Diploma thesis project on Study, and modeling of pedestrian walk with regard to the improvement of stability and comfort on walkways. The thesis was implemented both in the University of Modena & Reggio Emilia, Italy during an Erasmus+ internship (most of the basic version), as well as in the Technical University of Crete with the supervision of Assistant Professor Fabrizio Pancaldi and Professor Michalis Zervakis. A state-of-the-art paper on the subject will soon be published. Abstract: The static stability of footbridges or pedestrian walkways can be effectively assessed through several approaches developed in the fields of mechanical and civil engineering. On the other hand, the dynamic stability of pedestrian walkways represents an underexplored field and only in the last 2 years, the comfort of such structures has been investigated. A walkway under the tendency to oscillate, provokes panic and insecurity of the users and needs to be appropriately addressed in order to guarantee the safety of pedestrians.In this thesis, we introduce an innovative algorithm for modeling and simulation of the human walk using Gaussian Mixture Models. Our model satisfies the requirements of simplicity, ease of use by engineers and is suitable to accurately assess the dynamic stability of walkways. Furthermore, we implement a simulator that can be used to provide reliable prediction and assessment of floor vibrations under human actions. Evaluation results are promising, showing that our simulator is capable of supplementing the experimental procedure in future research.

1. Study and modeling of pedestrian walk with regard to the
improvement of stability and comfort on walkways
Pavlos Paris Giakoumakis
School of Electrical & Computer Engineering
Diploma Thesis Presentation
Diploma Thesis project - Erasmus+ internship at University of Modena & Reggio Emilia
(UNIMORE)

2. The Pedestrian Walkway
• Bridges designed for pedestrian crossings between two points.
Gained prominence in contemporary architecture of the 21st
century because of the growing urban expansion and the push
towards a “greener” mobility
• Lightness and slenderness are substantive aesthetic
characteristics in the design of a modern footbridge
• The loads these structures must support are generally low
(about 400-500 𝑘𝑔/𝑚2
)
• Thousands of pedestrians crossed the Millennium Bridge in
June 2000 to celebrate the grand opening
• Closed for modifications two days later because of unexpected
lateral vibration due to resonant structural response.
Pedestrians crossing the bridge unconsciously matched their
footsteps to its lateral oscillation, exacerbating it
The Knokke Footbridge in Deerlyck, Belgium
The Millennium Bridge in London, UK

3. Vibration Serviceability
• Ensuring the human comfort when crossing such a walkway
• Walking, jogging, jumping, and running are common sources of dynamic excitations leading to
vibration problems
• The vibrations’ frequencies must be distant from the low frequencies perceived by humans
• The pedestrian is a typical receiver of vibrations in the walkway but also represents the excitation
source that this work will focus on, due to the induced dynamic load
• The evaluation of vibrations induced by humans is still widely debated and poorly understood despite
the importance of vibration serviceability in structural design
• This problem would be surpassed by an acceptable uniform model of the human walk

4. Thesis Contributions
• Calibration of Walk database that consists of valid measurements for 𝟐𝟏𝟓 pedestrians
• Extraction of the appropriate variables describing the human gait
• Statistical modeling of variables based on all-samples database
• Estimation of the statistical model describing the parameters
• Development of a gait simulator based on the statistical model – Extraction of variables for a defined
number 𝑛 of steps
• Establish a uniform model of human walk as well as an experimental procedure and reference dataset
for future research

5. Presentation Outline
Introduction
•The Pedestrian Walkway
• Vibration Serviceability
•Thesis Contribution
•Thesis Outline
Background &
RelatedWork
•The Bipedal Walking
•Problem Definition
•Modeling of Human
Walking Forces
ProposedApproach
Experimental Framework
Variable Modeling
Parameter Modelling
The Simulator
Experimental Setup
Instruments Used for the
Analysis
Evaluation of Simulation
Conclusion & Future
Work
Conclusion
Future Work

