**4.1.1 Definition of footprint stability**

DEFINITION 2. (Footprint Stability) Footprint stability is described by the variability of the footprint stability features. We extract some of the stability features of footprint *FS (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).

$$F\_S(F) = \sum\_{i=1}^{7} \delta(f\_i) / \,\mu(f\_i) \tag{8}$$

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

#### **4.1.2 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.

Human Walking Analysis, Evaluation and Classification Based on Motion Capture System 377

LTO LFA LTV LIC LHR

50 60 70 80 90 100 10 20 30 40 50

stance Left swing phase Left stance phase

Terminal swing

Midswing

support

Loading

Right stance phase Right swing phase

Preswing

0 10 20 30 40 50 60 70 80 90 100

Double

Left leg Right leg

Such as

LTO = left toe off;

RHR = right heel rise; LTV = left tibia vertical; LIC = left foot initial contact;

RTO = right toe off;

LHR = left heel rise; RTV = right tibia vertical.

Loading

Left

Pre-

support Right single support

swing Initial swing

RIC RHR

Fig. 20. Walking cycle in normal gait

LFA = left foot adjacent to right foot;

RFA = right foot adjacent to left foot;

**4.2.2 Effects of aging on cycle stability** 

Table 2. Variability of walking cycle features

Table 2 shows the variability of walking cycle features per decade of age.

**Age 20+ 30+ 40+ 50+ 60+** RIC 0.0000 0.0000 0.0000 0.0000 0.0000 LTO 1.6664 1.2287 1.2272 0.9334 1.2268 LFA 1.2846 1.7323 1.3302 1.0199 1.1198 LTV 1.6354 1.8543 1.4046 1.2759 0.9008 LIC 1.2822 1.0320 1.0900 0.9751 0.7036 RTO 1.3563 1.2323 1.2866 0.7376 0.8605 RFA 0.9731 1.3764 1.4480 1.0737 0.8847 RTV 1.0459 1.2067 1.3846 1.3705 1.0909 *FS*(*C*) 9.2440 9.6627 9.1713 7.3861 6.7870

RIC = right foot initial contact;

response Mid-stance Terminal stance

Left single support Double

swing Initial swing

response Mid-stance Terminal stance

Mid-

RTO RFA RTV RIC

Terminal swing


Table 1. Variability of footprint features

#### **4.2 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.

Fig. 19. Footprint stability with age

#### **4.2.1 Definition of cycle stability**

DEFINITION 3. (Cycle Stability) Cycle stability is described by the variability of the cycle stability features. The cycle stability *FS(C)* is defined as Equation (9).

$$F\_S(\mathbb{C}) = \sum\_{i=1}^7 \delta(c\_i) / \,\mu(f\_4) \tag{9}$$

where *δ(ci)* is the standard deviation of the time when event i occurs, *ci* = {LTO, LFA, LTV, LIC, RTO, RFA, RTV}, and *μ(f4)* is the mathematical expectation of the speed *f4*.

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.

Fig. 20. Walking cycle in normal gait

Such as

376 Health Management – Different Approaches and Solutions

**Age 20+ 30+ 40+ 50+ 60+**  CycleTime 0.0437 0.0293 0.0333 0.0464 0.0716 LStepLength 0.0825 0.0779 0.0568 0.0766 0.0811 RStepLength 0.0645 0.0790 0.1099 0.0586 0.0645 Speed 0.0833 0.0736 0.0819 0.0709 0.0880 WalkingBase 0.1783 0.1721 0.1495 0.1269 0.1343 LToeOut 0.4047 0.1579 0.2407 0.2383 1.3606 RToeOut 0.3246 0.2175 0.3473 0.5013 1.0242 *FS(F)* 1.1817 0.8073 1.0195 1.1190 2.8243

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

DEFINITION 3. (Cycle Stability) Cycle stability is described by the variability of the cycle

4

(9)

 

7

1 ( ) ( )/ ( ) *S i i FC c f* 

where *δ(ci)* is the standard deviation of the time when event i occurs, *ci* = {LTO, LFA, LTV,

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

LIC, RTO, RFA, RTV}, and *μ(f4)* is the mathematical expectation of the speed *f4*.

important information to determine the positions of the human body.

stability features. The cycle stability *FS(C)* is defined as Equation (9).

Table 1. Variability of footprint features

Fig. 19. Footprint stability with age

**4.2.1 Definition of cycle stability** 

shown in Fig. 20.

**4.2 Cycle stability** 


#### **4.2.2 Effects of aging on cycle stability**

Table 2 shows the variability of walking cycle features per decade of age.


Table 2. Variability of walking cycle features

Human Walking Analysis, Evaluation and Classification Based on Motion Capture System 379

( ) [ ( ( ))] [ ( ( ))] [ ( ( ))] [ ( ( ))] [ ( ( ))] [ ( ( ))]

Thus, the orbital stability of lower limbs includes 108 indexes. For examples, Fig.22 and Fig. 23 show the joint angles of hips (*θ7, θ8*), knees (*θ9, θ10*) and ankles (*θ11, θ12*) in sagittal plane during a single walking cycle, their angular velocity (*ωθ9, ωθ10, ωθ11, ωθ*12) and angular

*iii*

*ZZZ*

*Si i i*

 

 

acceleration (*αθ9, αθ10, αθ11, αθ12*) respectively.

