seated whole body vibration modelling using inertial
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Seated Whole Body Vibration Modelling using Inertial Motion Sensors Seated Whole Body Vibration Modelling using Inertial Motion Sensors
James Luke Coyte University of Wollongong
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Recommended Citation Recommended Citation Coyte, James Luke, Seated Whole Body Vibration Modelling using Inertial Motion Sensors, Doctor of Philosophy thesis, School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, 2020. https://ro.uow.edu.au/theses1/1094
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Seated Whole Body Vibration Modelling using Inertial Motion
Sensors
James Luke Coyte
Supervisors:
Professor Haiping Du
Associate Professor Montserrat Ros
Associate Professor David Stirling
This thesis is presented as part of the requirement for the conferral of the degree:
Doctor of Philosophy
This research has been conducted with the support of the Australian Government Research Training
Program Scholarship
The University of Wollongong
School of Electrical, Computer and Telecommunications Engineering
16 April 2020
2
Abstract
Operators and drivers of heavy vehicles that are a source of vibrations are at an elevated
risk due the prolonged exposure to whole body vibrations. The pathological effects of
whole-body vibrations are well defined in the literature and include the following: lower
back pain, fatigue and degenerative disorders. The modelling and measurement of the
biomechanical response of the seated human body is extensively used in the areas of
ergonomics and automatic automotive suspension control system technologies.
In recent times there have been rapid advancements in wearable sensor technologies in
terms of the size, weight, power, connectivity and data bandwidth.
Microelectromechanical systems are capable of sensing inertial motion data, have
become increasingly miniaturised. This allows for the practice noninvasive sampling
inertial motion data and applying whole body vibration modelling to a broader scope.
Currently, there are two main gaps in the current state of the art research on whole body
vibration modelling that this thesis aims to fill. The first is that many studies do not deal
with the issue of real time modelling whole body vibration using streaming data. The
second is the lack of use of machine learnt pattern-based methods for forecasting body
model parameters.
This thesis proposes a flexible set of hybrid strategies to solve the filtering problem for
whole-body vibration modelling in real time. The outcome is a solution is given that
solves the modelling in all directions, is robust to sensor drift error and maintains
performance when the dimensionality of the sensor data is reduced.
This thesis proposes solutions to biomechanical modelling using machine learnt pattern
recognition approaches. Firstly, a sequential model is proposed that receive a stream of
inertial motion data to predict the likelihood of volatile movements like jerks and rocking
motions. The second is a model is proposed that is learnt using the paradigm of deep
learning using thousands of spectrograms. This model has advantages over analytical
methods including the ability to accurately approximate a higher order model without
over fitting.
3
Acknowledgments
I would like to express my sincere gratitude to my principal supervisor Professor Haiping
Du, Associate Professor Montserrat Ros and Associate Professor David Stirling, for the
continuous advice of my Ph.D study and research, for their patience, motivation,
enthusiasm, and immense knowledge. Their guidance has helped me in all the time of
research and writing of this thesis. I could not have imagined having a better advisor and
mentor for my Ph.D studies.
Throughout my tenure as a doctoral candidate at the University of Wollongong, my
sincere thanks also go to my employment supervisors for invaluable career development
and supervision including Mr Wayne Coyte, Dr Sasha Nikolic, Associate Professor Raad
Raad, Professor Phillip Ogonbona, Dr Winson CC Lee, Mr David Fulcher, Dr Faisel
Tubbal, Dr Alvaro Casas Bedoya and Professor Benjamin Eggleton.
Last but not the least, I would like to thank my family: Chris Coyte and Sandra Coyte,
Trang Dao, Melanie Coyte, Tu Dao and Ha Nguyen .
I wish to acknowledge Dr Donghong Ning who contributed to the engineering and
fabrication of the parallel robot.
4
Certification
I, James Luke Coyte, declare that this thesis submitted in fulfilment of the requirements for the
conferral of the degree Doctor of Philosophy, from the University of Wollongong, is wholly my own
work unless otherwise referenced or acknowledged. This document has not been submitted for
qualifications at any other academic institution.
James Luke Coyte
16 April 2020
5
List of Abbreviations
ADWIN
APRMS
ARMAX
CNN
DBSCAN
DOF
DPMI
EKF
EM
FRF
GMM
HMM
HVR
LBP
MEMS
MSE
NRMSE
OPTICS
PF
PSD
RMG
RMSE
RTG
SD
SEAT
Adaptive Window
Apparent mass
Autoregressive moving average
Convolutional Neural network
Density Based clustering with Noise
Degree(s) of Freedom
Driving Point Mechanical Impedance
Extended Kalman Filter
Expectation Maximisation
Frequency Response Function
Gaussian Mixture Model
Hidden Markov Model
Human–Vehicle–Road (model)
Lower Back Pain
Microelectromechanical system
Mean Squared Error
Normalised Root Mean Square Error
Ordering points to identify the clustering structure
Particle Filter
Power Spectral Density
Rapid Mixer Granulator
Root Mean Square error
Rubber Tyred Gantry Crane
Standard Deviation
seat effective amplitude transmissibility
6
SMC
STHT
VDV
WBV
Sliding Mode Controller
Seat to Head Transmissibility
Vibration Dose Value
Whole Body Vibration
7
Table of Contents
Abstract ............................................................................................................................. 2
Acknowledgments ............................................................................................................. 3
List of Figures ................................................................................................................. 12
List of Tables .................................................................................................................. 16
1. Introduction ...................................................................................................... 17
1.1. Research Challenges and Significance ..................................................... 18
1.2. Research Aim and Objectives ................................................................... 18
1.3. Contributions and Outcomes ..................................................................... 18
1.4. Thesis Outline ........................................................................................... 20
1.5. Declaration of Works presented in external publications ......................... 21
2. Literature review .............................................................................................. 22
2.1. WBV exposure Standards and Health ....................................................... 24
2.1.1. WBV dosage metrics and guidelines ................................................. 24
2.1.2. Vibration exposure in various vehicles .............................................. 26
2.2. Whole Body Vibration Response Analysis ............................................... 28
2.2.1. Impedance .......................................................................................... 33
2.2.2. Absorbed power and energy .............................................................. 33
2.2.3. Transmissibility ................................................................................. 34
2.2.4. Comparison between impedance and transmissibility methods ........ 35
2.3. Estimation of Whole-Body-Vibration Frequency Response .................... 35
2.3.1. H2 Frequency Response Function estimator ...................................... 37
2.3.2. HV Frequency Response Estimator .................................................... 37
8
2.3.3. Alternative Frequency Response Estimation Methods ...................... 39
2.3.4. Multi-axial vibration response ........................................................... 39
2.4. Whole Body Vibration Sensing Systems .................................................. 40
2.4.1. Inertial bite bar sensors ...................................................................... 41
2.4.2. Seat pad inertial motion sensors ........................................................ 41
2.4.3. Skin mounted MEMS IMUs .............................................................. 42
2.4.4. Optical sensor-based motion capture ................................................. 45
2.5. Applications of Modelling to vehicle Dynamic Systems.......................... 46
2.6. Biomechanical Models .............................................................................. 47
2.6.1. Non analogue models ........................................................................ 50
2.6.2. Rotary models .................................................................................... 50
2.6.3. Non-linear models ............................................................................. 51
2.6.4. Parameter search methods ................................................................. 51
2.7. Vehicle Dynamics Control Systems ......................................................... 52
2.8. Discussion ................................................................................................. 53
2.9. Conclusion ................................................................................................ 60
3. Experiment Methodology ................................................................................. 61
3.1. Introduction ............................................................................................... 61
3.2. Overview of the Parallel Robot ................................................................. 61
3.3. Inertial Motion Data Capture .................................................................... 64
3.4. Experimental Procedure ............................................................................ 69
3.4.1. Experiment Setup ............................................................................... 69
3.4.2. Pilot Platform Test Results ................................................................ 70
3.1. Experimental Platform Data Structure ...................................................... 76
9
4. Chapter 4 Biomechanical Whole-Body Response Model Estimation using an
Exact Noise and Process model ...................................................................................... 77
4.1. Introduction ............................................................................................... 77
4.2. Model identification with EKF ................................................................. 79
4.2.1. Model Definition ............................................................................... 79
4.2.2. EKF Prediction and update steps ....................................................... 80
4.2.3. Drift Detection ................................................................................... 82
4.3. EKF-ADWIN implementation .................................................................. 83
4.3.1. EKF System Model Definition and Decomposition .......................... 83
4.3.2. Process noise covariance update step ................................................ 84
4.4. Experimental Validation ........................................................................... 85
4.5. Results and Discussion.............................................................................. 86
4.6. Conclusion ................................................................................................ 90
5. Approximation methods for estimation of biomechanical models of whole-body
vibration .......................................................................................................................... 92
5.1. Introduction ............................................................................................... 92
5.2. Whole Body Vibration Model................................................................... 94
5.3. Particle Filter Strategy .............................................................................. 96
5.4. Experiment ................................................................................................ 98
5.5. Results and Discussion.............................................................................. 99
5.6. Conclusion .............................................................................................. 103
6. Whole Body Vibration Response Identification Through Unsupervised Learning
for Sensor Dimension Reduction .................................................................................. 104
6.1. Introduction ............................................................................................. 104
6.2. Identification of clusters.......................................................................... 105
6.3. Constrained State Estimation .................................................................. 110
10
6.4. Discussion of results ............................................................................... 118
6.5. Conclusion .............................................................................................. 119
7. A Sequential Seated Whole-Body Biodynamic Model .................................. 120
7.1. Introduction ............................................................................................. 120
7.2. Problem Background Design .................................................................. 121
7.2.1. Mixture Models ............................................................................... 121
7.2.2. Hidden Markov Model Training ...................................................... 122
7.2.3. Biomechanical Modeling ................................................................. 123
7.1. Results and Discussion............................................................................ 126
7.2. Conclusion .............................................................................................. 128
8. Artificial Neural Network Approach to Cross Coupled Whole-Body Vibration
Modeling ....................................................................................................................... 129
8.1. Problem Background............................................................................... 129
8.2. ANN Whole-Body Vibration Model Design .......................................... 130
8.2.1. Spectrogram Construction and Preprocessing ................................. 130
8.2.2. Deep Neural Network Architecture ................................................. 131
8.2.1. Network Training ............................................................................. 132
8.3. Results and Discussion............................................................................ 135
8.4. Conclusion .............................................................................................. 138
9. Concluding Discussions ................................................................................. 139
9.1. Conclusions on the Real Time Estimation of Biodynamic Models ........ 139
9.2. Conclusions on the use of Machine Learnt Models ................................ 139
9.3. Future Work Recommendation ............................................................... 140
10. List of References ....................................................................................... 141
11. Appendices ................................................................................................. 164
11
11.1. Appendix 1 .......................................................................................... 164
11.1.1. Human Ethics Research Council Approval Letters ......................... 164
12
List of Figures
Figure 1-1 A flow chart of the research formulation of problems and solutions this thesis ........ 20
Figure 2-1 Definition of the seated position and coordinate systems [37]................................... 25
Figure 2-2 Reproduced from the Australian Standard Health Guidance Zones [1]. .................... 27
Figure 2-3 APMS response of 5 test subjects, reproduced from [35] .......................................... 31
Figure 2-4 Head position and Transmissibility frequency response [39]. .................................. 32
Figure 2-5 Block diagram of FRF estimators [97]. ...................................................................... 36
Figure 2-6. Block diagram of single axis excitation cross-axial response [107]. ........................ 40
Figure 2-7. Block diagram of two-axis excitation and the cross- coupled in-line and cross-axial
response [107]. ............................................................................................................................. 40
Figure 2-8. Package of a seat pad accelerometer [2]. .................................................................. 41
Figure 2-9. A Seat pad accelerometer [2]. ................................................................................... 42
Figure 2-10 Comparison of optical and inertial motion capture [129]. ....................................... 46
Figure 2-11 a a)1DOF b) 2DOF and c) 4DOF prismatic whole body vibration models [4]. ....... 47
Figure 2-12. Prismatic series lumped parameter model [52]. ...................................................... 49
Figure 2-13. Lumped model with paralell elements [152]. .......................................................... 49
Figure 2-14. Lumped parameter backrest model [148]. ............................................................... 49
Figure 3-1 A computer aided design model of the parallel robot and the fabricated device [191].
