**5. 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.

#### **5.1 Footprint symmetry**

386 Health Management – Different Approaches and Solutions

Age 20+ 30+ 40+ 50+ 60+

LHIP 0.0001 0.0000 0.0001 0.0001 0.0003 RHIP 0.0001 0.0001 0.0001 0.0001 0.0003 CENT 0.0001 0.0000 0.0001 0.0001 0.0003 LKNE 0.0002 0.0005 0.0001 0.0006 0.0003 RKNE 0.0002 0.0002 0.0001 0.0004 0.0013 LANK 0.0012 0.0010 0.0001 0.0001 0.0009 RANK 0.0013 0.0006 0.0001 0.0001 0.0005 LHEE 0.0028 0.0027 0.0001 0.0002 0.0013 RHEE 0.0027 0.0018 0.0000 0.0001 0.0011 LTOE 0.0004 0.0001 0.0001 0.0001 0.0002 RTOE 0.0004 0.0002 0.0001 0.0001 0.0002

LELB 0.0125 0.0023 0.0122 0.0046 0.0055 RELB 0.0111 0.0030 0.0080 0.0039 0.0055 CENT 0.0021 0.0009 0.0016 0.0018 0.0024 LKNE 0.0109 0.0077 0.0038 0.0035 0.0066 RKNE 0.0109 0.0076 0.0024 0.0034 0.0054 LANK 0.0563 0.0405 0.0088 0.0091 0.0495 RANK 0.0678 0.0318 0.0594 0.0062 0.1520 LHEE 0.1522 0.1369 0.0055 0.0200 0.0775 RHEE 0.1447 0.0892 0.0073 0.0847 0.0662 LTOE 0.0336 0.0135 0.0067 0.0079 0.0158 RTOE 0.0400 0.0201 0.0129 0.0569 0.0204

LSHO 0.0071 0.0030 0.0041 0.0061 0.0037 RSHO 0.0071 0.0033 0.0047 0.0054 0.0035 NECK 0.0062 0.0028 0.0038 0.0051 0.0031 LELB 0.0162 0.0046 0.0230 0.0080 0.0069 RELB 0.0137 0.0045 0.0094 0.0080 0.0077 CENT 0.0059 0.0029 0.0054 0.0349 0.0449 LKNE 0.0293 0.0179 0.0214 0.0119 0.0161 RKNE 0.0299 0.0187 0.0095 0.0136 0.0204 LANK 0.0765 0.0505 0.0342 0.0432 0.0562 RANK 0.0915 0.0544 0.4998 0.0270 2.2505 LHEE 0.1604 0.1196 0.0812 0.0901 0.0887 RHEE 0.1662 0.1154 0.0477 0.2507 0.0980 LTOE 0.1567 0.0687 0.0274 0.2517 0.0965 RTOE 0.2266 0.1225 0.0816 0.2483 0.1541 *F(Ak)* **1.5449 0.9497 1.0028 1.0980 3.2639**

**Position** 

**Velocity** 

**Acceleration**

Table 3. Selected features in three types

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.

Fig. 32. Features of footprint for symmetry analysis

Human Walking Analysis, Evaluation and Classification Based on Motion Capture System 389

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

DEFINITION 6. (Cycle Symmetry) Cycle symmetry is described by the variability of the

Here *δ(Ci)* is the standard deviation of the feature *Ci* and *μ(Ci)* is its mathematical expectation, and *μ(Si)* is the mathematical expectation of the feature Speed. *LCi* or *RCi* is the element of {LTO, RTO; LHR, RHR; LFA, RFA; LTV, RTV}. Since the use of the relative value

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.

> **Age 20+ 30+ 40+ 50+ 60+** *S(TO) 0.05567 0.10183 0.04917 0.03336 0.05064 S(FA) 0.01323 0.01264 0.0179 0.01706 0.01161 S(HR) 0.03106 0.01222 0.03407 0.01832 0.02689 S(TV) 0.00201 0.03435 0.01366 0.01486 0.00648 S(C)* **0.10198 0.16104 0.11479 0.08359 0.09562**

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

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

( )/ ( ) ( )/ ( ) / ( ) *i i i ii*

*LC LC RC RC S*

 

(23)

 

symmetrical composition of 5 keywords in one side, they are TO, FA, TV, IC, HR.

cycle symmetric features. The cycle symmetry *S(C)* is defined as Equation (23).