Fig. 23. Angular velocity and acceleration of joint

Fig. 22. Joint angles

*FJ Z Z Z*

 

( 7 ~ 12)

(13)

 

 

*i*

 

 

Fig. 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.

Fig. 21. Cycle stability with age

#### **4.3 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).

#### **4.3.1 Definition of orbital stability**

DEFINITION 4. (Orbital Stability) Orbital stability is described by the variability of the orbital stability features. The orbital stability *FS(O)* of lower limbs is composed of three parts, position stability *FS(P),* segment stability *FS(S)* and Joint stability *FS(J),* as shown in Equation (10).

$$F\_S(O) = F\_S(P) + F\_S(S) + F\_S(f) \tag{10}$$

where position stability *FS(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).

( ) [ ( ( ))] [ ( ( ))] [ ( ( ))] [ ( ( ))] [ ( ( ))] [ ( ( ))] ( 1 ~ 4,7,8) *Si i i ii i F P pZ vZ AZ p Z vZ AZ i* (11)

*(x)* and  *(x)* are the maximum and minimum value of x respectively. Segment stability *FS(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).

$$\begin{aligned} F\_S(S) &= \delta[\vee(\wp\_i(Z))] + \delta[\vee(\alpha\_{\psi i}(Z))] + \delta[\vee(\alpha\_{\psi i}(Z))] \\ &+ \delta[\wedge(\wp\_i(Z))] + \delta[\wedge(\alpha\_{\psi i}(Z))] + \delta[\wedge(\alpha\_{\psi i}(Z))] \\ & & (i = 9 \sim 14) \end{aligned} \tag{12}$$

Joint stability *FS(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).

$$\begin{aligned} F\_S(f) &= \delta[\vee(\theta\_i(Z))] + \delta[\vee(a\_{\partial i}(Z))] + \delta[\vee(a\_{\partial i}(Z))] \\ &+ \delta[\wedge(\theta\_i(Z))] + \delta[\wedge(a\_{\partial i}(Z))] + \delta[\wedge(a\_{\partial i}(Z))] \\ & & (i = 7 \sim 12) \end{aligned} \tag{13}$$

Thus, the orbital stability of lower limbs includes 108 indexes. For examples, Fig.22 and Fig. 23 show the joint angles of hips (*θ7, θ8*), knees (*θ9, θ10*) and ankles (*θ11, θ12*) in sagittal plane during a single walking cycle, their angular velocity (*ωθ9, ωθ10, ωθ11, ωθ*12) and angular acceleration (*αθ9, αθ10, αθ11, αθ12*) respectively.

Fig. 22. Joint angles

378 Health Management – Different Approaches and Solutions

Fig. 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 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

DEFINITION 4. (Orbital Stability) Orbital stability is described by the variability of the orbital stability features. The orbital stability *FS(O)* of lower limbs is composed of three parts, position stability *FS(P),* segment stability *FS(S)* and Joint stability *FS(J),* as shown in Equation (10).

where position stability *FS(P)* is defined by standard deviation of the maximums and minimums of displacement, velocity and acceleration (knees, heels, and toes), as shown in

( ) [ ( ( ))] [ ( ( ))] [ ( ( ))]

Segment stability *FS(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

( ) [ ( ( ))] [ ( ( ))] [ ( ( ))]

Joint stability *FS(J)* is defined by standard deviation of the maximums and minimums of displacement, velocity and acceleration (hips, knees and ankles), as shown in Equation

*Si i i*

 

 

*FS Z Z Z*

 

[ ( ( ))] [ ( ( ))] [ ( ( ))]

*ZZZ*

*iii*

*Si i i*

 *(x)* are the maximum and minimum value of x respectively.

*F P pZ vZ AZ*

 

[ ( ( ))] [ ( ( ))] [ ( ( ))]

*p Z vZ AZ*

*ii i*

( ) ( ) () () *FO FP FS FJ S S SS* (10)

( 1 ~ 4,7,8)

( 9 ~ 14)

 

 

*i*

 

  (11)

(12)

*i*

 

Fig. 21. Cycle stability with age

**4.3.1 Definition of orbital stability** 

strongly periodic movements like walking (Marin, 2006).

**4.3 Orbital stability** 

Equation (11).

in Equation (12).

*(x)* and 

(13).

Fig. 23. Angular velocity and acceleration of joint

Human Walking Analysis, Evaluation and Classification Based on Motion Capture System 381

Fig. 27. Position stability, segment stability and joint stability, and orbital stability with age

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.

**4.4 Dynamic stability based on dynamic time warping (DTW)** 

Fig. 26. Each item of orbital stability with age