..................................................................................................................................................... 62
Figure 3-2. a side view of the parralel robot and the seat. ........................................................... 63
Figure 3-3 a spring mass capacitor accelerometer schematic [117]............................................. 64
Figure 3-4 a schematic of the cantilever MEMS gyroscope [117]. ............................................. 64
Figure 3-5, The definition of the axes for the IMU sensor nodes [193]. ..................................... 65
Figure 3-6 The Xsens IMU Modules [192]. ................................................................................ 65
13
Figure 3-7 The interanl DSP and sensor fusion architecture of the Xsens IMUs [192]. ............. 66
Figure 3-8 A sample of the magnetic field distortion compensaton for angle estimation on the
Xsens IMU. .................................................................................................................................. 68
Figure 3-9 The location of the chassis IMU. ............................................................................... 69
Figure 3-10. The parallel hexapod robot with seated test subjects. ............................................. 70
Figure 3-11 Transmissibility of a test subject from multiple IMUs ............................................. 71
Figure 3-12 shows the signal coherence between IMUs for Acceleration. .................................. 72
Figure 3-13. Neck Joint angle estimated from Xsens [194]. ........................................................ 73
Figure 3-14 A sample PSD plot of the motion platform with a rotational excitation source
Regime 6. ..................................................................................................................................... 74
Figure 3-15 PSD plot of the motion platform with a rectlinear excitation source. ...................... 74
Figure 3-16 A block diagram of the data stutcure of the sampled inertial motion data. .............. 76
Figure 4-1 Flow chart of the EKF-ADWIN process. ................................................................... 84
Figure 4-2 The layout of the IMUs and parralel hexapod robot. ................................................. 85
Figure 4-3. A comparison between the EKF-ADwin damping rstio tracking performance to and
ARMAX model parameter estimation. ........................................................................................ 87
Figure 4-5. Simulated model transmissibility during validation run. .......................................... 88
Figure 4-6. Comparison of the model to time series H1estimation. ............................................ 88
Figure 4-7. Asymptote function of natural frequency for EKF-ADWIN and EKF. .................... 89
Figure 4-8. Asymptote function of damping ratio for EKF-ADWIN. ......................................... 89
Figure 4-9. Surface plot of EKF-ADWIN Asymptotes. .............................................................. 90
Figure 5-1. The configuration of the IMUs. ................................................................................. 98
Figure 5-2. A Sample of the excitation motion source measured at the motion platform. .......... 99
Figure 5-3. A sample of the state space estimation data at the sternum. ................................... 100
Figure 5-4. A sample of the acceleration IMU tracking at the head. ......................................... 101
Figure 5-5. A sample of the angular velocity IMU tracking at the sternum. ............................. 103
Figure 6-1. Complex graph of ARMAX model DBSCAN cluster assignments. ....................... 106
Figure 6-2. (a) Analysis for clustering (b) Implementation and testing of the state estimator .. 107
14
Figure 6-3. Complex graph of ARMAX model EM cluster assignments .................................. 108
Figure 6-4. ARMAX Cluster assigments real imaginary graph, DBSCAN1 and DBSCAN2
Denote the cluster class label ..................................................................................................... 109
Figure 6-5. ARMAX Cluster assigments real imaginary graph, DBSCAN1 and DBSCAN2
Denote the cluster class label ..................................................................................................... 110
Figure 6-6. The measured and unkown plant. ............................................................................ 111
Figure 6-7. Sample cluster assignments natural frequency ........................................................ 112
Figure 6-8. Cluster assignments for damping ............................................................................ 112
Figure 6-9 Cluster assignment against a time series of acceleration data .................................. 114
Figure 6-10. flow chart of the estimation process. ..................................................................... 114
Figure 6-11. Transmissibility verification of the constrained EKF, where FRF is the estimated
frequency response function ...................................................................................................... 117
Figure 6-12, The comparsion of the estimation vs the validation data. ..................................... 118
Figure 7-1 A sample of the jerk measurement ground truth dataset. ......................................... 123
Figure 7-2 the assigned mixture models. ................................................................................... 124
Figure 7-3 A sample of the acceleration waveform with the superimposed cluster inference. . 125
Figure 7-4 a sample of the Time Series angular velocity with the superimposed cluster
assignments ................................................................................................................................ 126
Figure 7-5 the trained HMM for the sequence of seat to head states ......................................... 127
Figure 8-1 The high level Signal archetcure of the model for biodynamic parameter prediction
................................................................................................................................................... 130
Figure 8-2. An example of STFT Surface plot for data sampled from the chest connected IMU.
................................................................................................................................................... 131
Figure 8-3 the overall structure of the artifical neural network model. ..................................... 133
Figure 8-4. The convolution units of each artifical nueral network structure tested in this
chapter. ....................................................................................................................................... 134
Figure 8-5 the Convolutional Unit used for the CNN. ............................................................... 134
Figure 8-6 Whole body vibration model Residual nueral network training. ............................. 135
Figure 8-7 Whole Body vibration residual network loss. .......................................................... 136
15
16
List of Tables
Table 2-1. Active areas of WBV research [35]. ........................................................................... 23
Table 2-2. Subjective definitions of omnidirectional vibration comfort levels [3]. ..................... 26
Table 2-3. Comparison of reported vibration dosages e*denotes estimation using Eq.(2-3). ..... 29
Table 2-4 Summary works in seating suspension systems with an integrated biodynamic
modeling and response analysis ................................................................................................... 55
Table 2-5 A summary of biodynamic models in the literature. ................................................... 57
Table 3-1 properties of the parralel robot .................................................................................... 62
Table 3-2 Summary of the relevant signal properties of the IMUs [192]. ................................... 67
Table 3-3 RMS accelertion and velocity of each test regime. ..................................................... 75
Table 4-1 Pseudo code of ADWIN function. ............................................................................... 82
Table 5-1 Pseudo code for the particle filter [202]. ..................................................................... 95
Table 5-2 Pseudo code for systematic resampling [229]. ............................................................ 97
Table 5-3. Summary of the tracking goodness of fit performance for each location for the rotary
components of motion. ............................................................................................................... 100
Table 5-4. Summary of the tracking goodness of fit performance for each location for the
rectilinear components of motion. ............................................................................................. 102
Table 6-1 Pseudo code the constrained estimator ...................................................................... 115
Table 6-2. A summary of the tracking accuracy of the estimator. ............................................. 118
Table 7-1 State edge probabilities. ............................................................................................. 127
Table 7-2 Summary of the inference results of each models on the evalutation data set .......... 128
Table 8-1. The training parameters used for both architectures ................................................. 132
Table 8-2 A summary of the various error rates per archetecture. Minimum values are
highlighted in bold. .................................................................................................................... 137
Table 8-3 Comparison of the standard deviations of each model .............................................. 138
17
1. Introduction
This thesis explores new approaches to solve problems in the modelling and
measurement of the biomechanical response of the seated human body. With
the advancements in wearable sensing systems, the improvements in
precision inertial instrumentation and the recent explosion in the
accessibility to high fidelity data, there is a contribution to be provided in the
domain of ergonomics and automatic automotive suspension control system
technologies. There are challenges associated with in the acquisition of high
volume and applying the data to make decision and to tune a control system.
Furthermore, machine learnt pattern-based approaches to construct a
biodynamic model are introduced. Through this thesis a set of soft computing
strategies are proposed to solve estimation problems with whole body
vibration modelling using networked inertial motion sensors.
Vibrations are stationary mechanical oscillations that are transferred through a solid medium.
The term whole body vibrations refer to the transmission of mechanical vibrations through the
human body [3]. Operators and drivers of heavy vehicles that are a source of vibrations are at an
elevated risk due the prolonged exposure vibrations [4]. The pathological effects of WBV has
been documented in the literature and include the following lower back pain, fatigue and
degenerative disorders [5].
There is a recent trend in the increase in the usage of wearable sensing technologies and an
addition with the increase in the usage, reduction in cost and improvement in fabrication
technologies of MEMS. In addition technologies the integration with other technologies like the
Internet of Things (IoT) [7], there is a possibility to gather rich and voluminous data about users
[8] providing a newfound opportunity for innovative applications to many areas of whole body
vibration monitoring analysis and control. The modelling and monitoring of the seated human
response to Whole Body Vibration (WBV) has become an increasingly active topic of research
interest due to applications in workplace health and safety [5, 6].
In the existing experimental based research, the main sensing technologies applied in the
analysis of WBV response dynamics include the use of piezo-electronic inertial sensors [2],
Micro Electromechanical systems (MEMS) [9] and optical systems [10].
18
1.1. Research Challenges and Significance
A key challenge are the complexities and the apparent idiosyncratic nature of whole body
vibrations [11, 12]. It has been consistently demonstrated in the literature that the dependent
variables in the human body response to vibration expresses a degree of nonlinearities,
complexities, and cross coupling. However, the literature does not report of a consensus on a
unifying model to fit all cases.
There are three key challenges that this thesis attempts to overcome: finding a balancing
between performance and robustness, overfitting and the curse of dimensionality.
The curse of dimensionality is an observed problem in statistics and machine learning problems
where the inclusion of additional parameters results in a higher probability of error [13]. With
recent advancements in terms of the size weight and power of wearable sensor technologies for
smart sensing systems, there is an increase in the available of channels of data. Although one
could assume the additional sensor nodes would improve the accuracy, the increase in variables
will result in a higher complexity problem to solve.
Overfitting is an issue that can afflict the performance of whole body vibration models [14]. As
to be discussed in chapter 2, much of the literature utilise biomechanical models in the 3DOF or
less. Although it a larger DOF model would intuitively seem like a better candidate, however
the model fitting process can create a complications with the noise in the fitting data but when
the model is evaluated against an independent distribution of data its performance degrades.
In the context of providing useful information to a vehicular control system, a higher fidelity
model with real time identified parameters. Having a closer fit of a biodynamic model allows
for the control system design criteria to weigh in favor of performance over robustness [15].
1.2. Research Aim and Objectives
This thesis aims to formulate a model of whole-body vibration response estimation that is viable
for use in vehicle seat suspension control systems and performs under the influence of inter and
intra subject variabilities. Compared to the overwhelming majority of works focused on the pure
modelling and analysis of vehicular whole body vibration in the literature, this thesis proposes
self-tuning and low computational cost architecture’s with the application to be deployable on
automotive embedded hardware.
1.3. Contributions and Outcomes
This thesis has presented the following contributions:
19
- Chapter 2 has made recommendations and given insights on the trends on the state of
the art in the literature. This chapter provides a comprehensive review on the literature
and the study of whole-body vibration analysis state or the art modelling and control.
- Chapter 3 has presented a design of an experimental procedure for collecting inertial
motion data from networked IMUs. This provides a foundation for performing
controlled experiments. As well as a rich data set that provides a test and evaluation
ground truth.
- An instantaneous whole-body vibration estimator is presented in chapter 4 using an
exact whole-body vibration model. The contribution is a viable solution to
incrementally estimating a prismatic 1DOF whole body vibration model. The estimator
is evaluated experimentally to validate it performs under real world conditions like
sensor drift.
- An approximation filtering strategy is presented in chapter 5, with results demonstrating
that it is stable for estimating the whole-body vibration model that is omnidirectional.
- In chapter 6, A Gaussian Mixture model and Dirichlet distribution gaussian mixture
model based HMM approach is presented to solve the problem of time invariant state
transitions between biomechanical events not modelled by the classical equations of
motion like rocking and jerk motions.
- In chapter 7, a deep learning spectrogram regression model for whole body vibration
models is presented to solve the problem of cross coupling. It also demonstrates the
ability to learn the internal inter and intra subject variabilities in higher DOF models.
The flow chart in Figure 1-1 illustrates the flow of the contributions of the work in the and how
each chapter uncovers the next problem to solve.
20
1.4. Thesis Outline
This thesis is outlined as follows, the literature search on seated whole-body vibration analysis
using sensors and numerical methods of modelling is presented in chapter 2. In Chapter 3 an
exact numerical method of estimating the instantaneous biodynamic model of the human body
with streaming inertial motion sensors. In chapter 4 an automatic approximation method is
presented. In chapter 5 a method of unsupervised learning is utilized to train a biomechanical
model. In chapter 6 a sequential model is presented. The conclusions and discussion are
presented in chapter 8.
Figure 1-1 A flow chart of the research formulation of problems and solutions this thesis
21
1.5. Declaration of Works in External Publications
The candidate declares that part of the works in this thesis is published in and are referenced
within this thesis:
[5] J. L. Coyte, D. Stirling, H. Du and M. Ros, “Seated Whole-Body Vibration Analysis,
Technologies, and Modeling: A Survey,” in IEEE Transactions on Systems, Man, and
Cybernetics: Systems, vol. 46, no. 6, pp. 725-739, June 2016.
[16] J. L. Coyte, H. Du, D. Stirling and M. Ros, “A Drift Detecting Anti-Divergent EKF for
Online Biodynamic Model Identification,” 2015 IEEE 17th International Conference on High
Performance Computing and Communications, 2015 IEEE 7th International Symposium on
Cyberspace Safety and Security, and 2015 IEEE 12th International Conference on Embedded
Software and Systems, New York, NY, 2015, pp. 1116-1121.
22
2. Literature review
The modelling and measurement of the biodynamic response of the seated human
body in recent times has been an active research topic, with major applications to
ergonomics and automotive suspension control system technologies. This chapter
presents a comprehensive literature survey of topics including the state-of-the-art
research in vibration signal processing and modelling of the biodynamic response
of the seated human to vibrations. This chapter reviews recent sensing systems that
are reported to measure the motion of the seated body. The data processing
techniques that are currently accepted are surveyed and these include impedance,
transmissibility measures, frequency response function estimation, and model
development. A review of applications of biodynamic response analysis and
modelling to seating vibration isolation technologies and vibration monitoring
systems are presented within this chapter. This literature review chapter provides
a discussion on the direction that the future research in this field will aim toward
based on the trends in the recent research and the introduction and application of
new technologies.
Epidemiological studies have found evidence that suggests occupational exposure to low
frequency vibrations is associated with the following adverse physiological conditions:
degenerative diseases of the vertebrae, motion sickness, tiredness, fatigue [17], lower back pain
[18], digestive problems, impairment of vision and an increased risk of certain cancers [19].
Other short term effects of WBV include the reduction of human reaction time [20] and
impairment of balance [21].
The most common illness linked to WBV exposure is lower back pain caused by intervertebral
disc diseases [22]. Lower back pain has also recently been identified as one of the most
burdening health conditions. An analysis [23] of the Global Burden of Disease Study 2010
concluded from a study on 291 conditions that lower back pain was ranked 1st based on the
years living with the disorder and ranked 6th based on the metric of overall burden, the disability
adjusted life year.
Although vibration exposure and the development of lower back pain show a strong causal link
[24], the risk of causing lower back pain has been reported to depend on other conditions
present during the exposure to WBV. These include trunk posture [25-28], body mass
distribution [24], muscle activity/response, the condition of the machinery and suspension [29]
23
and the occupant’s age [30]. Drivers working in transport, delivery, construction and
agricultural industries have been reported to have higher rates of lower back disorders [24] even
when the assessed vibration exposure dosage levels are below the standard threshold [1, 31-33].
