A nonlinear time series analysis on the effect of foot injury on gait

dynamics

MIHAI DUPAC1, DAN B. MARGHITU2

1Department of Design and Engineering,

Bournemouth University,

Talbot Campus, Fern Barrow, BH12 5BB, Poole, Dorset,

2Department of Mechanical Engineering,

Auburn University,

1418 Wiggins Hall, Auburn, AL 36849,

USA

Abstract: - The effects of foot injuries regarding bilateral asymmetry and gait dynamics are still poorly

understood. Previous work discussed rehabilitation, postural control, and asymmetry, with the models being

mainly validated for upper body translations and no or minimal assessment on rotation. The aim of this study

was to assess the effect of foot injury on gait dynamics. For this, a wearable sensors system for data collection

of the key variables of the of human movement was considered. The dynamics of motion – recorded in the

plane of motion using a laser sensor – was assessed using a new projective method which considers the axial

rotations, translation, and in-plane rotation patterns for normal human gait vs. simulated gait pathologies. A

nonlinear timeseries analysis, along with a Poincare map, phase space, time delay, Lyapunov exponents, and

false nearest neighbors (FNN) method have been considered in order to convey the periodicity of the data

collected for a healthy individual with and without a simulated injury. The Lyapunov exponents which quantity

the degree of separation of nearby trajectories are used to differentiate between the chaotic and non-chaotic

behavior. The positive sign of the largest Lyapunov exponents for all data indicated “the exponential separation

of nearby trajectories as time evolves”, that is, the chaotic behavior of the system.

Key-Words: - Nonlinear analysis, Lyapunov exponents, Poincare map, gait dynamics, wearable sensors

Received: March 15, 2024. Revised: October 17, 2024. Accepted: November 13, 2024. Published: December 11, 2024.

1 Introduction

The use of wearable sensors [1,2] and motion

capture having reflective markers [3] has proved to

be viable solutions in the dynamic analysis of gait

and postural control. Force plates can also be used

to collect data of a small number of strides [4],

however this approach requires expensive force

plates which are adequate to measure the ground

contact point and contact force for a static

simulation, but not for a dynamical approach.

Inertial measurement units (IMUs) using

accelerometers and gyroscopes although ideal to

capture human are unable to maintain long-term

accuracy due to sensor drifting issues [6]. The

influence of some other sensors such as camera

calibration and markers size on the performance of

video-based motion capturing systems was reported

in [5].

A new symmetry angle (SA) index has been

considered in [10] as a substitute for the symmetry

index (SI) in assessing asymmetry. Joint bracing

was considered to assess movement differences of

limb joints for an asymmetric gate dynamics [11].

Inflated ground reaction forces and symmetry index

(SI) values have been reported by Herzog et al. [12]

in the assessment of gait asymmetry as a result of

dividing a reference average by a very small (close

to zero) reference value.

The physical constraints in fast and low speed

running have been analyzed in [14] by evaluation of

body energy storage and transfer. A musculoskeletal

approach of connected links [7] representing

different parts of the body evaluated the associated

length, mass center position, and moment of inertia

using regression equations [31]. Gait assessment

due to injury causing one-sided affection has been

discussed in [16] by objective measurements of gait

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2024.4.16

Mihai Dupac, Dan B. Marghitu

E-ISSN: 2769-2477

146

Volume 4, 2024

quality and gait function or self-reported function. A

combined gait metric (CGAM) that can be regarded

as a benchmark to evaluate and differentiate

kinematic and kinetic parameters has been presented

in [17]. The newly developed CGAM metric

successfully distinguish between multiple

asymmetry differences at different walking

velocities.

A ratio index (RI), as well as a statistical approach

has been considered in [9] to assess gait asymmetry.

The study [8] discussed the use of various

coefficients such as symmetry index (SI), ratio

index (RI), SA and GA, and their consistency in

assessing gait.