6. The Bipedal Walking

7. The Walking Cycle
• Two phases during a single step:
 Swing phase: the period when the foot is off the
ground
 Stance (or contact) phase:The period during which
the foot contacts the floor
• Two stages during the walking process:
 Double-support stage: both feet are in contact with
the ground
 Single-support stage: one foot is in contact, while the
other one is off the ground

8. The Vertical Load Induced During a Walk
• Two sources of randomness:
 Intersubject variability: Different persons have different key
parameters in induced forces (weight, step frequency, walking
speed, etc.)
 intrasubject variability: A person produces forces that are
different at each footfall
• Only the vertical force component will be addressed:
 Has the highest magnitude of all
 By far the most significant in vibration serviceability analysis
• The vertical force typically has two peaks and a trough
• Main factors considered affecting peak force’s amplitude:
the weight of the person and the frequency steps

9. Model Design

10. Variables Describing Human Walk
• The human walk is described through four variables
 𝑭𝒌:The mean vertical force applied at the 𝑘𝑡ℎ
step - related to the weight and gait of the pedestrian
 𝚫𝒕𝒌:The time interval between the 𝑘 − 1𝑡ℎ
and 𝑘𝑡ℎ
step
 𝒍𝒌:The length traversed between the 𝑘 − 1𝑡ℎ
and 𝑘𝑡ℎ
step
 𝜽𝒌:The direction of the 𝑘𝑡ℎ
step (angle between the 𝑘 − 1𝑡ℎ
and 𝑘𝑡ℎ
step)
• Human steps are never identical
 Pedestrian walk is modeled as a series of steps where each parameter of a given step is stochastically related
 The model is memoryless and unable to describe some events i.e. a stumble or a collision

11. The Model
The mathematical model uses a Markov chain application called random walk
 The position of each pedestrian is a sequence - each step’s position depends exclusively on the position of the
previous step
 The pedestrian is walking towards an endpoint
 The state of the Markov chain at the 𝑘𝑡ℎ
step is described as 𝑺𝒌 = {𝑭𝒌, 𝚫𝒕𝒌, 𝒍𝒌, 𝜽𝒌}
 The Markov chain is continuous in the state space 𝑆 and discrete in time
 The steps are independent and identically distributed (i.i.d.)
 Transition probability density function (pdf): 𝑝(𝑆) = 𝑝(𝐹, Δ𝑡, 𝑙, 𝜃)

12. The Model
• The correlation between the 4 variables describing the
human walk was extracted by Martina Fornaciari*
 Correlation coefficients of distinct variables have been
assessed and then averaged over the number of students in
order to devise general indications
• The 𝐷𝑡-𝐹, 𝐷𝑡-𝑙 and 𝐹-𝜃 correlations are important - any
other correlation can be considered negligible
• 𝐷𝑡, 𝐹 and 𝑙 are correlated, while 𝜃 can be considered as
statistically independent from the others
𝐷𝑡 𝐹 𝑙 𝜃
The average matrix correlation between the variables as
extracted by Martina Fornaciari*
Thus, the pdf 𝒑(𝑺) can be written as:
𝒑 𝑺 = 𝒑 𝑭, 𝚫𝐭, 𝐥, 𝛉 = 𝐩 𝐅, 𝚫𝐭, 𝐥 𝐩(𝛉)
*Martina Fornaciari, “Sviluppo di un modello matematico per la camminata bipede: Applicazione nello studio di stabilità delle passerelle ciclopedonali”,
Università di Modena e Reggio Emilia (UNIMORE), 2018