Most epidemiological studies recommend that further research be performed, aimed at
developing a deeper understanding of how to reduce lower back pain. In the workplace,
suspension systems have been commonly reported as being adjusted incorrectly or not
functioning optimally, due to mechanical damage. This can result in the amplification of the
magnitude of the root mean square (RMS) vibration acceleration, to which the seated occupant
is exposed [34].
Table 2-1. Active areas of WBV research [35].
Category Description
Impedance Measurements are made of the
force and motion at the input
driving point of the biomechanical
system.
Transmissibility Measurements of the motion of
the input and output to determine
the frequency response through
the biomechanical system.
Modelling The design of a mathematical
model that will estimate the
motion – frequency response of
the body based on the
experimental data obtained from
the impedance and
transmissibility measurements
Recently, there has been a focus on limiting the exposure to vibration within the guidelines set
in multinational standards [1, 31-33] as a part of workplace health and safety policies. There has
24
also been an increase in research in ergonomics and vehicle dynamics to reduce the transmission
of vibrations. Research in the broader field of the analysis of WBV can be classified into three
main categories, as shown in Table 1. These categories have further applications in terms of
ergonomics, biomechanics and vehicle dynamics. Most of the experimental research which
analyses WBV involves recording measurements of the acceleration of vibrations in a
laboratory environment. As can be seen from Table 2-1, depending on the location of the
instruments, the two main measures of vibrations are impedance and transmissibility [35]. The
impedance is the measurement of the vibrations delivered ‘to the body’ whereas the
transmissibility is a measure of vibrations being transmitted ‘through the body’ [36]. Measures
of impedance that are commonly employed in the literature are described in section 2.2.1 and
section 2.2.2. Impedance measures include Apparent Mass (APMS) Driving Point Mechanical
Impedance (DPMI) and absorbed power. The third main category is modelling where the effects
of vibration on the human body, is represented with a system of mathematical equations
representing the motion of the body, works on the applications of modelling to seating vibration
isolation systems in this area is presented in section V.
Considering that a large volume of research has been undertaken in this area, this chapter
provides an overview of the existing research problems and solutions from both well established
and emerging research. This chapter discusses the trend, which the emerging research and
application of new technologies are following. This chapter is organised as follows: Section 2.1
presents an overview of the WBV standards and the literature on quantification of WBV.
Section 2.2 contains a review of the literature on the analysis of the seated biodynamic response
to vibration and the signal processing techniques that are required to assess and to estimate the
biodynamic response. Section 1.1 presents an overview of the recent technologies applied to
measure WBV. Section 2.5 presents a review of applications of biodynamic modelling research
into the modelling of the biodynamic response. The discussion and conclusion are given in
Sections 2.8 and 2.9 respectively.
2.1. WBV exposure Standards and Health
2.1.1. WBV dosage metrics and guidelines
WBV is defined as vibration that affects all parts of the body [3]. The directions and axes that
the vibration can be transmitted through the human body and the contact surfaces through which
vibration can be transmitted to the seated occupant are given in Figure 2-1.
25
Methods used for quantifying the dosage of WBV exposed to an occupant are given in several
international industrial standards and safety manuals [21]. The Australian Standards [1], British
Standards [31], International Standards [33] and a European Directive on vibration exposure
[38], define a criterion for assessing the vibration dosage, the vibration dose value (VDV). The
VDV is calculated by Eq.. where 𝑎𝑤(𝑡) is a weighted instantaneous acceleration, 𝑇 denotes the
duration of the vibration exposure. The units of the VDV are in m/s1.75. The total VDV for
separated tasks is given in Eq. ). The standards [1, 31, 33] provide a guideline for a range of
VDV values that pose a significant health risk, as shown in Figure 2-2. Equation (2-3) gives the
estimated vibration value which is to be used when the weighted times series acceleration is
unknown but the RMS acceleration is known [39]. Where 𝑎𝑤 is the frequency weighted
acceleration and 𝑇 denotes the time of the measured vibration exposure, in seconds.
𝑉𝐷𝑉 = √∫ [𝑎𝑤(𝑡)]4𝑑𝑡
𝑇
0
4
(2-1)
𝑉𝐷𝑉𝑡𝑜𝑡𝑎𝑙 = √∑𝑉𝐷𝑉𝑖
4
𝑖
4
(2-2)
𝑒𝑉𝐷𝑉 = 1.4 × 𝑎𝑤 × √𝑇4
(2-3)
Figure 2-1 Definition of the seated position and coordinate systems [37].
26
Figure 2-2 shows the graph of the health guidance regions, with the likely risk zone in red and
the low-risk zone in green. Table 2-2 shows the perceived discomfort levels at each level of
vibration intensity. The orange region is the standard defined caution zone [1]. A threshold
described as the action level and the exposure limit [40] . The action level is a level of dosage
where there should be close monitoring of the exposed occupant. The exposure limit is the
upper level of allowable vibration exposure. The action level criterion is defined at 0.5 𝑚/𝑠2
r.m.s. or a VDV of 9.1 𝑚/𝑠1.75, the exposure limit is defined at 1.15 𝑚/𝑠2 r.m.s. or a VDV of
21 𝑚/𝑠1.75. Since many occupations require a variety of tasks to be performed, such as driving
earth moving machinery on various surfaces or driving different vehicles, a dose value is often
used which normalizes the vibration exposure as a continuous eight-hour value. The equation
for the eight-hour normalized vibration dosage is given in Eq.(2-4) [3] for a series of 𝑁 intervals
of vibration exposures. 𝑛 denotes the number of the exposure interval within the series, 𝑎𝑤𝑛 is
the weighted RMS acceleration and 𝑡𝑛 is the period of interval 𝑛.
𝐴(8) = √1
8∑ 𝑎𝑤𝑛
2 𝑡𝑛
𝑛=𝑁
𝑛=1
(2-4)
Acceleration Weighted RMS Perceived Comfort level
Less than 0.315 m/s2 not uncomfortable
0.315 m/s2 to 0.63 m/s2 a little uncomfortable
0.5 m/s2 to 1 m/s2 fairly uncomfortable
0.8 m/s2 to 1.6 m/s2 uncomfortable
1.25 m/s2 to 2.5 m/s2 very uncomfortable
> 2 m/s2 extremely uncomfortable
2.1.2. Vibration exposure in various vehicles
Occupants of heavy industrial machinery or automotive vehicles can be subjected to vibration
(mechanical cyclic loading) from a variety of sources at different contact points. Sources of
Table 2-2. Subjective definitions of omnidirectional vibration comfort levels [3].
27
vibration can be the engine and/or the driving surface [41]. Examples of vibration sources in
vehicles and machinery include road surface, vehicle activity and engine vibration. The severity
of the vibration can vary significantly depending on the condition of the suspension system, the
condition of the road, the task the vehicle is performing and the design of the vehicle and seat
[21]. A source of WBV that is not normally considered is that caused by low frequency
infrasound [42].
An audit of WBV exposure was conducted on operators of agricultural vehicles [43] performing
typical tasks. By comparing the exposure intensity and duration data from this study to the
guidelines for vibration dose values from AS 2670.1—2001 [1] and in Figure 2-2, it is apparent
that the spraying, cultivating and transporting operations of a diesel tractor will cross over into
the risk zone. A study on skidder (tractors with grapples) operators’ exposure to WBV [2]
measured WBV and shown that typically the skidder will have a weighted acceleration value of
1.83m/s/s when unloaded and 2.1m/s/s when loaded. Exposure durations of 4 to 5 hours at this
magnitude of intensity would result in the VDV crossing into the caution health risk zone. A
study on vibration exposure in stackers [44] (tractors with a scoop) investigated the resultant
VDV associated with different driving surfaces and tires. The VDV reported was shown to vary
by 3.5 𝑚/𝑠1.75 depending on the type of traction chains used. Other research has also reported
the VDV to be in the risky range for tractors and heavy construction vehicle operators for most
common occupational activities [45-47].
Figure 2-2 Reproduced from the Australian Standard Health Guidance Zones [1].
28
Experimental studies on the VDV of seated bus drivers [48] driving around a metropolitan route
found the dosage to be 9.4 m/sec1.75 for urban road surfaces and 10.5 m/sec1.75 for rough surfaces
and was also reported that the mass of the driver did not have a statistically significant influence
on the VDV. A summary of reported VDVs, vibration intensities and durations in the literature
is presented in Table 2-3. The numerical data from these studies demonstrate that many
occupational vehicles vibration exposure crosses over into the high-risk zone. Some studies [49]
on trucks found the VDV is generally within the safe range in accordance with the standards
[33], whereas others report an unsafe level depending on the driving surface conditions [50].
Many of the studies listed in Table 2-3, did not report on the uncertainty of the logged
acceleration data [51].
2.2. Whole Body Vibration Response Analysis
Past research into WBV response has had the common objective of determining the biodynamic
response of the human body reliably and accurately. The frequency response data is useful for
analysing seating qualities and to assess the performance of the vibration isolation system. The
response data can be further used to define a body model system which can be utilised to predict
the motion of the human body.
The quality of the model produced from the biodynamic response measurements is dependent
on the accuracy of these frequency-dependent measurements [52]. The most common methods
used for analysing the frequency response of the human body to vibration are the apparent mass
(APMS) Driving Point Mechanical Impedance (DPMI) and transmissibility methods such as
Seat to Head Transmissibility (STHT). The transmissibility response is obtained by measuring
the acceleration from the input of the system such as the seat cushion or the input excitation
signal to an output measurement point like the torso or head. This method quantifies the
vibrations that are transmitted from the seat through the body which is known as STHT[40]
[39]. When defining a model to predict the vibrations, often the STHT, APMS or both responses
will be used to obtain the model parameters [35, 53] . The resonant frequency of the human
body, where the maximum point of the STHT and APMS response is located is, on average,
approximately 5 Hz [54], and is considered to be a critical property of the biodynamic response
as it indicates the frequency range and the amplification of the transmission of a vibratory
acceleration through the human body.
The biodynamic response has been reported to be dependent on intra-subject variables. These
variations can arise from changes in the subject’s torso and head posture [39, 55]. Studies have
been undertaken to quantify the relationship between the arm position and the biodynamic
response [56]. Although there has been a consensus in the research [57] that the posture is a
29
significant factor in the biodynamic response, this has not been considered by international
standards [33]. The following intra-subject variabilities influence the biodynamic response:
body posture, position, and orientation [58]. Further external variables which influence the
response include seat design, foot rest, head rest, back rest and clothing [39].
A problem reported is that averaging an individual test subject’s biodynamic response data
tends to mask the individual responses to the intra-subject variabilities [59, 60]. The use of
averaging has been shown to provide misleading interpretations of biomechanical data [35]. In
Figure 2-3, an example given by Mansfield [35], shows the normalized APMS for 5 different
test subjects with different resonance frequencies, and demonstrates how the geometrical
average shown by the bold line has the lowest peak value. This average response would
incorrectly imply that the body system has a larger damping factor at the dominant roots of the
body system than the measured data set, by observing that the magnitude at the resonant
frequency has been reduced. The recommendation [35] is to not average the frequency response
functions for several test subjects but to determine the mean resonance frequency and associated
peak magnitude for each individual test subject.
Table 2-3. Comparison of reported vibration dosages e*denotes estimation using Eq.(2-3).
Vehicle and operation Duration
(hours)
Vibration
RMS
acceleration
(m/s2)
(e*) VDV
(m/sec1.75)
(e*)
A(8)
Value
(m/sec2)
Skidder loaded [2] 7 1.83 23.057* 1.718*
Skidder unloaded [2] 7 2.1 26.459 1.964*
Bus urban [48]
Bus Rough Surface [61]
5.5
0.17
0.41
1.51
9.4 (±0.24)
10.5
0.34*
0.53
Tractor [62] 0.17 31 26 4.475*
Rapid Mixer Granulator (RMG) [62] - 20 35 3.464*
30
Rubber Tyred Gantry (RTG) Crane [62] 0.016667 24 26 --
Sprayer [43] 10.1 0.53 – 0.69 11.426 0.775*
Tractor Sprayer [43] 8.9 0.36 – 0.78 10.676 0.823*
Ploughing [43] 8.9 0.49 – 0.93 13.299 0.9809
Cultivating [43] 8.9 0.53 – 1.39 17.98 1.099*
Garbage truck [63] 5 0.31 2.64 2.087*
front-end loader [44] 8 0.31
(±0.02)
9.2 0.47
Commercial Aircraft Landing
(front/rear) [64]
0.01667 0.6/0.9 3.2/5.2
0.041*
Truck [50] 1 1.418 15.377* 0.501*
A variable that has been reported to have significant influence on the biodynamic response is
the backrest angle and subject posture. Investigation into the resonance frequency for different
seated postures [65-68], where the test subject was slouched or sitting erect etc. shows that the
resonant frequency could deviate by up to 1.2Hz. The results show that when the subjects
changed posture from erect to slouched, the principal resonance frequency decreased. The
position of the hands was reported to have an influence over the location of the resonance
frequencies [69]. The effects of lower limb motion were reported to add additional damping to
the body system [70]. In the worst-case scenario, the lower limb motion was reported to cause a
15% difference in the APMS response.
Non-linearities have been observed in the biodynamic response to vibration and this results in
the resonance frequencies changing depending on the magnitude of the input excitation signal
[71]. Studies have suggested that this non-linearity is influenced by body deflections and muscle
tension [72]. This means that the elasticity of the muscles depends on the magnitude of the
vibration motion. The human body tends to act like a softening system, that is, the resonance
frequency reduces as the vibration magnitude increases [55]. The degree to which the body
system “softens” has been reported to be dependent on the body posture, with subjects seated in
31
an erect and forward leaning posture having a higher level of softening in the torso under higher
magnitudes of vertical vibration [59]. The materials of the seat cushion have been shown to
exhibit a softening effect, an increase in the elasticity of the seating material was reported to
decrease the resonance frequency as well as the APMS amplitude at the resonance frequency
[73-75]. The softening system phenomenon has also been reported to occur in the fore-aft
direction [76-78]. The transmissibility response in the fore-aft direction has been reported to
exhibit a more sophisticated response than the vertical response with up to three resonant peaks
[79, 80].
