which enabled the device to be statically calibrated to estimate the ±1g values. An example is shown in Fig. 9.

Adjusting acceleration data
Theoretically, the means of X and Y should be '0', that of Z should be '-1'. But actually, it is not true. The coordinate of acceleration data is not parallel with the space coordinate because the accelerometer is set up in gradient, see Fig. 10 so we need to adjust the coordinate. The method of adjusting is to rotate the accelerometer in correct position, as shown in Fig.  11. Adjusting rule is that the means of X and Y should be '0'. Firstly, calculate rotation matrix R with Homogeneous Coordinate used Equation (5), and then adjust the acceleration data to vertical position by Equation (6).
An example of adjusting is shown in Fig. 12. www.intechopen.com

Data integration
Data integration combines data from multiple sources to form a coherent data store. Metadata, correlation analysis, data conflict detection, and the resolution of semantic heterogeneity contribute toward smooth data integration.

Converting the coordinates
The coordinate of acceleration data is different from that of motion data, so we should match the acceleration data and motion data in the same coordinates. We define the coordinates, as shown in Fig. 13, x: anterior, y: left, z: superior.

Aligning the acceleration data with the motion data
The acceleration data and motion data come from different systems, the sampling rates are different, the time of starting measurement are not same, as shown in Fig. 14 Fig. 14. Title of figure, left justified so we must find which point of motion data is correlative with which point in acceleration data. The aligning method has two steps. 1. Computing the acceleration with motion data by Equation (7), as shown in Fig. 15.
2. Comparing the acceleration between the computing data and accelerometer data by minimum of relativity. The result of aligning is shown in Fig. 16.

Data reduction
Data reduction techniques can be used to obtain a reduced representation of the data while minimizing the loss of information content. To obtain a reduced representation of the data set, such as speed, average span, frequency, and so on, data reduction techniques can be applied, for examples, aggregation operations and conception hierarchy generation. We propose a hierarchical structure shown in Fig. 17.

Walking stability analysis
Although there is now a wealth of literature pertaining to the maintenance of stability when standing, there is a relative paucity of information regarding the biomechanics and physiology of walking stability. Various models are currently in development, but a unified model of walking stability does not exist yet. Unlike standing, the balance of walking is a kind of dynamic balance (Menz, 2000). Standard deviation is used to define the variability or randomness of the walking pattern. Less the amount of variability means better neuromuscular control and walking stability.
We extract a set of hierarchical features from FL model, such as walking cycle features, footprint features.

Footprint stability
Footprint analysis is a typical method of walking research as shown in Fig. 18.

Right step length
Stride length

Left step length
Right toe out (Footprint Stability) Footprint stability is described by the variability of the footprint stability features. We extract some of the stability features of footprint F S (F), such as the variability of cycle time f1, left step length f2, right step length f3, speed f4, walking base f5, left toe out f6, and right toe out f7, as shown in Equation (8).
where δ(f i ) is the standard deviation of the feature f i , and μ(f i ) is the mathematical expectation of the feature f i .

Effects of aging on footprint stability
Now, the variability of the footprint features per decade of age is calculated, as shown in Table 1. The footprint variability in last row of Table 1 is sum of the above items. It can be seen that the variability is increasing with the age, see Fig. 19. That is to say, the footprint stability is declined with the age, and especially there is a dramatic increasing over 50 years old. But the twenties are exceptional, maybe because the twenties walk more springily than the elders.

Cycle stability
The gait cycle is defined as the time interval between two successive occurrences of one of the repetitive events of walking. The detection of the human gait period can provide important information to determine the positions of the human body.
where δ(c i ) is the standard deviation of the time when event i occurs, c i = {LTO, LFA, LTV, LIC, RTO, RFA, RTV}, and μ(f 4 ) is the mathematical expectation of the speed f 4 . Generally, 7 events are used to identify major events during the walking cycle (Whittle, 2007). For the symmetry of left side and right side, 10 events are employed in this paper, as shown in Fig. 20.    21 shows that the cycle variability is declined with the age almost. This is not same as a common assumption. The reason is that the elder walking more rigidly and inflexibly.

Orbital stability
Orbital stability defines how purely periodic systems respond to small perturbations discretely after one complete cycle (i.e. one stride) and may be better suited to study strongly periodic movements like walking (Marin, 2006).