Unstable and stable, asymmetric and symmetric,

periodic motions have been observed in [22], and

chaotic trajectories have been detected in [23] using

a bifurcation analysis of the upper part of the body.

A nonlinear time-series analysis was considered in

[32] asses dynamic walking. Nonlinear behavior of

a walking model has been concluded in [24], and

chaotic dynamics of a bipedal model in [25]. The

chaotic dynamics observed in [25] could be part of

an interdisciplinary research for diagnosing gait

pathologies [4].

Actual recommendations for assessing asymmetric

and symmetric periodic or chaotic motion have been

reinforced in [8] due to missing standards and

establish criteria to differentiate between relatively

similar or contradictory results. Therefore, new

investigations to better assess gait dynamics while

overcoming the limitations of previous studies

should be considered.

This paper presents the effects of foot injuries on

asymmetry and gait dynamics. For this, a wearable

sensors system including,

- a foot-mounted pressure sensors that records the

ground contact time, contact point and contact force

- a laser pointer (attached to the subject) to project a

laser spot in the plane of motion

- a high-speed camera that records the laser spot

motion related to dynamics of the human gait in the

plane of motion.

have been used.

A nonlinear timeseries analysis, including Poincare

map and Lyapunov exponents have been considered

for the assessment of the data. The evaluation of key

variables in the assessment of gait dynamics - based

on a new projective method which considers the

axial rotations, translation and in-plane rotation

patterns of the gait in the plane of motion - is

performed by analyzing the trajectory of the motion

in Matlab.

2 Materials and Methods

2.1 Variables to be measured.

Following a review of the literature regarding

human gait dynamics a list of key variables to be

measured such as ground contact time, contact

force, trajectory, and associated devices have been

considered [1, 2, 7, 30]. The dynamics have been

assessed in the plane of motion with respect to the

sagittal plane that divides the human body into right

and left sides, taking into account the axial rotations,

translation and in-plane rotation patterns of the gait.

2.2 Summary of equipment

Once the variables to be measured have been

considered, i.e., ground contact time, contact force,

and trajectory, a range of devices have been selected

to allow data collection. A brief description of the

equipment used, including the manufacturer details.

is presented in Table 1.

Table 1

Outline of the equipment including manufacturer

name and details of the devices

Description

Manufacturer

Details

Flexiforce

WB201

Sensor

Tekscan

Trimmable 3-pin male

connector sensor in

three force ranges

Wireless Flexi

Force WELF 2

Tekskan

Wireless Economical

Load & Force

Measurement System

2 (WELF™ 2)

Digital JVC

GC-PX100

camcorder

JVC

12.8 MP Camcorder -

1080p (record 500

frames per second)

Laser Pointer

GBBG

Light rechargeable

Laser 303 pointer

Camera

Tripod

Vantage M10

Nest Vantage M10

Video Camera Tripod

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2024.4.16

Mihai Dupac, Dan B. Marghitu

E-ISSN: 2769-2477

147

Volume 4, 2024

Treadmill

PRECOR

TRM885 Treadmill

with ITF technology

2.2.1 Laser Sensor Pointer

Since the walking is to be assessed in the plane of

motion with respect to the sagittal plane the

trajectory of the laser spot is fundamental in the

understanding of the dynamics of the system. To

address this, the laser pointer (Fig 1c) was attached

to the back of the subject (at the intersection of the

sagittal and transverse plane as shown in Fig 3) via a

special type of clamp collar device attached to the

belt.

The laser pointed was firmly clamped the collar

as such it can only move with the body. In the initial

position (before start walking or running) the laser

spot projected by the laser pointer on the ground is

located behind the subject on the line defined by the

intersection of the sagittal plane and the plane of

motion (running belt surface) at a distance of 30 cm

from the coronal plane.

The laser pointer - made by hard aluminum withe

an anodized black surface treatment - has the

working voltage of 3.7V, wavelength of 532nm,

range of 500m-1000m, and an adjustable focus and

an APC line circuit control. The setup of the

displacement sensor, i.e., laser pointer, can be seen

in Fig. 3b.