Figure 2-3 APMS response of 5 test subjects, reproduced from
[35]
Studies on the relationship between the biodynamic response and anthropomorphic
characteristics have shown that the body-mass index and the age of the subjects had the greatest
influence over the variations [81], on average by 1.7Hz depending on the age or body mass
index. Body mass has been shown to have a greater influence over the biodynamic response
than the sitting posture and backrest angle [82].
32
Gender and
anthropometric
effects on the biodynamic response have been investigated in the vertical direction. Using mean
APMS response, the variation between gender was reported to have significant differences [83],
with males tending to have a higher rate of increase than females in the stiffness coefficient as
body mass increases. Other works have found an insignificant difference in the APMS response
[81, 84] between genders. An experimental study on the APMS response between two groups of
test subjects sorted by gender [69], found a more defined second resonant peak in the female
group at around 15Hz at all postures and a larger magnitude of APMS at frequencies higher then
15Hz. However, this gender effect was reported to be not statistically significant during an
exposure to excitation frequencies less than 15Hz. Other studies on the APMS [85] and STHT
[86] have shown that the resonant frequencies correlate strongly with gender, body mass index
[87], body fat percentage and hip circumference. Females were demonstrated to have a larger
APMS magnitude at the second resonance frequency [85], whereas males had a greater degree
of non-linear softening. Experimental results from [88] report an increase in the softening as
mass increases.
Figure 2-4 Head position and Transmissibility frequency response [39].
33
2.2.1. Impedance
The impedance measurement methods measure the vibration input to the biomechanical system
using force and motion. This measurement is usually taken from the contact point on the seat
cushion. The APMS is derived from Newton’s second law of motion. The definition of the
APMS [40] is given as the apparent mass – complex ratio of applied periodic excitation force at
frequency 𝑗𝜔, 𝐹(𝑗𝜔) to the resulting vibration acceleration at that frequency, 𝑎(𝑗𝜔), measured
at the same point and in the same direction as the applied force”. As the human body is not an
ideally rigid system, it will have complex frequency components. The disadvantage of using this
method to measure whole body vibration is that the apparent mass will be a function of the mass
of the body, so it is difficult to make a valid comparison without normalisation of the results of
different test subjects. Equation (2-5) is the APMS, with mass 𝑚(𝑗𝜔) as a function of
frequency; F denotes the force at the seat and �̈� denotes the acceleration at the seat. The APMS
can also be normalized by dividing APMS by the weight of the test subject.
𝑚(𝑗𝜔) =
𝐹𝑠𝑒𝑎𝑡(𝑗𝜔)
�̈�𝑠𝑒𝑎𝑡(𝑗𝜔)
(2-5)
In practice, a force platform is often used to measure the driving point force to the body system
(between the seat and the thighs). The force input into the biodynamic system can be used to
calculate the DPMI and the APMS [89]. The DPMI is defined as a ratio of the force over
velocity as a function of frequency. Equation (2-6) calculates the DPMI where 𝐹(𝑗𝜔) denotes
the force at the seat cushion as a function of frequency.
𝐷𝑃𝑀𝐼 =
𝐹(𝑗𝜔)
�̇�𝑠𝑒𝑎𝑡(𝑗𝜔)
(2-6)
2.2.2. Absorbed power and energy
The absorbed power is obtained by calculating the mean of the dot product of the force and
velocity vectors at the seat as given in Eq. (2-7). This method of measuring vibration energy
dissipation has been applied in several investigations of the evaluation and assessment of WBV
[90-92]. Absorbed power and energy is a scalar quantity, it is efficient to calculate the total
energy in 6 degrees of freedom (DOF) of the biomechanical system using a summation.
Absorbed power provides a measurement of the dissipation of vibration energy with respect to
magnitude, frequency, direction and duration [90]. The definition of absorbed power in
frequency domain is given in Eq. (2-8), where 𝐺𝑣𝐹(𝑗𝜔) is the cross-spectrum between the
velocity and force at the driving point of the WBV [93].
34
|𝑃(𝑡)|̅̅ ̅̅ ̅̅ ̅̅ = 𝑃𝑇𝑟(𝑡)̅̅ ̅̅ ̅̅ ̅̅ = 𝐹(𝑡) ∙ �̇�(𝑡)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ (2-7)
|𝑃(𝑗𝜔)| = 𝑅𝑒{𝐺𝑣𝐹(𝑗𝜔)} = |𝐺𝑣𝐹(𝑗𝜔)|𝐶𝑜𝑠(𝜙(𝑗𝜔)) (2-8)
2.2.3. Transmissibility
Transmissibility is the measurement of the vibration transmitted through two or more points.
Often the first point used is the input or the driving point of the system and other point locations
may include arms, spine and the head [35]. The STHT is widely used as an objective measure of
the seating dynamics. The STHT is defined[40] as seat-to-head transmissibility is complex non-
dimensional ratio of the response motion of the head to the forced vibration motion at the seat-
body interface . STHT is defined in Eq. (2-9) where �̈� denotes the acceleration. The
transmissibility response is obtained experimentally by measuring the acceleration from a
driving point of the system such as the seat cushion or the input excitation signal. This
measurement is very common in the fields of ergonomic research as it quantifies the vibration
that is transmitted through the body. This frequency response can only be directly evaluated
when the excitation signal closely matches a sinusoidal form otherwise further spectral analysis
will need to be performed given in the following section.
𝑆𝑇𝐻𝑇 = 𝐻(𝑗𝜔) =
�̈�ℎ𝑒𝑎𝑑(𝑗𝜔)
�̈�𝑠𝑒𝑎𝑡(𝑗𝜔)
(2-9)
The seat effective amplitude transmissibility (SEAT) factor is used to measure the performance
of seating suspension and as a predictor of the comfort of the seated occupant. The weights of
acceleration are determined from ISO2631. �̈�𝑠𝑒𝑎𝑡 is weighted acceleration at the seat and �̈�𝑓𝑙𝑜𝑜𝑟
is the acceleration at the vehicle cabin floor [39]. The SEAT factor has been recommended as an
objective measure for comfort [94].
𝑆𝐸𝐴𝑇 =
(�̈�𝑠𝑒𝑎𝑡)𝑅𝑀𝑆
(�̈�𝑓𝑙𝑜𝑜𝑟)𝑅𝑀𝑆
(2-10)
The performance indicators used in the literature on seated body modelling are the accuracies of
the estimated STHT, DPMI and the APMS impedance responses, compared to a measured
reference impedance response [36]. The values of the resonant peak will vary depending on
which impedance calculation method is used. The STHT is the ratio of the frequency-dependent
acceleration from the seat to the head, whereas the apparent mass is the ratio of the force and
acceleration at the head.
35
2.2.4. Comparison between impedance and transmissibility
methods
The apparent mass methods are known as the “to the body” response and the transmissibility
methods are known as the “through the body” response [36]. Both transmissibility and
impedance methods have been used extensively, however which method gives a more accurate
or valid representation of the response of the body system to vibration has not been proven.
Biomechanical models have been designed using data exclusively from one measurement
method or using both [35]. For a 1-DOF model of a linear mass spring damper system, the
transmissibility at the base and the APMS at the base will have identical resonance frequencies
[35].
A derivation of the mathematical relationship between the APMS and segmental
transmissibilities based on the Newtonian equations of motion in [93]. Equation (2-11) is based
on Newton’s second law of motion applied to the body system. The sum of forces at the input
driving points, 𝐹𝑧𝑖 and other boundary points, 𝐹𝑧𝑜, are equated to the distributed mass integral of
acceleration at a specific location, which is denoted by ∫ �̈�𝑑𝑚. Equation (2-11) is then factored
in terms of APMS and transmissibility by performing a division on Eq. (2-12) by the input
acceleration �̈� . M denotes the APMS as a function of frequency. This expression can be applied
to scenarios where the transmissibility of a segment of the human body is to be determined or
used to estimate the STHT equation from measured APMS data or vice-versa. Further
polynomial fitting may need to be performed when using Eq. (2-12) [95] as the coefficients of
the frequency response function for transmissibility are not normally known.
2.3. Estimation of Whole-Body-Vibration Frequency Response
In practice, frequency response estimation will be required in order to transform the signals
from the time domain to the frequency domain [96]. The differences between the biodynamic
impedance response when estimated using a random or sinusoidal signal have been
demonstrated to be negligible [97]. The following sections present the concepts and derivations
of the main frequency response function estimation algorithms that are widely accepted to be
∑𝐹𝑥𝑖 +∑𝐹𝑥𝑜 = ∫ �̈�𝑑𝑚
(2-11)
∑𝑀𝑥𝑖(𝑗𝜔) +∑𝑀𝑥𝑜−𝑖(𝑗𝜔) = ∫𝑇𝑧𝑑𝑚
(2-12)
36
suitable for the analysis of WBV. The cross-spectral density method is employed to evaluate a
transfer function of the impedance or transmissibility in the frequency domain [3, 39].
Equations (2-13) and (2-14) show the frequency response transfer functions of the APMS and
transmissibility. In Eq. (2-13) 𝑆�̈�0𝐹𝑣(𝑗𝜔) denotes the cross-spectral density between the force at
the output and the acceleration at the input and 𝑆�̈�0(𝑗𝜔) denotes the Power Spectral Density
(PSD) of the acceleration. The transmissibility function is given in Eq. (2-14), where 𝑆�̈�0�̈�(𝑗𝜔)
is the cross-spectral density between the output and the input acceleration.
𝑀𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙(𝑗𝜔) =
𝑆�̈�0𝐹𝑣(𝑗𝜔)
𝑆�̈�0(𝑗𝜔)
(2-13)
𝑇𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙(𝑗𝜔) =
𝑆�̈�0�̈�(𝑗𝜔)
𝑆�̈�0(𝑗𝜔)
(2-14)
The cross-spectral density method applied to estimate the transfer function of the impedance
and the transmissibility response has been commonly employed in measuring the single axis
rectilinear vibration response. Recently, with the introduction of 6-DOF IMUs, multiple inputs
and multiple outputs (MIMO) systems have been applied to include cross coupling between
different planes within the body system.
The DPMI and APMS can be defined as a ratio between the cross power spectral density of the
system input and output, and the auto spectra of the input. The advantage of this method is that
the data at the input and the output are correlated and this reduces the effects of the noise on
the data [35]. The frequency response estimated by this method is referred to as the 𝐻1
estimator. The H1 estimator can be expanded further to be applied to MIMO systems. The linear
system where noise is added to the output is given in Eq. (2-15), where {𝑁} is a vector
containing an extraneous contaminating noise [98]. H is the matrix containing the FRF that is to
be estimated, where Y and X are in the output and input column vectors, respectively. The
number of rows in the input vector, 𝐿𝑥 , and the number of rows in the output vector, 𝐿𝑦, the
Figure 2-5 Block diagram of FRF estimators [98].
37
dimensions of the FRF matrix H is, therefore, 𝐿𝑥 × 𝐿𝑦.
𝑌 = [𝐻][𝑋] + [𝑁]𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 (2-15)
Equation (2-15) is multiplied by the Hermitian transpose of the input vector,{𝑋}𝐻, then an
expected value operation is performed which results in Eq. (2-16) where 𝐺𝑦𝑥 is the cross spectra
of the output and the input, 𝐺𝑛𝑥 is the auto spectrum of the noise and the input.
[𝐺𝑦𝑥] = [𝐻][𝐺𝑥𝑥] + [𝐺𝑛𝑥] (2-16)
[�̂�1] = [𝐺𝑦𝑥][�̂�𝑥𝑥]−1
(2-17)
2.3.1. H2 Frequency Response Function estimator
Another FRF estimation method is the 𝐻2 estimator algorithm that reduces the noise added to
the input of the system, however this estimator is rarely used for biodynamic vibration analysis
as the measured output has an additive uncorrelated noise. The general equation for the 𝐻2
operator algorithm is given in Eq. (2-18), where 𝐺𝑖𝑜(𝑗𝜔) is the cross-spectrum between the
input and the output and 𝐺𝑜𝑜(𝑗𝜔) is the auto cross-spectrum of the output. The 𝐻2 estimator is
less commonly used in WBV research.
𝐻2(𝑗𝜔) =
𝐺𝑜𝑜(𝑗𝜔)
𝐺𝑜𝑖(𝑗𝜔)
(2-18)
2.3.2. HV Frequency Response Estimator
Under conditions where the system has MIMO and uncorrelated noise at both the input and the
output, the 𝐻𝑣 estimator is recommended to estimate the frequency response function [99].
Research into multiple excitation input WBV analysis has applied the 𝐻𝑣 estimator algorithm
with dual axis fore-aft excitation [100] and has concluded that it is an appropriate FRF
estimation algorithm due to the multiple sources of vibration delivered by the seat backrest and
cushion. In multi axial WBV analysis, this condition will arise when estimating the cross-axial
FRFs where the frequency response along one axis will be dependent on the FRF along other
axes, especially when there are significant excitation sources along the different axes. The
coupling along the sagittal plane of the human body has been reported to have a significant
degree of cross-coupling [101] and the 𝐻𝑣 estimator is recommended to be used when cross-
coupling is to be considered in the analysis.
The derivation of the MIMO 𝐻𝑣 estimator assumes the system to be linear and with a noise
38
component on the input and the output. Equation (2-19) denotes the definition of a linear system
that the 𝐻𝑣 estimator is applied to, where [N] is a vector containing an extraneous contaminating
noise [98]. H is the matrix containing the FRF that is to be estimated, where Y and X are in
output and input vectors, respectively.