4

1

*i* 

of sagittal plane, so the IC in here is meaningless.

variety has no obvious trend as shown in Fig. 34.

**5.2.2 Effects of aging on cycle symmetry** 

Table 5. Variation of cycle features

Fig. 34. Variation of cycle feature

**5.3 Orbital symmetry** 

**5.2 Cycle symmetry** 

**5.2.1 Definition of cycle symmetry** 

 **S(C)** =

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.

#### **5.1.1 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).

$$\mathcal{S}(\mathbf{F}) = \sum\_{i=1}^{4} |\delta(LA\_i) \mid \mu(LA\_i) - \delta(RA\_i) \nmid \mu(RA\_i)| \tag{22}$$

Here *δ(Ai)* is the standard deviation of the feature *Ai*, and *μ(Ai)* is the mathematical expectation of the feature *Ai*. *Ai* 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.

#### **5.1.2 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.


Table 4. Variation of footprint features

Fig. 33. Variation of footprint features

#### **5.2 Cycle symmetry**

388 Health Management – Different Approaches and Solutions

Walk ratio is the value of step length divided by step rate and step rate is the number of

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


*LA LA RA RA*

 

(22)

Here *δ(Ai)* is the standard deviation of the feature *Ai*, and *μ(Ai)* is the mathematical expectation of the feature *Ai*. *Ai* is the element of {LStepLength, RStepLength; LToeout,

The *δ/μ* of footprint bilateral features per decade of age is calculated, shown as the upper

It can be seen that the variation of footprint feature is mostly less than 0.02 excepted on

**Age 20+ 30+ 40+ 50+ 60+**  *S(StepLength) 0.01818 0.00865 0.00238 0.01286 0.00210 S(Toeout) 0.01660 0.04051 0.04079 0.16155 0.05628 S(StepFactor) 0.01803 0.00859 0.00247 0.01157 0.00298 S(WalkRatio) 0.00814 0.01854 0.00939 0.00293 0.00638 S(F)* **0.06095 0.0763 0.05504 0.18892 0.06774**

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.

4

1

*i*

RToeout; LStepFactor, RStepFactor; LWalkRatio, RWalkRatio}.

part in Table 4. The last row of Table 4 is the *S(F)* of above items.

**5.1.2 Effects of aging on footprint symmetry** 

steps per minute someone walks.

**5.1.1 Definition of footprint symmetry** 

Walk ratio, as shown in Equation (22).

 *S(F)* =

feature *Toeout* as shown in Fig. 33.

Table 4. Variation of footprint features

Fig. 33. Variation of footprint features

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.

#### **5.2.1 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).

$$\mathbf{S(C)} = \sum\_{i=1}^{4} \left| \delta(\text{LC}\_i) \right> \left/ \mu(\text{LC}\_i) - \delta(\text{RC}\_i) \right> \left/ \mu(\text{RC}\_i) \right| / \left/ \mu(\text{S}\_i) \tag{23}$$

Here *δ(Ci)* is the standard deviation of the feature *Ci* and *μ(Ci)* is its mathematical expectation, and *μ(Si)* is the mathematical expectation of the feature Speed. *LCi* or *RCi* 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.

#### **5.2.2 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.


Table 5. Variation of cycle features

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.

Fig. 34. Variation of cycle feature

#### **5.3 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

Human Walking Analysis, Evaluation and Classification Based on Motion Capture System 391

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

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

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> <sup>15</sup> <sup>16</sup> <sup>17</sup> <sup>18</sup> <sup>19</sup> <sup>20</sup> <sup>5</sup>

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> <sup>15</sup> <sup>16</sup> <sup>17</sup> <sup>18</sup> <sup>19</sup> <sup>20</sup> <sup>0</sup>

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> <sup>15</sup> <sup>16</sup> <sup>17</sup> <sup>18</sup> <sup>19</sup> <sup>20</sup> <sup>0</sup>

Person Group P20

A total of 162 attribute needs for a similar calculation, to get the related discrepant values

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

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

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

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

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

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

like Fig. 35. These results were used to do statistical analysis about gait symmetry.

indicators to make a choice. In other words, it must do some feature selection.

may result in larger bias, then it can think over use standard deviation.