(a) (b)

Fig. 1. Summary of equipment (a) Wireless Flexi

Force WELF 2 handle, (b) Flexiforce WB201

Sensor, (c) Laser sensor (pointer), and (c) Digital

camcorder – JVC GC-PX100.

2.2.2 Ground Force Sensor - Flexiforce A201

Sensor

To build up a more accurate image of the gait

dynamics device it was necessary to better

understand the ground contact point and contact

force for a simulated lower leg injury versus a

healthy uninjured leg. The lower leg injury was

simulated by placing a pebble stone inside the shoe.

The dimensions and shape of the pebble stone is

shown in Fig. 4. The sensors chosen for this task

were piezo-resistive devices of a printed

construction from Tekscan, Inc as shown in Fig. 1b.

These units are flexible, and are having a thickness

of 0.208 mm, a length of 197 mm, a width of 14

mm, and an active sensing area of 9.53 mm

diameter. The units made by polyester have 3-pin

Male Square Pin (center pin is inactive) connectors

with the pin spacing at 2.54 mm. The typical

performance as presented in the FlexiForce™

Wireless Economical Load & Force Measurement 2

(WELF™ 2) datasheet has the linearity < ±3 of

full scale, repeatability < ±2.5, hysteresis < 4.5%

of full scale, and drift < 5% per logarithmic time

scale. The force sensor has been inserted between

the insoles and the foam of the trainer sole, thus

protecting the sensors from direct contact with the

ground.

Fig. 2. Equipment setup including the Precor

TRM885 Treadmill with ITF technology, Star 75

Camera Tripod with attached JVC GC-PX100

camcorder.

The Wireless ELF 2 system comes with three

WB201 FlexiForce sensors (one in each of the three

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DOI: 10.37394/232028.2024.4.16

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E-ISSN: 2769-2477

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Volume 4, 2024

available force ranges), that is, WB201-L: Low (0-

25 lb; 111 N), WB201-M: Medium (0-150 lb; 667

N), and WB201-H: High (0-1000 lb; 4450 N). For

this study the WB201-H sensor was used, i.e., it was

inserted inside the running trainer between the

insoles and the foam of the trainer sol and connected

to the battery-operated WiFi transmitter shown in

Fig 1a.

2.2.3 Battery operated WiFi transmitter

To capture the data being generated by the sensor, a

Tekscan Wireless ELF 2 handle (Fig 1a) operating

under Windows system was considered. The device

is capable of transmitting data at two selected

frequencies, a low 200 Hz frequency transmitting

data at maximum 65 m distance named WELF 2 -

basic system, and a high 6000 Hz frequency

transmitting data at maximum 25 m distance named

WELF 2 - High Speed system. The WELF 2 – basic

system could support up to 16 transmitters for a

maximum distance of 50 m, while the WELF 2 -

High Speed system could support up to 8

transmitters for a maximum distance of 25 m.

The Tekscan Wireless ELF 2 device (Fig 1a) having

the dimensions 46.4mm x 26.7mm x 95.3mm, is

small and lightweight enough (95 grams) to be

placed above the ankle. The device contains its own

batteries (3 AAA alkaline batteries) and can be in

operation for up to 3 hours at the selected frequency.

To ensure repeatability the state of charge i.e.,

output voltage, of the batteries has been checked at

the start of each test.

2.3 Equipment setup

The equipment was set up in a simple way that can

be easily replicated. A picture of the setup including

the Precor TRM885 Treadmill with ITF technology,

the Star 75 Camera Tripod, and the JVC GC-PX100

camcorder is shown in Fig. 1. The tripod/camcorder

was placed at 3 m from the end of the treadmill, and

the orientation of the camera was 40 degrees with

the vertical direction as shown in Fig. 1.