[𝑁] = [𝐻][𝑋] + [𝑌] (2-19)
The first step in deriving the 𝐻𝑣 estimator is to determine the error auto spectrum matrix, 𝐺𝑛𝑛
which is defined by Eq. (2-20). Superscript H denotes the Hermitian transpose matrix operation.
𝐺𝑥𝑥 is the auto spectra of the input, 𝐺𝑦𝑥 is the cross-spectra of the output and the input and 𝐺𝑥𝑦
[98].
[𝐺𝑛𝑛] = [𝐺𝑦𝑦] + [𝐻][𝐺𝑥𝑥][𝐻]𝐻 − [𝐻][𝐺𝑥𝑦] − [𝐺𝑦𝑥][𝐻]
𝐻 (2-20)
Equation (2-20) can then be factored into a composed matrix as given in Eq. (2-21). The identity
matrix is abbreviated as [I].
[𝐺𝑛𝑛] = [𝐼|𝐻] [
𝐺𝑦𝑦 𝐺𝑦𝑥𝐺𝑥𝑦 𝐺𝑥𝑥
] [𝐼𝐻𝐻]
(2-21)
Matrix Eq. (2-21). Is then decomposed into the terms of an eigenvector matrix denoted by [𝑈]
and a matrix with eigenvalues in the diagonal elements denoted by [Λ].
[𝐺𝑛𝑛] = [𝑈]𝐻[Λ][𝑈] (2-22)
The Rayleigh quotient operation is applied to Equation (2-22), which results in the solution for
the 𝐻𝑣 estimator [98].
𝐻�̂� =𝐺𝑦𝑥
|𝐺𝑦𝑥|√𝐺𝑦𝑦
𝐺𝑥𝑥
(2-23)
The geometric average of the 𝐻1 and 𝐻2 estimators is also equal to the 𝐻𝑣 estimator [98] as per
Eq. (2-24).
𝐻�̂� = √𝐻1̂𝐻2̂
(2-24)
The 𝐻𝑣 estimator expressed in terms of orthogonal axes, applied to seated WBV can be
expressed in the form given in Eq. (2-25) [80], where k denotes the axis. The input excitation
39
and the output response are denoted by a and q, respectively.
𝐻𝑘(𝑗𝜔) = 𝐺𝑎𝑘𝑞𝑘(𝑗𝜔)√𝐺𝑞𝑘
(𝑗𝜔)
𝐺𝑎𝑘𝑞𝑘(𝑗𝜔)
, (𝑘 = 𝑥, 𝑧) (2-25)
2.3.3. Alternative Frequency Response Estimation Methods
In the literature for vibration modelling of other source like in structural engineering, and
machine vibration monitoring, other numerical transforms have been applied to analysis the
frequency response in cases where the frequency response is time variant.
The Continuous Wavelet Transform (CWT) [102] is given in Eq. (2-26), where ψ denotes the
wavelet function and 𝑥(𝑡) denotes the signal of interest. Continuous wavelet functions are used
when the scale and frequency of a signal is time invariant.
𝐻(𝜏, 𝑠) =
1
√|𝑠|∫ 𝑥(𝑡)ψ (
t − 𝜏
𝑠) 𝑑𝑡
∞
−∞
(2-26)
The Discrete counterpart to the CWT is the Discrete Wavelet Transform (DWT) [103] has been
applied to vibration monitoring applications [104]. The discrete wavelet transform definition is
given in Eq. (2-27), where 𝑥[𝑚] is the discrete signal of interest.
𝐻[𝑛, 𝑎𝑗] =1
√𝑎𝑗∑ 𝑥[𝑚] ∙
𝑁−1
𝑚=0
(𝑚 − 𝑛
𝑎𝑗)
(2-27)
The Short time Fourier transform STFT has seen applications in areas outside of Whole-body
vibration analysis where the spectrum of vibrations is time variant. STFT [105] where Ω is the
windowing function and 𝜏 is the slow time axis of the FFT and t is the time of the input signal
x(t).
𝐻(𝜏, 𝜔) = ∫ 𝑥(𝑡)Ω(𝑡 − 𝜏)𝑒−𝑗𝜔𝑡𝑑𝑡
∞
−∞
(2-28)
2.3.4. Multi-axial vibration response
The contributions from the vibration sources in the fore-aft and lateral directions to the vertical
vibration response have been investigated recently. Most existing seating suspension control
systems isolates vibrations from the vertical axis only, and the seated body is exposed to
vibration inputs from multiple directions. The STHT and impedance cross-coupled vibration
response with multiple excitation sources have been investigated [36, 72, 80, 100, 101, 106-
40
109]. The main outcome to be achieved from obtaining the cross-coupled biodynamic response
is to assist with the development of an improved body model [106]. It has been shown that the
𝐻1 and 𝐻𝑣 have an identical STHT response to a single axis vibration source, however during
conditions where there are multiple excitation sources, it has been suggested that the system
have a noise added to the input and output measurements so the 𝐻𝑣 estimator method is
recommended to be used to estimate the frequency response [90].
The definition of the cross-axial biodynamic response, 𝐻𝑖𝑗 , is the transfer function of the body
system. When 𝑖 = 𝑗, the function represents the in-line response and when 𝑖 ≠ 𝑗,the function is
of the cross-axial response. [�̈� �̈� �̈�]′ is the input acceleration excitation signal and [𝑞𝑥𝑞𝑦𝑞𝑧]′ is
the output acceleration [106]. In practice, an FRF estimation method like those mentioned in
Section 2.2.2.3 will be required in order to determine the response data, 𝐻𝑖𝑗 .
{
𝑞𝑥𝑞𝑦𝑞𝑧} = [
𝐻𝑥𝑥 𝐻𝑥𝑦 𝐻𝑥𝑧𝐻𝑦𝑧 𝐻𝑦𝑦 𝐻𝑦𝑧𝐻𝑧𝑥 𝐻𝑧𝑦 𝐻𝑧𝑧
] {�̈��̈��̈�}
(2-29)
2.4. Whole Body Vibration Sensing Systems
The performance of the body model and the accuracy of the estimated biodynamic response are
highly reliant on the accuracy and validity of the measurements acquired from the sensing
system. The sensors selected for the measurement of WBV depend on vibration analysis for
Figure 2-6. Block diagram of single axis excitation cross-axial response [108].
Figure 2-7. Block diagram of two-axis excitation and the cross- coupled in-line and cross-axial
response [108].
41
impedance measurements. Typically, a seat force plate or a pad accelerometer will be used to
measure the motion. For transmissibility measurements, as the sensors will need to be located at
specific points on the body, sensors like MEMS accelerometers or displacements sensor are
often used. For impedance measurements, force sensors and accelerometers are commonly
employed [110].
2.4.1. Inertial bite bar sensors
The STHT has been measured experimentally with the use of bite bar piezo electronic
accelerometers mounted to a bite bar, to directly measure the accelerations in the test subjects,
head during WBV [111-114]. The disadvantages of using bite bar mounted sensors for STHT
measurement is that they need held within the test subject’s mouth, which may cause discomfort
to the test subject, especially if the test is executed over an extended time. Bite bars, however,
provide a means of obtaining repeatable values and valid measurements over a wide range of
frequencies up to 100Hz [39].
2.4.2. Seat pad inertial motion sensors
Figure 2-8. Package of a seat pad accelerometer [2].
42
The traditional approach to measuring the vibration at the seat cushion being delivered to the
body is to use seat pad accelerometers. Seat pad accelerometers are standardized in design, as
specified by ISO 7096 [39, 115] as shown in Figure 2-9. Seat pad accelerometers are housed in
a flexible disk to allow them to deform with the seat cushion and are fitted between the
occupant and the seat cushion. Seat pad accelerometers typically consist of a tri-axial
piezoelectric accelerometer [116] Seat pad accelerometers serve the main purpose of providing
a quantitative assessment of ride quality and the dosage levels of vibrations. While using seat
pad accelerometers, the APMS response analysis will require units of force, so either an
estimation of seat force will need to be performed or additional force sensors will need to be
included.
Seat pad accelerometer sensors have been used to provide measurements of the vibratory
accelerations in 6-DOF [2, 43, 45, 48, 62, 63, 116, 117]. All studies that have used a single seat
pad transducer have measured the acceleration data for the purpose of assessing the dosage of
vibration and validating the performance of seating suspension systems.
A force plate is often used to measure the driving point force to the body system (between the
seat and the thighs). The force input into the biodynamic system can be used to calculate the
DPMI, APMS [89]. Thin film pressure sensors consist of a variable resistor which is dependent
on the force [73] and these have been applied to provide a measure for calculating the APMS
response.
2.4.3. Skin mounted MEMS IMUs
In recent times, improvements in silicon MEMS/CMOS-integrated technologies have allowed
for greater miniaturisation and a reduction in the cost [118]. Inertial Measurement Units (IMU),
which are based on Microelectromechanical Systems (MEMS) technologies, and which are used
for measuring whole body vibrations have become an effective and popular way to measure 6-
DOF vibrations. The skin mounted IMUs have further advantages. Due to their small mass and
Figure 2-9. A Seat pad accelerometer [2].
43
size, they can be embedded into seats or affixed to the skin easily [119]. MEMS technology
sensors have a much wider band of measurable frequencies ranging from sub Hz to 102Hz than
piezoelectronic IMUs, which typically attenuate sub Hz frequencies. MEMS are also a more
suitable option for measuring vibration signals [116]. Due to their compact dimensions and the
emergence of wireless technologies, MEMS IMUs have the potential to be embedded in
machinery and safety clothing, for the purpose of continuously monitoring WBV and hand-arm
vibrations in the workplace [120-123]. Future research to improve wearable IMU technologies
include the simplification of the data processing algorithms and improvements in the fixation
methods [124].
MEMS IMUs contain an accelerometer, a gyroscope and a magnetometer. The accelerometer is
composed of spring mass damper systems and the acceleration acting upon the mass causes it to
displace, which will vary its electrical capacitance. Gyroscopes measure the angular velocity.
They work on the principle of sensing the Coriolis acceleration acting on a vibrating mass in
proportion to the rate of rotation along an axis orthogonal to the vibratory axis [118].
Acceleration errors caused by misalignment of the accelerometers when mounted onto a seated
body has been reported to occur [9]. This error is given in Eq. (2-30), which is based on the
Abbe error equation [125]. This misalignment can occur due to the axes of acceleration not
being aligned to the skin surface where the sensor is mounted due to inclination and curvatures.
With the use of 6-DOF IMUs which incorporate the angular velocity measurements, the
alignment can be corrected by applying Eq. (2-31) and Eq. (2-32).
∆�̈� = �̈�𝑠𝑖𝑛(∆𝜃) (2-30)
Past studies have demonstrated that with the use of multi-axis accelerometers, it is possible to
obtain the STHT frequency response of a biodynamic system in multiple planes [80, 106, 126].
An example of an experimental procedure consists of a sensing system with force plates used in
conjunction with tri-axial skin mounted accelerometers. The initial orientation is then measured
and each component of motion is then calculated [106]. Zheng et al.’s experimental procedure
consists of a sensing system with force plates used in conjunction with tri-axial skin mounted
accelerometers which also allows for the direct measurement of impedance measures such as
DPMS.
A disadvantage of using skin mounted tri-axial accelerometers, such as the system used by
Zheng et al. [106], compared to the 6-DOF IMU used by Deshaw et al. [9], is that the 6-DOF
IMU can continuously measure the changes in orientation which occur during the period the test
subject is exposed to the vibration. When using the tri-axial accelerometer, however, if the test
44
subject involuntarily changes posture or if the sensor orientation changes during vibration, then
the results may be invalid.
(�̈��̈��̈�)
𝐵
= (�̈��̈��̈�)
𝑀
+ 𝑅𝑀𝑂 (
00𝑔)
(2-31)
(�̈��̈��̈�)
𝐼
= (�̈��̈��̈�)
𝐵
∙ 𝑅𝑂𝑀
(2-32)
One problem which can be encountered when collecting vibration data from IMUs is the
introduction of motion artefacts caused by the materials between the sensors and the mounting
point. Research using skin mounted sensors has used different means of attaching the sensors.
These include the use of textile, adhesive and plastic materials [9]. Although skin mounted
IMUs are only capable of measuring the inertial motion on the surface of the skin, the
transmissibility through the body and specific locations around the body such as the spine [127],
can be estimated with the aid of a frequency correction function. Due to the damping and
stiffness properties of the body tissue and the fastening material between the sensors,
interference in the vibration measurement of the body system can occur. A technique to correct
this error is to define a frequency correction function that will compensate for the consequent
change in the frequency response [106, 126, 127].
Equation (2-33) is the Newtonian equation of motion for the combined material-body system
where subscript 𝑡𝑟 denotes the function of the “true” value at the point of interest i.e. at a
specific bone or on the surface of the skin. Subscript 𝑚 denotes the function that is directly
measured by the sensor, 𝑥𝑚(𝑡) is the measured value of displacement. The variables x, y and z
denote the acceleration vectors and g denotes the acceleration due to gravity.
The frequency correction function, which is the inverse transfer function of the local
tissue/material system, is given in Eq. (2-34). The frequency correction function represents the
impedance between the sensor and the rigid body material. Equation (2-34) is determined by
solving Eq. (2-33) for the solution that has the minimum effect from the material-body system.
The natural frequency damping ratio and frequency ratio are given in Equations (2-35), (2-36)
and (2-37), respectively. For the material-body system the parameter k is the stiffness, c is the
damping factor and m is the mass. 𝜔0 denotes the natural frequency of the material-body
system.