Definition of orbital stability
DEFINITION 4. (Orbital Stability) Orbital stability is described by the variability of the orbital stability features. The orbital stability F S (O) of lower limbs is composed of three parts, position stability F S (P), segment stability F S (S) and Joint stability F S (J), as shown in Equation (10).
where position stability F S (P) is defined by standard deviation of the maximums and minimums of displacement, velocity and acceleration (knees, heels, and toes), as shown in Equation (11).
(x) and  (x) are the maximum and minimum value of x respectively.
Segment stability F S (S) is defined by standard deviation of the maximums and minimums of segment angle, angular velocity and angular acceleration (thighs, shanks and feet), as shown in Equation (12).
Joint stability F S (J) is defined by standard deviation of the maximums and minimums of displacement, velocity and acceleration (hips, knees and ankles), as shown in Equation (13).

Outlier analysis in orbital stability
Because of the errors of computing, maybe there are few outliers in the features of orbital stability. For an example, the last 3 cycles of the displacement of right hip (p10) at vertical is shown in Fig. 25. There is an outlier in the last cycle, the left maximum point should be found instead of the right maximum point.
There are a variety of outlier detection approaches from several areas, including statistics, machine learning, and data mining. A kind of proximity-based outlier detection approach, called distance to k-nearest neighbor, is used to find the outliers in orbital stability. This approach is more general and more easily applied than statistical approached, since it is easier to determine a meaningful proximity measure for a data set than to determine its statistics distribution.

Effects of aging on orbital stability
After analysis all subjects between 20 to 70 years old, the item results of orbital stability are shown in Fig. 26. The position stability F S (P), segment stability F S (S), joint stability F S (J) and the whole orbital stability are in shown in Fig. 27. It is observed that, although each of orbital variability does not increase strictly with age, the variability of orbital features increases with age generally. As persons grow old, the orbital stability is becoming weakly.

Dynamic stability based on dynamic time warping (DTW)
Dynamic stability is the main reason for leading to falls for people, especially for elders. This section discusses age influence on dynamic stability based on dynamic time warping.

Definition of dynamic stability
Subject's walking is one kind of periodic movements and the same events will happen during different walking cycles, so the similarity of the data between adjacent cycles to assess subject's walking stability. This paper used dynamic time warping (DTW) to calculate this similarity, which is a method for flexible pattern-matching scheme. It translates, compress and expands a pair of patterns so similar features within the two patterns are matched (Li, 2003). Fig. 28 and Equation (14) show details.

Extracting of dynamic stability features
According to FL model, it's easy to get corresponding position, velocity and acceleration motion data of 19 points. Therefore, 19×3=57 features are extracted to describe human dynamic stability.

Effects of aging on dynamic stability
Equation (15) calculates the sum of single feature value by age.
where m is the number of subjects in the same age class, l k is the average leg length of subjects in the same age class, D j (S i ) is similarity value of the selected feature i by Equation (14), is a single stability value of the same feature i in the same age class A k , and k is the number of age class. According to Equation (14), similarity value of all selected features in the same age group was calculated by age.
where p is the number of features, in this case, p is equal to 32, and k is equal to 5. Fig. 29 shows trend on the change of dynamic stability. This figure tells us that if all dynamic features are employed to do statistics on such kind of stability, the dynamic stability is decreasing with ageing except the twenties. It seems that all of those features are desirable.

Feature selection on dynamic stability
The previous section did statistic on all 57 dynamic stability. To reduce the computational complexity, and more importantly, to get more important and contributing features, this paper did feature selection on those 57 dynamic features. On the one hand, it could simplify the method of data acquisition; On the other hand, it is more persuasive by analyzing those selected features. This section ties to find the best features reflecting the relationship between age and walking stability among those 57 ones, which include 2 57 -1 different kinds of feature combination. A classic method of feature selection, which is the cooperation of adaptive genetic algorithm and support vector machine, was used to do it.

Improved crossover operation
In order to avoid that better solutions with high fitness disappear in a standard genetic algorithm although the algorithm is accommodated again and again. This paper proposed a formulation to adjust crossover probability (p c ) between average fitness and maximum fitness, as shown in Equation (17).
where f max is the maximum fitness of current population, f avg is the average fitness of current population, f' is the larger fitness between two individuals in crossover operation.