TOE of Position

symmetry may include 3×18×3=162 indexes.

and number 11 in Z direction.

10 15 X- Variability

> 1 2 3 Y- Variability

0.005 0.01 0.015

Fig. 35. Example of calculated result on one attribute

from cycles, thereby reducing the sensitivity of indicators.

should be chosen carefully (Karaharju-Huisnan, 2001)

**5.5 Feature selection on dynamic symmetry** 

large number of features to make a choice.

**5.5.1 Selected symmetric features** 

Z- Variability

**5.4.2 Effects of aging on dynamic symmetry** 

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).

#### **5.4 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.

#### **5.4.1 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).

$$\text{S(D)} = \text{S(Mo)} + \text{S(SA)} + \text{S(IA)}\tag{24}$$

$$\text{Here, } \operatorname{S(Mo)} = \sum\_{i=1}^{8} \Lambda \Big( L\mathbf{P}(\mathbf{t})\_i \, \big( \operatorname{R}\mathbf{P}(\mathbf{t})\_i \big) \big), \operatorname{S(SA)} = \sum\_{i=1}^{5} \Lambda \Big( L\Phi(\mathbf{t})\_i \, \big( \operatorname{R}\Phi(\mathbf{t})\_i \big) \big), \operatorname{S(IA)} = \sum\_{i=1}^{5} \Lambda \Big( L\Theta(\mathbf{t})\_i \, \big( \operatorname{R}\Theta(\mathbf{t})\_i \big) \big).$$

And LP(t)∈{ p1, p3, p5, p7, p9, p13, p15, p17 }, RP(t)∈{ p2, p4, p6, p8, p10, p14, p16, p18 }, L(t)∈{ φ3, φ5, φ9, φ11, φ13 }, R(t)∈{ φ4, φ6, φ10, φ12, φ14 }, L(t)∈{ θ3, θ5, θ7, θ9, θ11 }, R(t)∈{ θ4, θ6, θ8, θ10, θ12 }.

The symbol *Δ* in the formulas will be described in next section. It is the DTW algorithm used to calculate the distance of two sequences with different length of them. Here consider all the symmetry properties included on upper body and lower body is want given the general definition of gait dynamic symmetry.

Considering 3-dimension, add *d={x, y, z}* to Equation (24), then we got Equation (25).

$$\mathbf{S}\_d(\mathbf{D}) = \mathbf{S}\_d(\mathbf{M}\mathbf{o}) + \mathbf{S}\_d(\mathbf{S}\mathbf{A}) + \mathbf{S}\_d(\mathbf{I}\mathbf{A})\tag{25}$$

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 *Sx(Mo), Sy(Mo), Sz(Mo)* and *Sx(SA), Sy(SA), Sz(SA)* and *Sx(JA), Sy(JA), Sz(JA),* not the *Sd(D).*

The dynamic walking symmetry also has velocity and acceleration data. We got attributes as follow: *VMo V(t), VSA Ω(t), VJA Ω(t), AMo A(t), ASA Λ(t), AJA Λ(t)*. 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.

### **5.4.2 Effects of aging on dynamic symmetry**

390 Health Management – Different Approaches and Solutions

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

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

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

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) = S(Mo) + S(SA) +S(JA)* (24)

*, S(JA) =*

5

1

*i*

() , () *i i*

*Lt Rt*

*.* 

() , () *i i*

*Lt Rt*

footprint, cycle and dynamic data.

symmetry may be abandoned (Murray, 1967).

angle symmetry and joints angle symmetry.