45
𝑚�̈�𝑚(𝑡) + 𝑐(�̇�𝑚(𝑡) − �̇�𝑡𝑟(𝑡)) + 𝑘(𝑥𝑚(𝑡) − 𝑥𝑡𝑟(𝑡)) = 0 (2-33)
𝐹𝐶(𝛽) =
1 − 𝛽2 + 2𝑗𝜁𝛽
1 + 2𝑗𝜁𝛽
(2-34)
𝜔0 = √𝑘
𝑚
(2-35)
𝛽 =𝜔
𝜔0 (2-36)
𝜁 =𝑐
2√𝑚𝑘 (2-37)
2.4.4. Optical sensor-based motion capture
Stereophotogrammetric motion capture systems are capable of providing highly accurate
position measurement of human motions [128]. Compared to inertial motion capture systems,
stereophotogrammetry systems carry a higher cost, have more restrictions on space, and have a
more complicated installation procedure. As a result, although they have been used in the
laboratory environment, optical motion capture systems have been rarely used in field
experiments. These systems, however, have high position and orientation tracking accuracy and
are widely accepted as being the gold standard of motion capture [129], and are often used a
reference for fine tuning MEMS IMU inertial motion capture systems. Figure 2-10 shows the
simultaneous comparison between inertial and optical motion capture technology motion
measurements of upper limb movements [130]. The velocity obtained from the optical motion
capture system has noise introduced from the numerical differentiation operation, whereas the
inertial motion sensor velocity is calculated using vector mathematics based on the gyroscope’s
angular velocity measurements.
Research into seated WBV motion capture has been undertaken using optical sensors [10, 131-
133], using reflective marker optical motion capture technology. Like the skin mounted IMUs,
stereophotogrammetry techniques can record the STHT in multiple planes.
46
2.5. Applications of Modelling to vehicle Dynamic Systems
In this section, literature in the area applying biodynamic modelling and biodynamic response
analysis to seating vibration isolation technologies will be discussed. The objective of deriving a
biodynamic model is to be able to predict the performance of seating suspension control systems
[134].
Body models can be categorised into three main classes: lumped parameter models, continuum
models and discrete models [135]. In this section all the literature on vehicle dynamic systems
considers the body model as a lumped parameter model, that can be represented in the
frequency domain as a transfer function or in the time domain as a differential equation of
motion, which is analogous to a mechanical system of spring masses and dampers. Although
there has been in increase in the recent volume in works on non-lumped parameter models, to
date there has been a lack of research applying them to seating suspension control systems,
possibly due to the increased complexity and efficiency of integrating them with control
systems. Examples of such biodynamic models include spline models[136, 137] finite element
analysis models [138-142] and regression models based on artificial neural networks [143-147].
Figure 2-10 Comparison of optical and inertial motion capture [130].
47
2.6. Biomechanical Models
Existing research into the analysis of the vibration response has modeled the human body using
lumped parameter models with varying configurations and degrees of freedom ranging from 1
up to 12 [148-150] . The DOF in this case refers to the number of motion vectors the system can
be composed of, whether they are rotary or rectilinear. The biodynamic response has been
modeled using both passive and active transfer functions. The passive models are defined as
mathematical models of simplified mechanical systems. For example, a single DOF model will
be composed of a single spring-mass-damper. It has been shown that in general, the 4-DOF
model provides more accurate estimation of the human body response to vibrations, than do the
higher DOF models and this likely due to the over fitting of these models [148].
The general equation of motion for a linear n-DOF lumped parameter model is given in Eq.
(2-38). Where [M] denotes the matrix of the values for each mass, [C] denotes the matrix of
damping factors and [k] denotes the matrix of stiffness values.
[𝑀]𝑑2𝑥
𝑑𝑡2+ [𝐶]
𝑑𝑥
𝑑𝑡 + [𝑘]𝑥 = 𝐹
(2-38)
The equations for a 2nd DOF model are frequently utilised in the vibration control literature and
are given as follows in the form of a state space model for brevity; where matrix A denotes the
state transition matrix as given in Eq. (2-39) which corresponds to the model shown in Figure
2-11 a) . B denotes the input matrix. M, k, c denotes the mass, spring stiffness and damping
factor of each element in the lumped parameter model.
Figure 2-11 a a)1DOF b) 2DOF and c) 4DOF prismatic whole body vibration models [4].
48
A = [
0 1 0 −1−𝑘2/𝑀2 𝑐2/𝑀1 −𝑘1/𝑀1 𝑐2/𝑀2
0 0 0 1𝑘2/𝑀1 𝑐2/𝑀1 −𝑘1/𝑀1 −(𝑐2 + 𝑐1)/𝑀1
]
(2-39)
The input matrix B is given in Eq. (2-40).
B = [0 0 0 −1/𝑀1] (2-40)
The Output matrix is given in Eq. (2-41).
C = [−𝑘2/𝑀2 −𝑐2/𝑀2 0 𝑐2/𝑀2] (2-41)
Where the general form of the state space transition and output equation is given in Eq (2-42)
and Eq. (2-43) respectively. D is a feedforward matrix which is 0 in the case of this model.
�̇�(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡) (2-42)
𝑦(𝑡) = 𝐶𝑥(𝑡) + 𝐷𝑢(𝑡) (2-43)
The majority of lumped parameter body modelling studies are focused exclusively on the
vertical motion [60]. One example of such a study designed a vertical backrest model using the
experimental data from thirteen test subjects of different body shapes and masses [151]. This
model considers the damping and friction of the backrest of the seat, as well as representing the
arm/steering wheel linkage as a second ground frame. The parameters of the biodynamic model
are calculated experimentally. This investigation introduces two models: one with hand support
and backrest support and the other with backrest support but without hand support.
A comparative evaluation was done by modelling data from both transmissibility and
impedance vibration measurement. A significant variation between the resonant peaks measured
using each method was found [52]. The frequency response of the linear 4-DOF simulated body
model using the DPMI to estimate the parameters was more accurate than the response
predicted using the same model with the STHT data.
It has been reported that angular vibrations around the pitch and roll axis at the seat base make a
significant contribution to the vertical vibrations transmitted through the backrest [77, 152-154].
A recommendation has been made to incorporate the angular components of motion into body
models which factor for the rotary stiffness and damping.
49
Figure 2-12. Prismatic series lumped parameter model [52].
Figure 2-13. Lumped model with paralell elements [161].
Figure 2-14. Lumped parameter backrest model [151].
50
Non-linear body models mathematically have been employed to represent the effects of
deformations, visco-elasticity and non-linear stiffnesses, as described above in section 2.2. The
4-DOF model of the human body which includes non-linearities of the body and the vehicle
suspension components [162], considers the effect of non-linear stiffness through the pelvis,
thighs and seat cushion as well as that which depends on the deflection of the seating
suspension. In many cases, however, the biomechanical models can provide a more accurate
prediction depending on the validity of the model parameters [148]. The visco-elastic stress-
strain properties of a polyurethane seat and non-linear stiffness has been modeled [74, 156].
2.6.1. Non analogue models
An artificial neural network model (ANN) algorithm for predicting the STHT, DPMI and
APMS of a 4-DOF model of the biodynamic response has also been used [143, 144]. It has been
proven that the ANN can accurately predict the model parameters even when the mass and
spring stiffness factors vary. The difficulty is that the neural network is trained using data from
the ideal linear 4-DOF body model, where the parameters are set to constant values. The
biodynamic response results are then produced using the identical model with different stiffness
and mass coefficients. It must also be noted that the body model used which are based on [52] to
obtain the training data itself has a significant error compared with experimental data. The
goodness of fit is 76.8% and the peak frequency error is at 0.4Hz for model [143], model [144]
reported a goodness of fit of 96%.
Compared to manufactured mechanical joints, biomechanical joints can exhibit more complex
motions. For example, the knee joint has motion in both rotation and translation when extended,
that is, the joint acts like two smooth curved surfaces sliding on each other. In this case, the use
of a multi-body spline model will provide a more accurate representation of the joint [136]. A
spline model is a representation of a geometric curve using a series of polynomials. Spline
models have been applied to a human spine vibration body model, and constructed using
dynamic splines [137] which represent the joints as a curve where the centers of articulation are
located in a frame that is free to move along the spline curve.
2.6.2. Rotary models
It has been found that angular vibrations around the pitch and roll axis at the seat base make a
significant contribution to the vertical vibrations transmitted through the backrest [77, 152,
153]. A recommendation has been made to incorporate the angular components of motion into
body models.
In the existing research, most lumped parameter models are purely prismatic and consist of a
51
single dimension, that is, they do not consider the effects of rotation of any elements within the
system. This means that the springs and dampers are not able to rotate. Research in the
ergonomics field has used multi-body system representations of the body to model the effects of
posture and body movements when exposed to whole body vibration. A study by Yoshimura et
al. [155] presents a 10-DOF body model system with revolute joints at each vertebra. The
contact point between the seat and the body is modeled using two prismatic joints. Other studies
that have modeled revolute joints in multi-body human to seat models are also given [137, 156].
This model takes into considerations the lower limbs and considers the effects of torque on the
upper and lower legs [70].
2.6.3. Non-linear models
Non-linear body models mathematically represent the effects of deformations, visco-elasticity
and non-linear stiffnesses, as described above in section 2.2. The 4-DOF model of the human
body which includes non-linearities [162] considers the effect of non-linear stiffness through
the pelvis, thighs and seat cushion as well as that which depends on the deflection of the seating
suspension. In many cases, however, the non-linear models can provide a more accurate
prediction depending on the validity of the model parameters [148].
The visco-elastic stress-strain properties of a polyurethane seat and non-linear stiffness have
also been modeled [74, 156]. By measuring the force-compression relationship of a sample of
the foam from the seat cushion, the stress-strain relation has been used to estimate the
relationship between force and deflection. In this model, the system equations of the seat and
human body include a term for deflection and contact distances between the body and the
cushion. Equation (2-44) has been proposed by Singh et al. [163] to model the visco-elastic
effects of materials, where the first sigma summation term represents the non-linear elastic
motion of the foam mass system and the exponential term represents the visco-elastic deformity
of the foam. 𝑘𝑗 denotes the stiffness coefficients and alpha is the reciprocal time constants of the
foam’s visco-elastic deformation.
𝑚�̈� + 𝑐�̇� +∑𝑘𝑗𝑥𝑗 +∫ ∑𝑎𝑖𝑒
−𝛼𝑖(𝑡−𝜏)𝑥(𝜏)𝑑𝜏 = 𝐹(𝑡)
𝑁
𝑖=1
𝑡
−∞
𝑀
𝑗=1
(2-44)
2.6.4. Parameter search methods
Most applications in the existing biodynamic modelling research do not require the body system
parameters to be known in real time as the biodynamic models are only used for post-processing
batch data. The most reported means of parameter identification has been to use a curve fitting
52
toolbox within a computing environment such as MATLAB. In the case of vibrational lumped
parameter biomechanical modelling, the parameters which need to be identified are the masses,
damping coefficients, stiffness coefficients and non-linear parameters. Seating control systems
have been optimised offline using a parameter search algorithm [164], but in particular
applications such as real-time vibration monitoring and control systems, the performance and
accuracy depend on the ability to predict the model parameters. Optimisation of seating control
systems has been undertaken. An approach of entropy compromise programming criteria has
been applied to a multi-degree of freedom linear [165] and a non-linear [166] biomechanical
model. The results reported 20 seconds delay in the identification of biomechanical parameters.
Research in the area of control systems has applied real-time parameter estimation for a 1-DOF
body systems with the assumption that the mass of the seated occupant can be measured
continuously [167]. Other work has implemented a sliding mode controller which considers an
initial set of identified parameters of the body model but will consider the elements of
uncertainty in the mass distribution of the biomechanical model [168-170]. A simple cell
mapping (SCM) method is considered in [171] to solve the multi objective biodynamic lumped
parameter problem.
2.7. Vehicle Dynamics Control Systems
Traditionally, biodynamic human body models have primarily been applied to predict motion
under situations that are not practical to replicate in an experimental environment, and to
optimize other vibration isolation systems [39]. In recent research, biodynamic models have
served in the design of model-dependent control systems [167]. Recently, there has been a great
deal of research into vertical seating suspension control system design with designed body-
vehicle model dependent control systems with varying degrees of freedom. Examples of such
systems categorised based on the methods of control action include passive [164, 172], active
[170, 173] and semi-active [168, 174, 175] systems. Passive suspension systems have been
applied which act in the ‘fore-aft’, or longitudinal direction [176] however most research has
been restricted to the control of vertical motion. Currently, only a few studies in seating
suspension control systems have considered the real-time body model. Past research has
identified body mass parameters in real-time [167], as well as body stiffness and damping
[174]. Literature on the design of seating suspension control systems have applied passive
constant coefficient 1-DOF models [173, 177], 2-DOF Models [174, 178] and 3-DOF models
[179, 180]. Another research area utilises a quarter-car model coupled with the biodynamical
model of the driver. Combining the two models in a single human–vehicle–road (HVR) model,
[162, 172].
53
The results from the literature in seating suspension control systems is presented in terms of a
power spectra, SEAT factors, transmissibility plots and time domain response results plots,
examples are given in the studies as shown in Table 2-4. The reported results are often difficult
to compare without resulting in a false analogy due varying conditions set within each
individual chapter. The reported results in works involving simulations tend to yield a much
lower SEAT factor than those in the experimental studies, that is the simulated cases are
overestimating the performance of the designed control system. Both experimental studies [169,
181, 182], have reported their respective HVR to improve performance in terms of the SEAT
factor. The fore aft body motion response has been modelled [183, 184] and a passive
suspension has been optimised in [176], however there is a small volume of works on
integrated human body and seat suspension models on non-vertical axes.
2.8. Discussion
The literature on the study of body modelling has revealed that there is a large amount of
research that has been undertaken in predicting vibrational STHT, DPMI and APMS. Currently
there is sparse research on estimating the body model parameters in real-time. The possible
reasons for this are the large search space for multiple DOF models, the application of the
model not requiring real-time knowledge on the parameters and difficulties in implementing a
reference point in an identification algorithm due to the associated problems with using sensors.