Improved mutation operation
Just as crossover operation, there are the same problems with mutation operation. If probability of mutation (p m ) is undersize, new individual can't be generated easily, inversely, genetic algorithm will be a pure searching process.
To solve this problem, this paper improved it as shown in Equation (18).
where f is the fitness of individual going to mutate. All other parameters have the same meaning as Equation (17).

Improved support vector machine (SVM)
According to information of age classification, SVM was used to separate datasets and assess fitness of specific feature combination during feature selection. This paper improves SVM in two parts: classification balancing and evaluation. A conventional SVM is to build a decision function f c (x) for each class C. Then use Equation (19) as the predicted class label.
However, this equation may fail to work in some skewed inseparable distribution. Therefore, it's improved as Equation (20), which suggests a function p c (f) to balance values of f c (x) in Equation (19).
Another problem is about evaluation. Generally, correctness is calculated by correctNumber/totalNumber, but it does not fit well to skewed distributions. It's improved as Equation (21)

Selected stable features
After 18 generations of GA, 32 walking stability features were selected with classification correctness from 89.4% to 94.7%. To compare number of markers between before and after feature selection, those 32 features are marked with 14 red markers, as shown in Fig. 30. It means that there is (30-14)/30=53.3% reduction on markers, which could simplify equipment to a large extent.  Table 3 shows these fours features with gray background in details. The difference between this method and previous one is the number of dynamic features. Therefore, the same statistic method will be applied on this one. Another, the similarity calculated by DTW is used to assess the stability of specific dynamic feature. Because DTW doesn't care about the data unite, all features from position, velocity and acceleration will be counted together. As a response to the assumption that elder the person is, less stable his gait is. This method assesses walking stability by searching the best contributing features and doing statistics on them. The result shows that walking stability truly becomes worse as ageing except the group of twenties.

Walking symmetry analysis
Gait symmetry analysis is a part of normal gait analysis. Our research is based on the Fourteen-Linkage Walking Model of human. The detail of this model can be seen in chapter 2. We all know, gait symmetry reflects the general characteristics of human walk gait, and it is an important indicator to assess the function of the human walk. Especially in the human aging process the recession of brain and central nervous system and physiological function will affect the lower limb gait of left or right side, and lead gait mutation.

Footprint symmetry
Here, we mainly aim to investigate the footprint properties of the left and right foot. See Fig.32, the symmetric properties were step length and toe out angle of two feet.

Right step length
Stride length

Left toe out
Walking base Besides step length and toe out angle, we also consider step factor and walk ratio.
Step factor is the value of step length divided by leg length. Walk ratio is the value of step length divided by step rate and step rate is the number of steps per minute someone walks.

Definition of footprint symmetry
DEFINITION 5. (Footprint Symmetry) Footprint symmetry is described by the difference of the bilateral footprint features. We extract some symmetrical features of footprint as S(F), which contains the difference of Step length about two feet, the difference of Toe out angle of right foot and left foot, the difference of bilateral Step factor and the difference of bilateral Walk ratio, as shown in Equation (22).
Here δ(A i ) is the standard deviation of the feature A i , and μ(A i ) is the mathematical expectation of the feature A i . A i is the element of {LStepLength, RStepLength; LToeout, RToeout; LStepFactor, RStepFactor; LWalkRatio, RWalkRatio}.
The δ/μ of footprint bilateral features per decade of age is calculated, shown as the upper part in Table 4. The last row of Table 4 is the S(F) of above items.

Effects of aging on footprint symmetry
It can be seen that the variation of footprint feature is mostly less than 0.02 excepted on feature Toeout as shown in Fig. 33.

Cycle symmetry
This section shows the main characteristics of the symmetry of the gait cycles. As movement of two legs turns symmetrically, the ten tags of walking cycle (see Fig. 20) in fact is a symmetrical composition of 5 keywords in one side, they are TO, FA, TV, IC, HR.

Definition of cycle symmetry
DEFINITION 6. (Cycle Symmetry) Cycle symmetry is described by the variability of the cycle symmetric features. The cycle symmetry S(C) is defined as Equation (23).
Here δ(C i ) is the standard deviation of the feature C i and μ(C i ) is its mathematical expectation, and μ(S i ) is the mathematical expectation of the feature Speed. LC i or RC i is the element of {LTO, RTO; LHR, RHR; LFA, RFA; LTV, RTV}. Since the use of the relative value of sagittal plane, so the IC in here is meaningless.