All of the attached equipment i.e., force sensor (Fig

1b), battery operated WiFi transmitter (Fig 1a), and

laser pointer (Fig 1c and Fig 3b) was fitted in a non-

invasive manner not to affect or significantly

influence the use of the foot and the recording of the

data, i.e., the trajectory of the laser spot on the

treadmill running belt surface (Fig 3a). The total

mass of the sensor system, i.e., force sensor and

battery-operated WiFi transmitter, attached to the

foot was found to be 148 grams.

(a)

(b)

Fig. 3. (a) Recorded trajectory of the laser spot for a

30 second time period (approximately 20 – 27

strides) for the gait at 2 mp, (b) Participant walking

on a treadmill having the laser pointer attached

2.4 System Testing

Following The system was tested with a healthy

individual - a 52-year-old male with a height of 1.80

m and a mass of 97 kg - who did not suffer from any

pathology that might adversely affect running style

or repeatability. The selection of the participant and

testing was conducted following the Bournemouth

University ethical approval including the use of

Participant Agreement Form and Participant

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Information Sheet provided in advance to the

participants.

The study required walking/running at 3 different

speeds on a Precor TRM885 Treadmill in order to in

order to assess some key variables such as the foot

ground reaction force and trajectory of motion. A

laser sensor (pointer) was attached to a belt around

the participant waist (Fig 4.) and the spot of the

laser pointer was projected on the treadmill running

belt (Fig 3 and Fig 4). The study consisted in

recording data related to dynamics of the participant

(trajectory of the laser spot) captured in the plane of

motion (on the treadmill running belt). The session

lasted approximately 60 minutes.

Fig. 4. Pebble stone used to simulate a lower leg

injury

The participant was informed about the devices to

be used and questioned if the walking and running

on the treadmill was comfortable. He was allowed

15 to 30 minutes to warm up by choosing his own

pace and cadence at which he felt most comfortable

and get used to the device to ensure that the setup

and the additional mass of the instrumentation

would not cause any issues. Healthy participants

have been considered for this preliminary study,

while disable individuals (lower limb amputees) will

be considered in further studies.

3 Results

The investigation of chaotic behaviour in medical

science and engineering represents essential aspects

of research emphasizing the effect and significance

of chaos (random and unpredictable behaviour in

systems in which “uncertainties increase at an

exponential rate”) within these disciplines [32].

There are various techniques to determine chaos in

dynamical systems related to biology and medicine.

Some of the most well-known techniques namely

phase space analysis, power spectrum, trajectory

tracing or bifurcation diagram are qualitative

analysis approaches requiring the interpretation of

the results. On the other side, the Lyapunov

exponents technique is a quantitative analysis

approach offering some significant advantages over

the previous mentioned techniques. To name some,

Lyapunov exponents can

a) be computed from time series obtained from

experimental data

b) reveal the stability of a system

c) identify the existence of chaos

c) indicate the exponential separation of nearby

trajectories

f) be robust to noise, change in data, and increase of

sample size

In this section, the dynamic evolution of the system

is investigated using the Lyapunov exponents

method.

Data from an exercise trial on a treadmill has been

collected for normal human gait (healthy individual)

and simulated gait pathology, i.e., healthy individual

with a simulated leg injury. The data represents the

trajectory of the laser sensor recorded for a 30

second time period (approximately 20 – 29 strides)

for the gait at 2 mph, 4 mph and 6 mph as shown in

Fig. 3.

The trajectory of the laser spot displacement

gathered for the gait at 2 mph for a healthy

individual and for a healthy individual with a

simulated injury is shown in Fig 5.

(a)

(b)

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Fig. 5. Trajectory of the laser spot gathered for the

gait at 2 mph for, a) healthy individual, and, b)

healthy individual with a simulated injury.