13. Gaussian Mixture Modeling
• Let a GMM with K components. The kth
component has:
 Mean: 𝒎𝒌 for the univariate case and 𝝁𝒌 for the multivariate case
 Variance/covariance: 𝝈𝒌 for the univariate case and 𝚺𝒌 for the multivariate case
 Mixture component weights: 𝝓𝒌 with the constraint that 𝑖=1
𝐾
𝜙i = 1 - the total probability distribution normalizes to 1
• pdf of a GM based on K components:
 Univariate case (modeling 𝜃, means, component weights, covariance tables):
𝒑 𝜽 = 𝒊=𝟏
𝐌
𝝓𝒊𝑵(𝒎𝒊, 𝝈𝒊) where 𝑁 𝑚𝑖, 𝜎i =
1
𝜎𝑖 2𝜋
exp −
𝑥−𝑚𝑖
2
2σi
2
 Multivariate case (modeling 𝐹-Dt−l):
𝒑 𝑭, 𝜟𝒕, 𝒍 = 𝒊=𝟏
𝑲
𝝓𝒊𝑵(𝝁𝒊, 𝜮𝒊) where 𝑁 𝜇𝑖, Σi =
1
2π 3 Σi
exp −
1
2
𝑥 − 𝜇𝑖
𝑇
Σi
−1
(𝑥 − 𝜇𝑖)
• The transition pdf 𝑝(𝑆) is therefore approximated as:
𝒑 𝑺 = 𝒑 𝑭, 𝚫𝐭, 𝐥 𝒑(𝜽)

14. Modeling of Human Walking Forces
• Challenges in Mathematical Modeling:
 High variability of force waveform shapes due to many parameters
 The dynamic forces induced by a single person are narrowband random processes that are not well understood and thus
are difficult to accurately represent
• Two types of models commonly used for human-induced walking excitation:
 Time-domain force models:
 Deterministic: Aim to generate a uniform force model for any individual without directly considering the natural variability between people.
Based on key assumptions such as perfectly periodic induced forces and both feet producing exactly the same force
 Probabilistic: Considers the intersubject variability of each person
 Frequency-domain force models
• Weaknesses of deterministic force modeling:
 Do not explicitly consider the inter-and intrasubject variability of human walking
 Classification of floors by their natural frequency is not considered accurate in certain circumstances e.g., a floor with a
fundamental frequency close to the 9-10 Hz threshold probably exhibits a mixed response
 The computed vibration response is usually compared to a tolerance limit.This binary assessment does not provide enough
information for the design engineer to make an informed decision

15. Probabilistic Force Models
• Introduced in the literature as a more reliable approach to deterministic
models
• Human activity is rather a random or stochastic process with
nondeterministic behavior (i.e., the next state is not predictable knowing
the current state)
• The intersubject and intrasubject variabilities are taken into
consideration i.e., an extended experimental procedure is required
• Instead of producing a single binary vibration response value, the
vibration response can be expressed as a probability that it will not
exceed a certain value
• There is a notable difference between step properties from several
studies in the literature
• Step properties have been proven to depend on gender, nation, culture
and the environment
• Several researches conclude that step frequency is typically expressed as
normal distribution
Normal distributions of the step frequency for
normal tempo walking as extracted in several
researches
Normal distribution of walking speed at 𝟏. 𝟖Hz of
step frequency

16. Modeling Plan
Parameter
modeling
concerns the
estimation of
the models’
parameters

17. Simulation Plan
Each Parameter (mu, sigma , phi) of a variable,
e.g Dt, is modeled by many components e.g.
low, medium or large. Each component is again
fitted by a GMM model

18. The Database
• Includes valid measurements of 𝟐 traverses in a 10𝑚2
walkway for 𝟐𝟏𝟓 pedestrians in a variety of
physical characteristics in a normal walking tempo
• Includes 𝟐𝟏𝟓x𝟑𝟐 cells – 32 is the maximum number of steps each pedestrian performs in 2 walkway
traversals
• Each pedestrian’s step (cell) contains:
 Vertical force applied to the floor (𝒇𝒐𝒓𝒄𝒆)
 Time that the foot was in contact with the ground (𝒕𝒊𝒎𝒆)
 Contact coordinates in 𝑥 and 𝑦 axes (𝒙_𝒄𝒐𝒐𝒓𝒅, 𝒚_𝒄𝒐𝒐𝒓𝒅)
Depiction of 𝑥 and 𝑦 axes