Effects of aging on cycle symmetry
Here described left and right foot gait cycle symmetrical properties change with aging groups. We calculated the δ/μ on average speed of every decade of age, as shown in Table 5. Also the last line of Table 5 is the different value of feet around the corresponding cycle time points. It can be seen that the variation of cycle feature of every group is less 0.20, but the variety has no obvious trend as shown in Fig. 34.

Orbital symmetry
Orbital symmetry is described by the variability of the locus of symmetric points of body, in particular in the sagittal plane. Before doing calculation, we must do some adjustment about dynamic data of right side and left side. Adjustment method is to move the left side data ahead about half stride cycle. That is, we let the LIC to match the RIC. The first removal data may put to last part according to stride cycle, or be doffed and then we got three steps bilateral data. It is clearly that selected features of symmetry may as indicators for evaluation of a person's gait. According to our work, the indices of motion data may be an immediate measure for quantification dynamic gait symmetry. The indices of up-body even the arm can be used as a factor to qualify the symmetry of human walking. The symmetry of gait should include footprint, cycle and dynamic data. Normal walking does not need consider anything, but walking is very complex control, including central command, physical balance and coordination control, involving segments about feet, ankle, knee, hip, torso, neck, shoulder, arm and joint coordination. Any aspect of the disorder may affect gait symmetry, and some abnormalities may be compensatory or conceal. Pathological gait is often characterized by asymmetry, the selected frequencies of the movements of the limbs may deviate considerably from their eigenfrequencies and symmetry may be abandoned (Murray, 1967).

Dynamic symmetry based on dynamic time warping
In theory, the dynamic symmetry of gait is described by the properties which dynamic changed. These dynamic properties consist of three parts: position point symmetry, segment angle symmetry and joints angle symmetry.

Definition of dynamic symmetry
Theoretically, dynamic symmetry is described by the variability of the walking symmetric bilateral features. The dynamic symmetry is also composed of three parts, position symmetry S(Mo), segment symmetry S(SA) and Joint symmetry S(JA), as Equation (24).

S d (D) = S d (Mo) + S d (SA) +S d (JA)
Here Mo means motion data which has 8 attributes, SA means Segment Angle data and JA means Joint Angle data, they both have 5 attributes. There is a total of 18 attributes. Actually, we use the index like S x (Mo), S y (Mo), S z (Mo) and S x (SA), S y (SA), S z (SA) and S x (JA), S y (JA), S z (JA), not the S d (D).
The dynamic walking symmetry also has velocity and acceleration data. We got attributes as Then, we also can do calculation with three velocity equations.
Similarly, it can easily be listed out three acceleration formulas. Thus, we have three sets of attributes, each containing 18 attributes. So, considering three directions, the dynamic symmetry may include 3×18×3=162 indexes.

Effects of aging on dynamic symmetry
We use DTW algorithm to calculate the discrepancy between right data and left data on all attributes of all tested persons. The example of calculated result is shown as Fig. 35. We can see that somebody was more asymmetry than others on this attribute, for example number 2 and number 11 in Z direction.

Feature selection on dynamic symmetry
The dynamic symmetry is concerned with some direction features more meaningfully. Also some features did not reflect out gait characteristics goodly. Therefore, it is necessary in a large number of features to make a choice. The existing gait symmetry indicators more used in accordance with the swing phase and stance phase, then neglected bilateral discrepancy of these phase while them were divided from cycles, thereby reducing the sensitivity of indicators. We analyzed all results of the calculations and found that the dynamic symmetry is concerned with some direction indicators more meaningfully. Also some indicators did not reflect out gait characteristics goodly. Therefore, it is necessary in a large number of indicators to make a choice. In other words, it must do some feature selection. Human walking is a complex procedure with both limb position variability and limb motion. When quantifying symmetry of walking gait, the used parameters and calculations should be chosen carefully (Karaharju-Huisnan, 2001)