Figure 6a shows the phase plane portrait for the

laser spot displacement data gathered for the gait at

2 mph for a healthy individual while Fig. 6a shows

the phase plane portrait for the laser spot

displacement data for the same healthy individual

with the lower leg simulated injury. The trajectory

shown in Fig. 6 is associated with the motion around

the attractor and shows the classical characteristics

of nonperiodic motion [19]. It is visible in both

cases that the phase trajectory does not close due to

the nonlinearities in the system, that is, the system

exhibits a complex dynamic such as chaos as seen in

Figs. 6a and Fig. 6b.

(a)

(b)

Fig. 6. Dynamical trajectories of the system in the

phase plane for a) healthy individual, and b) healthy

individual with a simulated injury

From Fig. 6 it can be seen that the dynamical

trajectory of the system does not converge with the

increase in the number of interactions, while the

associated trajectory of the simulated injury in phase

plane resembles mostly a nonperiodic wobbly curve.

The return map of the maxima (Poincare map)

viewed as the cross section of the trajectory in state

space provide an enhanced perspective of the

system periodicity [20].

Since a Poincare map detects the intersection points

of the trajectory for a specific section when the

intersection points of the previous section are

known, i.e., describe how the points of a section get

mapped back onto the section, the result helps to

identify the type of the attractor.

A finite set of points with the number of points

corresponding to the period of the attractor define a

periodic attractor, while a Poincare map with a

closed orbit define a quasiperiodic motion. For the

Poincare maps shown in Figure 7a and 7b the data is

not periodic or quasiperiodic. The Poincare map of a

chaotic attractor appears as a large/infinite number

of randomly grouped and respectively ungrouped

scatter dots, that is, the case in Fig. 7a and Fig. 7b.

To assess the walking/running variations in the gait

pattern i.e., the trajectory of the human gait captured

in the plane of motion, the state space of the

attractor was reconstructed using time delays and

nonlinear dynamics embedding dimension

techniques [21]. The purpose of the delays

embedding method was to unfold the state space

projection of the observed gait trajectory back to the

state space that represents the system.

Fig. 7. The Poincare map. The dots dispersion

indicates a chaotic and respectively a hyperchaotic

time series.

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For the reconstruction of the phase space, the time

delay used as important information to certify that

the delayed coordinates are not dependent on the

initial data and the dynamical properties of the

original system are preserved, is calculated as

depicted in Fig. 8.

The method of delay [21] was used to reconstruct

the state space X(t) from the time series by

󰇛󰇜  󰇟󰇛󰇜 󰇛  󰇜 󰇛  󰇜   󰇛  󰇜󰇜󰇠

A proper time delay T is selected at first local

minimum connecting the delayed time series and the

original data, that is the average mutual information

calculated for various time delays based on an

iterative process [21].

(a)

(b)

Fig. 8. The initial minimum average mutual

information for a) healty individual data is located at

the time delay T=17, and for b) healty individual

with a simulated injury data is located at the time

delay T=13

The time delay determined by the difference

between the actual and the delayed state of the

system using the average mutual information as a

function of time [26] is shown in Fig. 8. The time

delay calculated for the trial in Fig. 8a, i.e., healty

individual, is T=17, and for an helathy individual

with a simulated injury (Fig. 8b) is T=13.

The proper embedding dimension of the time series

is determined from the false nearest neighbours

(FNN) method by calculating the distance between

neighbouring points in order to unfold the

reconstructed walking/running attractor in a suitable

state space.

The FFN percentage was calculated at the highest

embedding dimension defined by number of

independent variables, until the dimension reached a

zero percent FFN [27]. From Fig. 9 it can be seen

that the total FFN percentage declines and dE is

chosen where this percentage approaches zero, that

is, the embedding dimension dE=3.

Selecting a minimum embedding dimension have

been considered a key element for decreasing the

noise associated the dynamical system.

Fig. 9. The percentage of false nearest neighbours

against the time series in Fig. 8a for the healthy

individual data located at the time delay T=17 shows

that dE=3.