() , () *i i*

*, S(SA) =*

5

1

And LP(t)∈{ p1, p3, p5, p7, p9, p13, p15, p17 }, RP(t)∈{ p2, p4, p6, p8, p10, p14, p16, p18 }, L(t)∈{ φ3, φ5, φ9, φ11, φ13 }, R(t)∈{ φ4, φ6, φ10, φ12, φ14 }, L(t)∈{ θ3, θ5, θ7, θ9, θ11 },

The symbol *Δ* in the formulas will be described in next section. It is the DTW algorithm used to calculate the distance of two sequences with different length of them. Here consider all the symmetry properties included on upper body and lower body is want given the general

 *Sd(D) = Sd(Mo) + Sd(SA) +Sd(JA)* (25) Here *Mo* means motion data which has 8 attributes, *SA* means Segment Angle data and *JA*

Actually, we use the index like *Sx(Mo), Sy(Mo), Sz(Mo)* and *Sx(SA), Sy(SA), Sz(SA)* and *Sx(JA),* 

The dynamic walking symmetry also has velocity and acceleration data. We got attributes as

*(t), AMo* 

 *A(t), ASA* 

 *Λ*

*(t), AJA* 

 *Λ(t)*.

*i*

Considering 3-dimension, add *d={x, y, z}* to Equation (24), then we got Equation (25).

means Joint Angle data, they both have 5 attributes. There is a total of 18 attributes.

 *Ω*

*(t), VJA* 

*LP t RP t*

**5.4.1 Definition of dynamic symmetry** 

*Here, S(Mo) =* 8

1

*i*

R(t)∈{ θ4, θ6, θ8, θ10, θ12 }.

*Sy(JA), Sz(JA),* not the *Sd(D).*

follow: *VMo*

definition of gait dynamic symmetry.

 *V(t), VSA* 

 *Ω*

Then, we also can do calculation with three velocity equations.

**5.4 Dynamic symmetry based on dynamic time warping** 

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.

A total of 162 attribute needs for a similar calculation, to get the related discrepant values like Fig. 35. These results were used to do statistical analysis about gait symmetry.

### **5.5 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)

### **5.5.1 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

Human Walking Analysis, Evaluation and Classification Based on Motion Capture System 393

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 *5*4×*n* matrices (assumed we have *n* valid test persons). The matrices data come

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

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.

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

min '

max min *i*

In the Equation (27), the *A* is a vector and *xi* 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 *xi'*. That is, the original vector is *A = (x*1*, x*2*, ..., x*n*),* and then the

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

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> <sup>0</sup>

Normalized AGE

48 43

Numerals are AGE Cluster Number = 7

> 65 62

40 <sup>38</sup> <sup>4546</sup> 46<sup>4748</sup> <sup>42</sup> 41

43

Clustering on Sagittal Direction

<sup>38</sup> <sup>36</sup>

*x A*

*A A*

(27)

*i*

*x*

data, and the vertical axis is normalized data of the symmetric attribute.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 36. Clustering on Position Elbow and Age

Normalized ELBOW Position

23

29 23 22 22

30 36

28 32 3131 30

22

23

22 21

from the calculation results of bilateral discrepancy about symmetric attributes.

In addition, here we had 48 valid test objects, that is to say *n=48.*

performance and efficiency.

means are as follows.

normalized vector is A' = (x1', x2', ..., xn').

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.

$$f\mathbf{a} = \frac{\overline{A} - A\_{\text{min}}}{A\_{\text{max}} - A\_{\text{min}}} \tag{26}$$

Here *A* also expresses the discrepant value on attribute *A* of all persons. So *A* is the average value of all tested persons on attribute *A*, the *Amin* is the minimum value of the row, and the *Amax* is the maximum value evidently.

It is clear, attribute *A* is the element of the set which includes all symmetric attributes about 8 position items, 5 segment angle items, 5 joint angle items and their velocity and acceleration data.

The following table shows the comparable rates on Z direction, or namely sagittal direction.


Table 6. Position attributes with velocity and acceleration


Table 7. Segment angle attributes with velocity and acceleration


Table 8. Joint angle attributes with velocity and acceleration

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.

#### **5.5.2 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

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

*A A fa A A*

Here *A* also expresses the discrepant value on attribute *A* of all persons. So *A* is the average value of all tested persons on attribute *A*, the *Amin* is the minimum value of the row, and the

It is clear, attribute *A* is the element of the set which includes all symmetric attributes about 8 position items, 5 segment angle items, 5 joint angle items and their velocity and

The following table shows the comparable rates on Z direction, or namely sagittal direction.