Selected symmetric features
Although the tested persons are all healthy and walking with normal gait, but the individual may have a great difference in gait symmetry. So the general average method may result in larger bias, then it can think over use standard deviation. Consider the three dimensional direction respectively, the gait symmetry data can available in three matrices. Then the rows of the matrix are the various attributes, and columns correspond to the tested persons of gait symmetry data. In this way we get three 54×n matrices and n is person number. The original data come from the calculation results of bilateral discrepancy about symmetric attributes, and the calculation method was described in above sections. For every attribute, that is the row of the matrix here, it can use the function fa to calculate the minimal rate of discrepant value.  There are 24 attributes in Table 6 to Table 8, and their rate less than 0.1 (see bold). In other words, these attributes have expressed better gait symmetry in this database. Also it can do same work on other two directions. These attributes can be selected out for future testing a person whether or not symmetrical of his gait.

Clustering on symmetry features for classification
We use dynamic time warping (DTW) algorithm to calculate the similarity between bilateral symmetric attributes. Consider the three dimensional direction respectively, the gait symmetry data can available in three matrices. Then the rows of the matrix are the various attributes, and columns correspond to the tested persons of gait symmetry data. In this way we get three 54×n matrices (assumed we have n valid test persons). The matrices data come from the calculation results of bilateral discrepancy about symmetric attributes. Now we plan to do some clustering analysis, we need an algorithm which is no high cost on machine, but better on efficiency. Since there is a lot of clustering algorithms, for the sake of quickly carry out research and obtain some results, we have chosen the affinity propagation clustering algorithm APCLUSTER (Frey & Dueck, 2007). The reason for using this algorithm is that it not only can do clustering on data but also can pass the original information rather than random values into the clustering processing, and meanwhile the algorithm has good performance and efficiency.
Using the APCLUSTER algorithm, we have done all the attributes clustering analysis in collusion with age or height or weight and so on. But in order to facilitate description, the following example is mainly discussed on age and gait symmetric data of sagittal direction.
In addition, here we had 48 valid test objects, that is to say n=48. Before the data entry the clustering algorithm function for computing, we also need to normalize the data, so that the two vectors may in the same range, and then the output graphics could be easier to see clearly with the clustering results. The normalizing ways and means are as follows.
In the Equation (27), the A is a vector and x i is its element. The minA is the minimum element of the vector and evidently the maxA is the maximum element of the vector. We get normalized element data as x i '. That is, the original vector is A = (x 1 , x 2 , ..., x n ), and then the normalized vector is A' = (x 1 ', x 2 ', ..., x n ').
Here are some examples of clustering of symmetric attributes. One clustering result is shown in Fig. 36. The inputs of the clustering algorithm are two vectors, one is the test person's gait kinetic data of Position Elbow and another is the corresponding age data.
In terms of our Fourteen-Linkage walking model, there are 54 symmetric attributes in one direction and they can be done clustering analysis. So there are 162 attributes in all for three directions. For each gait symmetric attribute to do clustering, the original symmetry data must be normalized to a vector. Take test person's age data to a vector, and take their gait data of one symmetric attribute to another vector. So the horizontal axis is normalized age data, and the vertical axis is normalized data of the symmetric attribute. Here m is the number of clustering centres or number of clustering groups, n i is the number of members of each group, and C i is the clustering centre of group i, while G i (j) denoted the member-point of the group i. The distance calculation in the Equation (28) can use the Euclid's, such as the Equation (36).
Of course, the distance can be calculated using other methods, such as the absolute value of errors. This will be discussed later. Use the Equation (28) with (29), we have calculated the values of all the clustering, choose the smallest ten values of the clustering attributes, and listed in Table 9. From the Table 9, we can see that the smaller the SA value of 10 attributes has a relatively closer clustering results. In the Equation (28), if using other methods to calculate the distance, such as the Equation (30) and (31), we can compare their 10 minimum value of corresponds to the attributes and found that many of them are same, only individual attribute of exceptions, shown in Table 10. We followed the value of SA from small to large, listed from top to bottom.
Although every time the results of clustering is not exactly the same, but in the course of dozens of experiments, the attributes of the top ten were similar broadly, only slightly changed before and after the order. The values of these attributes are the 10 smallest of SA which value was calculated by three different distance formula D 1 , D 2 , and D 3 . So that you can more clearly see that most of these attributes are overlapped. This indicates that the relationship of that using any distance formula and the selection of attributes was not so great. We can see Fig. 36 and Fig. 37 are different in shape and the latter are not so beautiful shape. These clusters are very closer to the center of all its members. Perhaps because of the presence of a large offset values, so most of the clusters are compressed in a small range. We can find that clustering results displaying isolated points of the cluster, actually these points show these persons with bad gait symmetry. It is maybe the method to classify asymmetry gait and symmetry gait.