19. The Final Database

20. • Computation of each step’s necessary variables
based on the database samples
 Interarrival time between steps (𝑫𝒕)
 Mean vertical force induced in the floor on each step
(mean force)
 The traversed distance between steps (length)
 The step angle along the gait horizontal direction (angle)
• Extraction of unified table X (32x4x215)
• Each subject provides measurements for 𝟐 walks on
the pathway
 When the second walk starts, the analysis resets to zero
and the first step must ignore the effect of previous steps
which belong to the first walk
Variable Extraction
Dt-meanF-length-angle
Pedestrians
Steps

21. Length & Angle Extraction
• The x_coord and y_coord tables are used
 𝑥1−2 and 𝑦1−2 denote the traversed x and y distances
between the first and second step:
𝑥1−2 = 𝑥2 − 𝑥1 and 𝑦1−2 = 𝑦2 − 𝑦1
 Length can be calculated as the Euclidean distance between
one step and the next:
𝑙𝑒𝑛𝑔𝑡ℎ1−2 = 𝑥1−2
2
+ 𝑦1−2
2
 Angle is the arctangent of the traversed 𝑦 and 𝑥:
𝑎𝑛𝑔𝑙𝑒1−2 = arctan
𝑥1−2
𝑦1−2
 Using this procedure, all lengths and angles traversed on each
step of a walk can be determined.
3 consecutive steps depicted as circles based on 𝑥_𝑐𝑜𝑜𝑟𝑑 and
𝑦_𝑐𝑜𝑜𝑟𝑑 coordinates

22. Variable Modeling

23. • Use of Gaussian Mixture Models (GMMs)
 No correlation of step angle (angle) with the other three random variables has been reported
 Correlation between Dt – Mean force – length
 A GMM of 𝟑 components has been fitted in these 𝟑 variables as well as a GMM of 𝟐 components for the step
angle - 430GMMs in total (215x2)
• Each GMM is characterized by 3 parameters:
 Mean table (mu): components x variables vector (3x3 for the 3 variables or 2x1 for the angle)
 Mixing probability – component proportion coefficients: array in the size of the GMM’s component number (3
or 2) – sums to 1, describes the a-posteriori estimates of the component probabilities
 Covariance matrix (Sigma): Symmetric and positive semi-definite vector describing the variance/covariances
between variables. Size: variables x variables (3x3 or single value)
Variable Modeling

24. The Bayesian (BIC) and Akaike (AIC) Information Criteria
• The Bayesian information criterion (BIC) is a criterion for model selection among a finite set of
models. It is based, in part, on the likelihood function, and it is closely related to the Akaike
information criterion (AIC).
• Mathematically BIC can be defined as: 𝐵𝐼𝐶 = ln 𝑛 𝑘 − 2ln(𝐿)
• AIC can be defined as: A𝐼𝐶 = 2𝑘 − 2ln(𝐿)
 𝑳:The maximized value of the likelihood function of the model
 𝒏:The number of data points
 𝒌: the number of parameters to be estimated
• A lower AIC or BIC value indicates lower penalty terms, hence a better model
• BIC considers the number of observations in the formula, while AIC works without it

25. The Default Index Method - Results
• AIC and BIC values for a variety of
components on 11 pedestrians
 In most cases, theAIC/BIC values vary
irregularly with the number of components
 This probably occurs due to the impact of
noise in the data
 The formal use of AIC/BIC criteria may lead
to complex models and overfitting
• Thus, the application of each method
needs modification as to derive models
of acceptable complexity

26. • A unified table with all database steps is created for each person
• A GMM is fitted on the 3 variables (F, Dt, l)
 The AIC/BIC values on 1-15 components are extracted
 A different result is produced each time a GMM is fitted
 Therefore, the same procedure is repeated 12 times (abstract) in order to get more representative results
 A mean index evaluation diagram is then created from all the 12 repetitions
 The score difference is computed on this diagram and is graphically depicted. Note that large improvement in
score implies large negative score difference
 The lowest pointy in the previous graph, depicting the largest improvement on the AIC/BIC scores, denotes the
selected number of components
• We test model fitting for multivariate problem as well as for the univariate case
Fitting a GMM on the Unified Walk data