 SHO ELB WRI HIP KNE ANK HEE TOE *Mo* 0.166 0.157 **0.030 0.033** 0.113 **0.073** 0.151 0.180 *VMo* 0.408 0.177 **0.059 0.029** 0.453 **0.063** 0.289 0.292 *AMo* 0.266 0.305 **0.042 0.024** 0.262 **0.046** 0.292 0.176

*SA* 0.191 **0.048 0.034 0.079** 0.132 *VSA* 0.147 0.109 **0.038 0.046** 0.225 *ASA* 0.220 **0.043 0.039 0.032** 0.168

*JA* 0.213 0.124 **0.031** 0.144 0.109 *VJA* 0.172 0.280 0.133 **0.069** 0.247 *AJA* **0.046 0.042 0.037 0.048 0.031** 

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

These attributes can be selected out for future testing a person whether or not symmetrical

We use dynamic time warping (DTW) algorithm to calculate the similarity between bilateral symmetric attributes. Consider the three dimensional direction respectively, the gait

UPPERAR FOREAR THIGH SHANK FOOT

SHOULDER ELBOW HIP KNEE ANKLE

min max min

(26)

the minimal rate of discrepant value.

*Amax* is the maximum value evidently.

Table 6. Position attributes with velocity and acceleration

Table 7. Segment angle attributes with velocity and acceleration

Table 8. Joint angle attributes with velocity and acceleration

**5.5.2 Clustering on symmetry features for classification** 

same work on other two directions.

of his gait.

acceleration data.

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 *5*4×*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.

$$\mathbf{x}\_{i} = \frac{\mathbf{x}\_{i} - \min A}{\max A - \min A} \tag{27}$$

In the Equation (27), the *A* is a vector and *xi* 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 *xi'*. That is, the original vector is *A = (x*1*, x*2*, ..., x*n*),* and then the normalized vector is A' = (x1', x2', ..., xn').

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.

Fig. 36. Clustering on Position Elbow and Age

Human Walking Analysis, Evaluation and Classification Based on Motion Capture System 395

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,

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 D1, D2, and D3. So that you can more clearly see that most of these attributes are overlapped. This indicates that the relationship of that

Attributes **SA using D1 Attributes SA using D2 Attributes SA using D3**

HIP P-VEL 0.035746144 HIP P-VEL 0.001258594 HIP P-VEL 0.000008664

ACC 0.036825407 HIP P-ACC 0.00276713 WRIST Posit 0.000090040

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

ACC 0.00017294 THIGH S-

ACC 0.000894828 ANKLE J-

ACC 0.001953265 SHANK S-

ACC 0.002879147 ANKLE P-

using any distance formula and the selection of attributes was not so great.

HIP J-ACC 0.035669813 HIP J-ACC 0.000523717 THIGH S-

J-VEL 0.03882457 WRIST Posit 0.005047818 WRIST P-

<sup>2</sup> ( ) () ( ) *i i D G j C abs y y ii G <sup>j</sup> <sup>c</sup>* (30)

ACC 0.000000074

VEL 0.000001067

ACC 0.000006696

ACC 0.000023426

VEL 0.000131324

ACC 0.000172132

2 <sup>3</sup> ( ) ( ) *i i DGj C y y i i Gj c* (31)

VEL 0.000598035 HIP J-ACC 0.000003165

VEL 0.00685193 HIP Joint 0.000225147

listed from top to bottom.

THIGH S-

ANKLE J-

THIGH S-

SHANK S-

WRIST P-

SHANK S-

gait and symmetry gait.