The emerging research on WBV motion capture has applied two main sensory technologies,
wearable IMUs and optical sensors, which are a relatively new application to the field of WBV
research. The research reviewed in section 2.4.3 about IMUs and section 2.4.4 about optical
motion capture systems reveal the trend in the increase in the use of such technologies. Each of
these technologies has their respective problems when capturing WBV. For optical motion
capture systems, further research could be undertaken in image signal processing, to improve
the mobility and inertial motion estimation, and for IMUs, further research in the field of signal
processing for mechanical systems can be undertaken to improve their position estimation
accuracy and improve their estimation of the human body’s FRF. There are several major topics
that are outside the scope of this chapter but are relevant to the modelling and analysis of WBV.
This includes modelling of the seat and seating suspension system, seating suspension control
system formulation that includes an integrated biodynamic model. The modelling of the human
body response to WBV in non-seated positions such as the supine and standing position were
not considered in this survey due to the limitation set by the scope. However, this is currently an
active topic of research, and most of the fundamentals on WBV modelling, sensors and
frequency response analysis can be applied. In the area of control system design this chapter
54
mentioned the aspects of incorporating the seated body model into the plant of the control, other
control systems problems were not discussed in this chapter.
55
Table 2-4 Summary works in seating suspension systems with an integrated biodynamic modelling and response analysis
First Author Seating System Biodynamic body model
Wan [134, 164] Passive spring mass damper system 4DOF lumped parameter model,
nonlinear suspension stiffness
Choi [169] Sliding mode controller (SMC) semi-active controller 6DOF body model
Hurmuzlu [170] SMC Point mass variable load
Maciejewski [181,
182]
Active Point mass variable load
Yingbo [175] H∞ reliable controller 4DOF Body Model
Kuznetsov [172] Passive 3DOF HVR model
Sun [179] H∞ active 3DOF
Du [185, 186] Active controller Point mass with variable load
Gudarzi [187] Micro synthesis based active controller 4DOF
56
Yao [174] Semi-active H∞ control with magnetorheological damper Point variable mass
Stein [176] Fore-aft passive suspension system Point mass
57
Table 2-5 A summary of biodynamic models in the literature.
Model Name DOF Characteristics Data set / Parameter optimization Verification/Reported results
Boileau [52] 4 Linear lumped parameter Experimental/sequential
search
Min error: 1.2% max error: 20%
Ippili [156] 2 Seat – occupant model,
nonlinear stiffness and cushion
viscoelasticity
Body data from [188], seat dynamic
parameters identified experimentally
optimized using least squares fit
Verified MATLAB working model 2D Simulation
Toward [189] 4 Nonlinear 4DOF + car quarter
model vert, considers changes
in posture, hand position and
vibration magnitude, Transfer
function parameters varied
using lookup table
Squared error minimization using
experimental data collected from laboratory
Simulation compared with experimental data
Papalukopoulos
[162]
2 car + 5
Body
Quarter car model +
biodynamic cushion
displacement model+loss of
contact considered
Data from study[190]. Simulation using random and sinusoidal signals
Kim [154] 6DOF XYZ
model
Rotary stiffness XYZ axes Measured APMS using mechanical shaker
seat / Curve fitting
Model developed from experimental data
Gohari [144] n/a ANN direct STHT estimation
from acceleration input
Experimental training data from shaker table Compares simulation results with [83]
Abdeen [143] 5 ANN Parameter estimator +
5DOF linear model
Stein [157] 4 Longitudinal model+ steering
wheel
Experimental data/ error minimisation Laboratory experiment random freq sweep
58
Yoshimura [155] 10 Spine model rotary simplified
to rectilinear
STHT from accelerometers error
minimisation
Compare model to experiment
Valenti [137] Spline
Formulism
spline model Simulated response Accuracy reported graphically
Srdjevic [166] n Parameter identification using
Comprise Programming,
nonlinear model
Comprise Programming using a fitness
function
Calculates an optimal parameter within a time span
of 20 seconds for 2000 samples
Kim [161] 4 Linear model, rotational joints Experimental data/ Optimised with GA Reported error least-square 1%
Wang[138] n/a Finite Element Analysis model
includes muscle forces
Bazrgari [139] n/a kinematics-driven finite
element passive-active system
Darling [191] 6 Lumped parameter Data from [52, 192] STHT, DPMI and APMS
curve fit
STHT Goodness of fit 90.84% DPMI Goodness of
fit:81.33%
Tufano [74] 2 Frequency dependant stiffness
model
APMS,STHT
Wang [159] n/a Head neck model/ Inertia and
Rotary joints
Experimental / MATLAB Optimisation
toolbox
Presented graphically
Lemerle [70] 1 Includes legs with inertia Curve fitting Compared with 1-DOF model without lower limb
inertia
59
60
2.9. Conclusion
The following findings were observed in the survey of research into WBV Research:
• There is a clear consensus in the industrial safety and epidemiology research that the
relation between vibration dosage and LBP is difficult to specify due to the additional
variables including age, posture, and uncertainties.
• Because of recent developments in the improvements in the low size weight and power
of MEMS technology, MIMO system frequency response estimation in 6-DOF has
become increasingly more practical to implement, even though it remains very difficult
to analyse due to the complexities of cross-coupled motion.
• A wide variety of approaches has been employed to design biodynamic models.
Because in the application of control systems it is desirable for the body model to be
robust, the main problems which have still not been solved is model over-fitting and the
identification of body model parameters in real time. Existing research in control
systems has modelled the human body mass with uncertain parameters and has achieved
robust performance. There is a lack of research investigating the performance of active
seating suspension control systems under the inter subject and intra subject variations.
• Comprehensive seated whole-body vibration analysis has been undertaken with both
optical and inertial motion capture systems. Both systems have their respective
advantages and disadvantages. Inertial motion sensors have the potential to replace
optical motion capture systems with further developments in software and MEMS
technologies.
• The application of soft computing to whole body vibration response modelling is an
emerging topic of research.
61
3. Experiment Methodology
This chapter presents the experimental methodology used for the analysis in this
thesis. The experiment was designed such that a rich data set which includes all the
anomalies and uncertainties associated with real world could be recreated in a
laboratory environment. A custom-built parallel hexapod robot was utilised for the
purpose of collecting repeatable datasets to provide inter and intra subject
variabilities. The platform used as it has a high repeatability and positional
precision. The data store and structure details are presented in this section that
provide the foundation for the analysis and modelling in the following chapters of
this thesis. A preliminary analysis is performed on the data sampled from the
parallel
3.1. Introduction
This Section outlines the experimental methodology and the initial pilot tests of the parallel
robot and wearable inertial motion sensors. The outcome is to provide the preliminary details
that provide a foundation for the following chapters. An experimental vibration simulator using
a parallel robot was custom designed by the researchers at the Centre for Intelligent
Mechatronics Research (CIMR) at the University of Wollongong. The parallel robot is capable
for delivering high force to payload ratio and has a high displacement precision. For all
experiments a set of wireless IMUs manufactured by Xsens were employed [193].
3.2. Overview of the Parallel Robot
A parallel hexapod robot or Gough-Stewart [194, 195] platform had been utilised for the
research in this thesis for generating the datasets. The Parallel robot is comprised of a solid base
platform connected to six prismatic electromechanical actuators. The robot allows for controlled
excitation signals that are repeatable upon each trial and test subject. A custom 6-DOF parallel
robot had been designed and built with six electrical prismatic actuator cylinders. An image of
the parallel robot with the complete seat setup is shown in Figure 3-2. The robot and the
computer aided design is given in Figure 3-1
The main performance specifications of the robot are given in Table 3-1. The parallel robot can
support a payload of up to 250 Kg. The Robot can deliver a constant acceleration up to a
frequency of 12Hz before it approaches roll off. The parallel robot can deliver vibrations in the
62
x y z and roll pitch yaw locations that is the full 6DOF of motion.
The electromechanical stepper actuators of the parallel robot are controlled by Panasonic
Programmable logic Controllers, a PID controller is implemented on a desktop computer using
the inverse kinematic algorithm in [196]. The electromechanical actuators consist of a linear
encoder giving the robot feedback control for precise displacement accuracy.
Property Value
Maximum Payload 250Kg
Mechanical roll off
frequency
12Hz
Figure 3-1 A computer aided design model of the parallel robot and the fabricated device [197].
Table 3-1 properties of the parralel robot
63
Figure 3-2. a side view of the parralel robot and the seat.
64
3.3. Inertial Motion Data Capture
In this thesis inertial motion sensors are selected for the use of collecting data. The IMUs
consist of Microelectromechanical Systems, which consist of a proof mass suspended off a
spring as shown in Figure 3-3. The operating principle of this system works on a when the
suspended mass experiences a force, the capacitor is dependent on the displacement of the
mass. Using the principle of capacitive transduction, it is possible to use an electronic
amplifier circuit to measure the force. In the case of a MEMS gyroscope as shown in Figure
3-4 the force is dependent of the Coriolis acceleration that is acting on the mass due to the
angular velocity on the mass [118].
The Xsens MTW IMU [198] was used in this thesis, which consist of a miniaturised wireless
6DOF MEMS based IMUs. The main signal properties are given in Table 3-2. In this thesis
the sampling rate was set to 60Hz. The Xsens IMU contain internal sensors and a drift
correction algorithm that compensates for magnetic field and thermal disturbances.
Figure 3-3 a spring mass capacitor accelerometer schematic [118].
Figure 3-4 a schematic of the cantilever MEMS gyroscope [118].
65
Figure 3-5 shows the definition of the coordinate system of the Xsens IMU. Using an
orientation estimator [198]. The Z axis is calibrated to by upward against the direction of
acceleration due to gravity and the X axis is aligned with the magnetic north.
Figure 3-5, The definition of the axes for the IMU sensor nodes [199].
Figure 3-6 The Xsens IMU Modules [198].
The Xsens MTw development kit is shown in Figure 3-6, where a) is the sensor module in its
enclosure. B) is the charging station and wireless transceiver module, C) is a wireless USB
dongle transceiver and D) shows the bracket for the IMU with the textile hook and strap
connector that is used to fasten the IMUs to the seated test subjects.
66
Figure 3-7 Illustrates the DSP block diagram of the Xsens MtW IMU. Both the gyroscope and
accelerometer are MEMS devices. The initial stage the gyroscope and accelerometer data are
passed through an analogue low pass filter, then is digitized to be passed through a discrete low
pass filter. Considering the bias and drift correction of the acceleration and gyroscope signals an
EKF if utilised by the on board MCU. This is due to the metallic cantilever beam components
inside the IMU being distorted by temperature and magnetic fields. The benefit of the on-sensor
processing is the possibility processing at a higher frequency before the data is transmitted over
the wireless channel.
The sensor performance specifications are given in Table 3-2. The key performance requirement
that is important for is the frame rate and bandwidth. The bandwidth gives an indication of the
response of the sensor to a step signal. A frame rate of 60Hz was chosen so that the principal
resonance peaks could be captured according to Nyquist’s sampling theorem as well as the
highest possible sampling rate due to the bandwidth limitation of the wireless Xsens MTw.
The magnetic field distortion compensation algorithm of the Xsens IMU has been implemented
and verified for this study and the results are shown in Figure 3-8, where the raw angular
estimation is compensated using the feedback from the magnetometer.
Figure 3-7 The interanl DSP and sensor fusion architecture of the Xsens IMUs [198].
67
Table 3-2 Summary of the relevant signal properties of the IMUs [198].
Accelerometer Gyroscope Magnetometer Barometer
Full Scale +- 160 m/sec +-2000deg /sec +- 1.9 Gauss 300-1100hPa
Non-linearity 0.5% of Full Scale 0.1% of Full Scale 0.1% of Full Scale 0.05% of Full Scale
Bias stability 0.1mg 10 deg/hr n/a 100Pa/year
Noise 200 μg/√Hz 0.01deg/s/√Hz 0.2 mGauss/√Hz 0.85Pa/√Hz
Bandwidth 184Hz 184Hz 10-60Hz n/a
Internal Sampling Rate 1000Hz 1000Hz n./a n/a
Output FPS 60Hz 60Hz 60Hz 60Hz
68
Figure 3-8 A sample of the magnetic field distortion compensaton for angle estimation on the
Xsens IMU.
69
3.4. Experimental Procedure
3.4.1. Experiment Setup
For this study human test subject were involved. As per Australian federal legislation on the
National Statement on Ethical Conduct in Human Research (2007), the University of
Wollongong human research ethics council reviewed the proposal of this study and have given
formal approval for the experiment to be conducted. See appendix A for the formal approval.
For this study four male test subjects were recruited, and several controlled excitations were
delivered to the test subject, the mean mass was 79.4Kg with SD of 7.1 Kg. The properties of
the excitation signal will be discussed with more detail in the next section. Although the parallel
robot has high precision displacement feedback an Xsens IMU was fastened to the parallel robot
to provide complete 6DOF inertial motion data for the platform base as per Figure 3-9. Each test
subject has a system of 10 Xsens MTW IMUs fastened with a textile hoop and loop secure lock,
an example of this setup is shown in Figure 3-10. As per Figure 3-10 the appropriate safety
measures were put in place for each test subject, like the anti-slip mat, a seat belt and a robot kill
switch on the operator table. The test subjects were recruiting from members within the research
lab at CIMR through email in line with the ethical clearance for this study.
Figure 3-9 The location of the chassis IMU.