27. Fitting on the Unified Walk Data
Repetition 1 Repetition 3
Repetition 2 Repetition 4
Repetition 5 Repetition 6 Repetition 7 Repetition 8
Repetition 9 Repetition 10 Repetition 11 Repetition 12

28. Fitting on the Unified Walk Data
MEAN SCORE DIAGRAM SCORE DIFFERENTIALS

29. • The largest AIC/BIC score improvement (biggest decrease) is spotted on the 𝟑-component GMM
• The same procedure is followed on random separate pedestrians to identify
the complexity index of the 3-component model
 TheAIC/BIC values on 1 − 15 components are extracted
 The same procedure is repeated 20 times (abstract) since a different result is produced each time a GMM is fitted
 A mean score diagram is created and the step difference graph is computed
 The lowest graph point, or the largest score difference in the AIC/BIC scores, denotes the selected number of
components
• It can be inferred that the 𝟑-component model offers the best optimality-complexity payoff in
most cases
• The same procedure is followed before every GMM fit
Fitting on the Pedestrian Walk Data

30. Fitting on a Random Pedestrian
MEAN SCORE DIAGRAM SCORE DIFFERENTIALS

31. Multivariate GMM of 3 Components
Fitted on Dt, mean force & length

32. Univariate GMM of 2 Components
Fitted on angle of step

33. Parameter Modeling

34. • Estimating GMMs’ parameters (mu, Sigma, weights) in order to extract a complete statistical model
that describes the human gait
 The parameters of the multivariate GMMs describing the 3 variables and the univariate GMMs describing the
angle of step are modeled
 These parameters are described as normal distributions
 These models will be used in the simulation phase to estimate the parameters of the final GMM describing a
random walk
• Modeling mu:
 Separate Gaussians for each variable’s mu have been fitted
 One Gaussian for each component of each variable
 Unification and sorting of mu’s in order to acknowledge the small values
 Therefore, 3 Gaussians have been fitted for each of the 3 multivariate GMM’s variables (Dt, meanF, length) and 2
more for the angle –Total: 𝟏𝟏 Gaussians for the mu modeling
Parameter Modeling

35. • Modeling mixing probability – component proportion coefficients (weights):
 A Gaussian describing the component proportion coefficient of each component has been fitted
 Correlation with mu values – During mu’s sorting, the coefficients are shifted to correspond to each component’s
mu
 Thus, 3 Gaussians fitted for the multivariate model (Dt, meanF, length) and 2 more for the univariate model
describing the angle –Total: 𝟓 Gaussians to describe the component proportion coefficients
• Modeling Sigma:
 A Gaussian for each critical value of Sigma has been fitted, i.e. for the multivariate GMM describing the 3
variables, a symmetric 3x3 matrix is required
 6 critical values of the 3x3 symmetric matrix Σ
 Unified covariance matrices containing the critical values have been created
 Concluding, 6 Gaussians describing the covariance matrix of the 3 variables (Dt, meanF, length) and 1 Gaussian
for the angle have been fitted –Total: 𝟕 Gaussians to describe Sigma
Parameter Modeling

36. Modeling mu - Logic
• Modeling mean values of the 3 components
describing 𝐃𝐭 - The same procedure is followed for
the mean force and length
 Each one of the first column’s values corresponds
to the mean value of each GMM component of Dt
 Modeling each of the 3 GMM parameter values
describing each variable (Dt, F, l)
 Note:The distribution of each variable (e.g. Dt)
has 3 modes, leading to 3 mean values in each
test case with varying amplitudes. Both large and
small-amplitude modes are important.Thus, we
organize 3 groups of test model parameters, i.e.
large, medium, small.
 Initially, ascending sorting of the variable’s mean
values is performed

37. Modeling mu - Logic
 Each element is distributed in a corresponding array
 3 arrays containing the mu’s of each GMM are created –
ascending order
 A Gaussian is fitted in each array
 Total: 3 Gaussians modeling the mu’s of Dt
 Hence, it is ensured that the low mu values are not ignored
in our modeling
 Modeling of angle is performed in a similar logic – with the
use of 2 Gaussians