KNEE

ACC 0.035657274 THIGH S-

ACC 0.035717594 THIGH S-

VEL 0.035719691 ANKLE J-

ACC 0.036075667 SHANK S-

VEL 0.037846491 WRIST P-

WRIST Posit 0.044141919 SHANK S-

Table 10. Comparison of three distance calculation

(a) Clustering on Acc of Joint Ankle and Age (b) Clustering on Vel of Segment Shank and Age Fig. 37. More examples of clustering

For all the clustering results how to evaluate them. We used the following formula to calculate a mean square error. The formula is as follows.

$$SA = \frac{1}{m} \sum\_{i=1}^{m} \left( \frac{1}{n\_i} \sum\_{j=1}^{n\_i} D\left(G\_i(j) - C\_i\right) \right) \tag{28}$$

Here *m* is the number of clustering centres or number of clustering groups, *ni* is the number of members of each group, and *Ci* is the clustering centre of group *i,* while *Gi(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).

$$D\_1\left(\mathcal{G}\_i(j) - \mathcal{C}\_i\right) = \sqrt[2]{\left(\mathbf{x}\_{\mathcal{G}\_i(j)} - \mathbf{x}\_{c\_i}\right)^2 + \left(y\_{\mathcal{G}\_i(j)} - y\_{c\_i}\right)^2} \tag{29}$$

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.


Table 9. Top ten minimum values of clustering results

(a) Clustering on Acc of Joint Ankle and Age (b) Clustering on Vel of Segment Shank and Age

For all the clustering results how to evaluate them. We used the following formula to

Here *m* is the number of clustering centres or number of clustering groups, *ni* is the number of members of each group, and *Ci* is the clustering centre of group *i,* while *Gi(j)* denoted the

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

Use the Equation (28) with (29), we have calculated the values of all the clustering, choose

Attributes of Z-dir No. SA using D1 Clustering

THIGH S-ACC 1 0.035657274 6 HIP J-ACC 2 0.035669813 6 ANKLE J-ACC 3 0.035717594 6 THIGH S-VEL 4 0.035719691 6 HIP P-VEL 5 0.035746144 6 SHANK S-ACC 6 0.036075667 6 WRIST P-ACC 7 0.036825407 6 SHANK S-VEL 8 0.037846491 6 KNEE J-VEL 9 0.03882457 6 WRIST Posit 10 0.044141919 5

1 1 1 1 ( ) *m ni*

*i j i SA DG j C m n*

2 2 <sup>2</sup>

the smallest ten values of the clustering attributes, and listed in Table 9.

Table 9. Top ten minimum values of clustering results

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized SHANK S-VEL

 2223 <sup>23</sup> 22222223

*i i*

<sup>1</sup> ( ) ( ) ( ) *ii ii DGj C x x y y i i Gj c Gj c* (29)

2829

3132 36 30

22

2321222323

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> <sup>0</sup>

Normalized AGE

(28)

Clustering on Sagittal Direction

Numerals are AGE Cluster Number = 6

<sup>31</sup> <sup>30</sup> 38<sup>40</sup> 4345 46<sup>48</sup> <sup>47</sup> <sup>4142</sup> <sup>43</sup> 5253545454 <sup>57</sup> <sup>62</sup> <sup>65</sup>

38

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> <sup>0</sup>

Normalized AGE

calculate a mean square error. The formula is as follows.

Clustering on Sagittal Direction

Numerals are AGE Cluster Number = 6

2829303132 36383840 41 42434345 464748 5253 5454 57 62 65

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized ANKLE J-ACC

22

23

<sup>22</sup> 21<sup>22222323</sup>

member-point of the group *i*.

errors. This will be discussed later.

Fig. 37. More examples of clustering

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.

$$D\_2\left(\mathcal{G}\_i(j) - \mathcal{C}\_i\right) = abs(y\_{\mathcal{G}\_i(j)} - y\_{\mathcal{C}\_i}) \tag{30}$$

$$D\_3\left(\mathbf{G}\_i(j) - \mathbf{C}\_i\right) = \left(y\_{\mathbf{G}\_i(j)} - y\_{c\_i}\right)^2\tag{31}$$

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 D1, D2, and D3. 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.


Table 10. Comparison of three distance calculation

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.

Human Walking Analysis, Evaluation and Classification Based on Motion Capture System 397

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

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

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

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

The basic part of this work was implemented in the Biomedical Information Technology

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**7. Acknowledgment** 

**8. References** 

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