Main conclusions
In this paper, we have proposed a so-called FL model to analyze the walking stability and symmetry of different age subjects while walking on a normal pace. The most important finding is that human walking stabilities are not strictly monotone decreasing with age. Walking stability of human beings varies with age, but does not reduce in the elderly people always.
1. The variability of footprint increases with age for subjects over 30 years old, and it dramatically increases for the elderly over 60 years old, showing much less footprint stability. 2. The variability of walking cycle declines with age. That is to say, the elderly subjects have more cycle stability. 3. The variability of orbital increases with age for all subjects. In other words, the elderly has weaker orbital stability. 4. Human dynamic stability decreases with age except the twenties, which proves previous assumptions. This data mining technology not only gets the contributing dynamic stability features, but also makes the data acquisition simpler.

Discussions in clinic
Aging effects on motor control have been implicated as a key factor in adjusting posture during walking. Sensory feedback and muscular strength play important roles in maintaining stability against the presence of unpredictable external perturbations or internal variations of gait. The footprint stability and walking stability of 20 years old subjects is less than that of over 30 years old. This cannot certify that 20 years old subjects have less quality of neuromuscular control. Nevertheless, they have much strength to control their walking pattern, so it shows a springily walking pattern. The orbital stability is strictly monotone decreasing with age. The orbital stability could express the ability of stability control more appropriately. As we mentioned above, in most comprehensive opinions, walking stability will decrease with ageing. But the cycle stability increase with age. Why? It seems to be more confused to understand. In fact, in the three kinds of stability, only the cycle stability describes the relative stability of walking, which is the relative relationship among occurrence sequence of cycle events in a walking cycle, while the other two kinds of stability are absolute stability of posture. That indicates that the elderly subjects have a rigid and inflexible walking pattern. The elderly improves his/her walking stability by maintaining cycle stability more carefully, because it needs less strength to control cycle stability than the other two. That is to say, young subjects have more powerful muscles to control walking balance, while elderly subjects improve their walking stability by keeping their fixed walking patterns carefully. This is one of the most important findings of this paper. One conclusion about gait symmetry is that, according to the attributes that selected out by APCLUSTER algorithm and our calculation analysis, we can classify some test objects in order to better meet the natural age groups. An appropriate grouping method to gait symmetry analysis will make the results of statistical analysis more meaningful. Another conclusion is that, when the symmetry evaluation of normal walking gait, compared the trunk with the limbs, the latter gave larger contribution. Thus, if we have a device to measure gait symmetry of normal people, it may be wrong to wear it at the waist. It may very appropriate if we paste the device somewhere in the lower extremities (for example, shank or ankle), of course pair-wise and it will be more effective.
The next study is to further deepen the existing clustering classification, including gait symmetry attributes and the relationship between weight and height in order to obtain meaningful results. It is our goal that combining gait symmetry attributes with a number of individual characteristics may construct a simple approach to determine a test object should belong to which group. Clinical gait analysis is aimed at revealing a key aspect of abnormal gait and impact factors, so as to assist the rehabilitation assessment and treatment, but also help to assist the clinical diagnosis, evaluation. We hope that we can evaluate the symmetry degree of a person gait accurately, not whether symmetry or asymmetry. In other words, it can't use one piece of value, but use a set of indices to evaluate. And then each index may indicate an aspect of gait symmetry. Further research needs to determine how these gait symmetry is related to actual fall risk. At least to a certain extent, the symmetry between the low-body such as legs seems to codetermine the stability of walking.

Acknowledgment
This work is supported in part by the Fukushima Prefectural Foundation for the Advancement of Science and Education (No.F-18-10), Japan, Shanghai, and Shanghai Leading Academic Discipline Project (J50103), China, and Pujiang Program from Science and Technology Commission of Shanghai Municipality, China. The basic part of this work was implemented in the Biomedical Information Technology Lab, the University of Aizu, Fukushima, Japan.