38. Normal Distributions Fitted on mu – Dt variable

39. Normal Distributions Fitted on mu – mean force variable

40. Normal Distributions Fitted on mu – angle variable

41. Modeling Sigma - Logic
 Symmetric 3x3 matrix – 6 critical values
 Each variance/covariance value is matched to an
array
 A Gaussian is fitted in each array
 Total: 6 Gaussians modeling Sigma
 Angle: 1 Gaussian in variance values is fitted

42. Normal Distributions fitted on each critical value of Sigma –
Multivariate Model

43. Modeling Overview
Variable Extraction
Variable Modeling
Parameter Modeling

44. The Simulator

45. • Extraction of the 𝟒 variables describing a gait of 𝒏 steps based on the statistical model
• A reversed procedure is followed
 The 3 parameters (mu, mixing probabilities and Sigma) are generated randomly using the Gaussians describing
them
 A multivariate GMM is fitted using the 3 generated parameters.ThisGMM constitutes the distribution that
describes the human walk
 𝑛 sets of the 𝟑 variables (Dt, meanF and length) are generated using this GMM
 A similar procedure (fitting a univariate GMM) is performed in order to extract the step angle using the
corresponding Gaussians describing the counterpart parameters
Developing a Simulator

46. • 𝟐 GMMs are fitted using the generated
parameters
• 𝒏 sets of variables are generated using these
GMMs (Dt, meanF and length using the
multivariate and angle of step using the
univariate)
• The merging of those sets describes a random
walk of 𝒏 steps
Extraction of Random Walk

47. Instruments Used for the Analysis
• Experimental campaign conducted at the University of Modena and Reggio
Emilia, Department of Engineering “Enzo Ferrari” (Modena, Italy)
• Instrumented floor equipped with a sensor system
• 10 plates occupying an area of 1𝑚2
, each, for a total of 10𝑚2
of available
surface on the walkway
• This approach allows to acquire the quantity of interest directly unlike other
available methods used in the literature
• Force sensors are installed beneath each vertex of the plates so that the
induced force can be measured
• The position of the applied force can be inferred through simple
trigonometric manipulation
• The output of the force sensors is acquired through an acquisition system
that consists of a control unit and a computer manufactured by National
Instrument.

48. • The simulated results approach the experimental quite
effectively
 Notable difference in the mean force values due to different body types
affecting the measurements
 Average speed of simulated gaits (1000 simulations): 𝟓. 𝟐𝒌𝒎/𝒉
 Average speed of human walk at a normal tempo: 𝟒. 𝟓𝟏 − 𝟓. 𝟒𝟑 𝒌𝒎/𝒉
 The two plots depict a simulated gait in comparison to an experimental
Evaluation of Simulation

49. • In this thesis:
 We presented a practical model to use at the design stage of a walkway that satisfies the requirements of
simplicity (the GMMs and Gaussians used require simple calculations), ease of use by engineers as well as accuracy
(a large number of experimental samples is used)
 We presented an innovative algorithm for modeling and simulation of human gait - No other research work has
usedGMMs for the modeling of human walk
 We conceived the human walk as a random process – only a few researches with a small number of data have
done so
 We modeled the human walk talking into account the intersubject and intrasubject variability
 We implemented a simulator using the developed models that can be used to provide reliable prediction and
assessment of floor vibrations under human actions
 Thus, using this approach, the experimental procedure can be omitted in future researches
Conclusion

50. • Apply the simulator in a digital pedestrian walkway developed by the Civil and Environmental
Engineering Department of University of Modena & Reggio Emilia (Dipartimento di Ingegneria "Enzo
Ferrari”) in order to conduct research on bridge oscillations and the resonance effect
• Extend the model to include real world occasions e.g., the presence of many pedestrians in the
walkway, the synchronization of step within groups, the contingency of a collision, a random roaming,
jogging, running, etc.
• Use of a bigger instrumented floor approximating the dimensions of a real footbridge to allow for
more steps at each stride
• Consider other factors e.g., gender, physical characteristics, nation, culture, the environment etc.
• Publication of research paper that includes this work
Future Work

51. Thank you!