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The Lyapunov exponents which quantity the degree

of separation of nearby trajectories are used to

distinguish between the chaotic and non-chaotic

behaviour in the state space, determine the nonlinear

structure of the attractor, and analyze the local

stability of the system [18,26].

The type of dynamical evolution of the system is

best shown by the sign of the Lyapunov exponents

[13], one positive exponent indicates and chaotic

motion, more than one positive exponent indicate

instability and hyperchaotic behavior, while

negative or zero exponents indicate a periodic

motion.

Fig. 10. Largest Lyapunov exponents calculation

for the gait at 2 mph for (a) healthy individual (blue

plain circular dots), and (b) same healthy individual

with a simulated injury (red empty circular dots)

To determine the sign of Lyapunov exponents and

to characterize the behavior of the system the

computation method developed in [15,21] was used.

The divergence of nearest neighbors in state space

[21] is estimated as

󰇛󰇜  

󰇛󰇜

where λ is the largest Lyapunov exponent estimated

󰇛󰇜  󰇛󰇛󰇜󰇜



with   representing the average over j.

Fig. 10 shows the values of the largest Lyapunov

exponents calculated for the laser spot trajectory

gathered for the gait at 2 mph, 4 mph, and 6 mph for

the healthy individual (blue plain circular dots) and

for the same healthy individual with a simulated

injury (red empty circular dots).

The value of the largest Lyapunov exponent

quantifies the exponential divergence of the

neighboring trajectories in the reconstructed state

space and reflects the degree of chaos in the system.

The sign of the largest Lyapunov exponents is

positive for all data denoting the exponential

separation of nearby trajectories as time evolves,

that is, the system is characterized by chaotic

behavior. So, one can conclude at this point the

chaotic behavior of each system.

The vertical ground reaction force is also an

important factor in analyzing and understanding gait

dynamics. The ground reaction force data from a

treadmill trial has been collected for the healthy

individual walking at 2 mph, 4 mph, and 6 mph.

With respect to the magnitude of the vertical ground

reaction similar conclusions to the one presented in

[29] have been obtained, that is “that stride

frequency, stride length and contact length”

increased at higher speed by applying greater forces

to the ground. The conclusion is also in good

agreement with the one obtained in [28] showing a

linear increase in ground force from 1.2 body

weight (BW) during walking to approximately 2.5

BW when running at 6.0 m/s.

4 Conclusion

This study provides new information about the

effect of foot injury on gait dynamics. The approach

was based on a new projective method which

considers the axial rotations, translation and in-

plane rotation patterns of the gait in the plane of

motion. Poincare map, phase space, FNN method,

Lyapunov exponents and correlation dimension –

which offer excellent quantifications for various

characteristics of gait dynamics - have been

considered to validate the assessment. The positive

sign of the largest Lyapunov exponents for all data

indicate “the exponential separation of nearby

trajectories with time”, that is, the chaotic behavior

of the system.

Acknowledgement:

This research was conducted at the facilities of the

Department of Design and Engineering,

Bournemouth University.

Source of Funding:

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DOI: 10.37394/232028.2024.4.16

Mihai Dupac, Dan B. Marghitu

E-ISSN: 2769-2477

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Volume 4, 2024

The research was funded by the Department of

Design and Engineering, Faculty of Science and

Technology, Bournemouth University.

Conflict of interest:

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

Contribution of Individual Authors to the Creation

of a Scientific Article:

Mihai Dupac, has organized and executed the

experiment presented in Section 2.

Dan B. Marghitu carried out the simulation

presented in Section 3.

Both authors discussed the results and contributed to

the final version of the manuscript.

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The authors equally contributed in the present

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problem to the final findings and solution.

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Scientific Article or Scientific Article Itself

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Conflict of Interest

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that are relevant to the content of this article.

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International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2024.4.16

Mihai Dupac, Dan B. Marghitu

E-ISSN: 2769-2477

155

Volume 4, 2024