70
3.4.2. Pilot Platform Test Results
For the results to be repeatable the parallel robot was set a to perform a desired vibration
sequence with feedback control. The RMS value of each trial per test subject is shown in Table
3-3, a combination of frequency swept excitations under various axes were performed. In total 9
regimes of vibration excitation sources were tested, with the initial one being a 0 source to
provide a noise and perturbation for the algorithm development in the later chapters. The inertial
motion data was sampled and stored in raw binary format then processed into a data structure
for use in the following chapters. The total duration of data recorded for all trials was
approximately 2 hours. For each of the models in the following chapters the recorded data was
split into a 1:9 ratio of training to validation data for evaluating the results of each model.
For illustrating and validating the performance requirements of the experimental setup a
frequency response analyses was performed. Figure 3-11 shows a sample of the STHT
estimation using the h1 frequency response estimation algorithm under regime 1 from Table
3-3. The results show the pelvis being a multimodal response whereas the head and sternum
contain a single mode and are nearly proportional to each other. The STHT response shows the
human bodies’ underdamped phenomena of the resonant frequencies. The corresponding
coherence is shown in Figure 3-12 which indicates most of the signal power is transferred at a
Figure 3-10. The parallel hexapod robot with seated test subjects.
71
value less than the resonance frequency. Figure 3-13 shows a sample of the neck angle joint
estimation using the Xsens IMU under the excitation regime number 7 from Table 3-3. For each
of the following chapters a sample
Figure 3-11 Transmissibility of a test subject from multiple IMUs
72
Figure 3-12 shows the signal coherence between IMUs for Acceleration.
73
Figure 3-13. Neck Joint angle estimated from Xsens [200].
The power spectral density of the parallel robot is given in Figure 3-15 when the platform is
under a rectilinear excitation regime 1. The power spectral density demonstrates the main
operating frequency range of the parallel robot is up to 10Hz where the peak energy transferred
by the platform is in the range from 5 to 8 Hz.
For the case when the platform is under a rotational excitation source, under regime 6, Figure
3-14 shows the power spectral density plot. Under rotation it can be observed that the peak
energy transferred is under a band from 3 to 4 Hz.
74
Figure 3-14 A sample PSD plot of the motion platform with a rotational excitation source
regime 6.
Figure 3-15 a PSD plot of the motion platform with a rectlinear excitation source.
75
Axis Rectilinear
Acceleration (m/sec)
RMS
Angular
Velocity
(deg/sec) RMS
Regime
Number
x y z x y z
0 0 0 0 0 0 0
1 0.160 0.160 0.931 0.624
0.672
0.565
2 0.177 0.138 0.943
0.634
0.710
0.573
3 0.191 0.122 0.956
0.630
0.743
0.571
4 0.531 0.127 0.091
0.619
0.582
0.577
5 0.186 0.499 0.156
0.629
0.604
0.571
6 0.685 1.035 0.408
1.218
1.691
7.862
7 0.288 0.234 0.803
0.638
5.887
0.764
8 0.164 0.142 0.496
0.617
3.034
0.677
Table 3-3 RMS accelertion and velocity of each test regime.
76
3.1. Experimental Platform Data and Analysis
The structure of the data collected from the IMUs are to be presented in this section. The data
collected from the test subjects is further processed to provide information on joint articulations,
calibrated orientations and the frequency response of each node.
In the following chapters a training data set is used that is independent from the test dataset. To
achieve this each trial of inertial motion data capture was shuffled into a training set or a testing
set category. Figure 3-16 shows the tree data store that is used and the data available for the
analysis and modelling in the following chapters of this thesis.
For the purpose of data analysis the following toolboxes and programming environments were
utilised: MATLAB Weka [201] Moa [202], ELKI [203, 204], Oracle Java Development Kit,
MATLAB Optimization Toolbox, MATLAB Deep Learning Toolbox and Xsens software
development kit [193].
Figure 3-16 A block diagram of the data stutcure of the sampled inertial motion data.
77
4. Biomechanical Whole-Body Response Model Estimation using
an Exact Noise and Process Model
This chapter describes an online biomechanical lumped parameter model
estimation algorithm applied to data acquired from a network of IMUs. The
predicted biomechanical model represents the response motion of a human seated
in a vehicle subjected to mechanical vibrations. The motivations for identifying the
body model parameters are, to aid vibration isolation technologies and to
continuously monitor the transmission of vibrations through the seated occupant.
The algorithm consists of a bank of parallel Extended Kalman Filters for
estimating each biodynamic parameter. During the posteriori state prediction
update step of the Extended Kalman Filter, an Adaptive Sliding Window concept
drift detection algorithm maintains a variable window of past priori estimation
errors. The model process error statistics are then updated based upon the
statistics of the priori state errors in the window. The algorithm updates the
estimator parameters incrementally and is suitable for streaming data. The
estimator is applicable for cases where the distribution of the biomechanical model
process noise is unknown or is abruptly varying, and when the model parameters
are unknown. This estimator was evaluated experimentally with data sourced from
a network of wearable wireless Inertial Measurement Units affixed to the test
subject. The test subjects were then exposed to an external vibrating excitation
source. The results validate that this algorithm provides accurate online estimates
of the human biomechanical model parameters for a second order transmissibility
model. The algorithm presented mitigates problems associated with the estimation
of biomechanical parameters such as biomechanical nonlinearities, process noise
drift and divergence.
4.1. Introduction
Biomechanical model identification is the identification of a mathematical model that can be
applied to estimate the motion–frequency response to mechanical vibrations of the body based
on the experimental data obtained from vibration transmission measurements or estimations
[97]. The practical applications of biomechanical human body models are for the prediction of
78
motion under situations that are not practical or safe to replicate in an experimental
environment, and to optimise vibration isolation systems such as vehicle and seating
suspensions [39].
In recent research, biodynamic models have served in the design of model-dependent control
systems [167], however in practice the biodynamic parameters have been shown to be
dependent on further variables which are hidden from the control system and body model, such
variabilities include seating posture and body type. The use of real time estimation has
significance for health and safety monitoring and vibration isolation technologies. There has
been a large volume of studies undertaken for body modelling under multiple degrees [148]
where the parameters are estimated after post processing a batch of biomechanical data.
The problem presented in this chapter can be generalised as a nonlinear process and observer
state estimation problem. Current existing methods for estimating states and system model
parameters using parallel Kalman Filters [206], Rao-Blackwellised Filter [207], Particle Filters
[208-210], Unscented Kalman Filter (UKF) [211], Invariant Extended Kalman Filters (IEKF)
[212] and Moving Horizon Estimators (MHE) [213]. On a spectrum, most state estimators have
a trade between high accuracy and higher computational complexity, The EKF is placed on the
end of low accuracy/complexity whereas the MHE is on the opposite end with higher
accuracy/complexity [214]. The UKF has been proven to have an equivalent complexity to that
of the EKF, however in some cases has been difficult to implement in practice due to
uncertainties in unscented transform parameters [207].
In this chapter a solution to biomechanical model parameter estimation is presented using the
Extended Kalman Filter ADaptive Sliding WINdow (EKF-ADWIN) algorithm that provides
incremental updates to the biodynamic model parameters. The advantages of the approach of an
ensemble between online data concept drift detection and the EKF, is that the benefit of the low
computational cost of the EKF is retained whilst the state convergence stability is improved.
The simplicity of the estimator is desired for practical applications such as wireless IMU sensor
networks. The contribution of this method is the design of a filtering strategy that allows for the
real time estimation of the body model considering for sensor measurement model errors as well
as biomechanical model errors. The usefulness and the advantages of the proposed estimator
have been demonstrated via a validation on experimental and on a numerical simulation.
This chapter is structured as follows, the preliminaries on model parameter identification with
EKFs is presented in section 4.2, the preliminaries on drift detection is given in section 4.2.3,
the implementation of EKF-ADWIN is presented in section 4.3, the experimental methodology
is presented in section 4.4, the discussion of the results is given in section 4.5 and the
79
conclusions are given in section 0.
4.2. Model identification with EKF
4.2.1. Model Definition
In this chapter a prismatic 1 DOF model is considered with automatically tuned parameters. The
definition of the system to be estimated by the EKF is similar to that of a standard linear
Kalman filter [215] except the process state transition equation and the measurement equation
are represented as nonlinear functions. The process and measurement model for the EKF are
given in Eq. (4-1) and Eq. (4-2) respectively. The process state transition equation given in
Eq.(4-1) , describes a dynamic system where the current state 𝑥𝑘 is dependent on a function of
the previous state and an external input, 𝑢𝑘−1. The additive zero mean Gaussian noises is
denoted by 𝑤𝑘 where the covariance of this noise is a constant defined by 𝑄. In Eq. (4-2), the
measurement equation, 𝑦𝑘 is the measurement vector with elements that contain values read
from any observable sensor source such as a bearing from a global positioning system or an
angle from an IMU. Function ℎ is the observer function that describes the relation between the
current state and current measurement vector. The zero-mean noise with a Gaussian distribution
is denoted by 𝑣𝑘, and has a covariance of 𝑅.
𝑥𝑘 = 𝑓(𝑥𝑘−1, 𝑢𝑘−1) + 𝑤𝑘−1 (4-1)
𝑦𝑘 = ℎ(𝑥𝑘) + 𝑣𝑘 (4-2)
Since the objective of the EKF is to estimate parameters that the state process state transition
function is dependent on, an additional equation will be defined as 𝜃𝑘 [207], which is a state
vector that contains the parameters to be estimated. The parameter state transition equation is
defined in Eq. (4-3), where Φ is a state transition matrix, usually an identity matrix, 𝜃𝑘−1is the
estimated parameter at the previous state. A random Gaussian noise of 𝑛𝑘−1 is introduced to
allow the parameters to grow to their prime values, the variance of this noise value is dependent
on the desired search space of the parameter 𝜃.
𝜃𝑘 = Φ𝜃𝑘−1 + 𝑛𝑘−1 (4-3)
The nonlinear state transition and the measurement equations are then augmented to incorporate
the state parameters by passing 𝜃 as an input to functions ℎ and 𝑓, Eq. (4-1) and Eq. (4-2) are
now rewritten in the form as Eq. (4-4) and Eq. (4-5).
80
𝑥𝑘 = 𝑓(𝑥𝑘−1, 𝜃𝑘−1, 𝑢𝑘−1) + 𝑤𝑘−1 (4-4)
𝑦𝑘 = ℎ(𝑥𝑘 , 𝜃𝑘) + 𝑣𝑘 (4-5)
The complete process state transition equation is defined in Eq. (4-6), by incorporating the
process state transition equation and the parameter transition equation, where both the process
model sates, and unknown parameters are now treated as time variant states.
[𝑥𝑘𝜃𝑘] = [
𝑓(𝑥𝑘−1, 𝜃𝑘−1, 𝑢𝑘−1) 00 Φ
] [1
𝜃𝑘−1] + [
Qk 00 1
] [𝑤𝑘𝑛𝑘]
(4-6)
For brevity, the augmented states will be defined by the variable 𝜉, given in Eq. (4-7).
𝜉 = [𝑥𝑘𝜃𝑘] (4-7)
The state transition equation is defined, in the form with the states 𝜉, in Eq. Function 𝑔 is a
nonlinear state transition function of states 𝜉 , it is defined by the state transition matrix from
Eq. (4-8).
𝑔(𝜉𝑘−1, 𝑢𝑘−1) = [
𝑓(𝑥𝑘−1, 𝜃𝑘−1, 𝑢𝑘−1) 00 Φ
] [1
𝜃𝑘−1]
(4-8)
The final form of the process state transition equation is given in Eq. (4-9). The measurement
equation as a function of states 𝜉𝑘 is denoted in Eq. (4-10) is determined by substituting Eq.
(4-7) into the measurement equation Eq. (4-5).
𝜉𝑘 = 𝑔(𝜉𝑘−1, 𝑢𝑘−1) + [Qk 00 1
] [𝑤𝑘−1𝑛𝑘−1
] (4-9)
𝑦𝑘 = ℎ(𝜉𝑘) + 𝑣𝑘 (4-10)
4.2.2. EKF Prediction and update steps
The EKF is consists of two key steps, the priori prediction of the state value and state
covariance and secondly an update step, where a correction is applied to the estimated
covariance and priori state based on the residual between the measurement vector and the
estimated value [216]. Equation (4-11) is the state covariance prediction step where 𝐹𝑘 is a
linearized state transition matrix computed by evaluating the Jacobian matrix in Eq. (4-13)
where 𝜉𝑘−1|𝑘−1 is the previously estimated state. The priori state estimate given in Eq. (4-11) is
calculated by evaluating function g for the previous state value.
81
𝑃𝑘|𝑘−1 = 𝐹𝑘−1𝑃𝑘𝐹𝑘−1𝑇 + 𝑄 (4-11)
𝜉 ̂𝑘|𝑘−1 = 𝑔(𝜉𝑘−1|𝑘−1, 𝑢𝑘−1) (4-12)
𝐹𝑘−1 =
𝜕
𝜕𝜉 𝑔(𝜉 )|
�̂�𝑘−1|𝑘−1
(4-13)
The second step is the update step. Equations (4-14) and (4-15) describe the update of the
Kalman gain, where 𝐻𝑘 is the Jacobian matrix of the measurement function ℎ given in Eq.
(4-18). The updated state covariance and state estimations are given in Eq. (4-16) and Eq.
(4-17).
𝑆𝑘 = 𝐻𝑘 𝑃𝑘|𝑘−1𝐻𝑘
𝑇 + 𝑅 (4-14)
𝐾𝑘 = 𝑃𝑘|𝑘−1𝐻𝑘|𝑘−1𝑇 𝑆 𝑘
−1 (4-15)
𝜉𝑘|𝑘 = 𝜉𝑘|𝑘−1 + 𝐾𝑘(𝑦𝑘 −𝐻𝑘𝜉𝑘|𝑘−1) (4-16)
𝑃𝑘|𝑘 = 𝑃𝑘|𝑘−1 − 𝐾𝑘𝐻𝑘|𝑘−1 𝑃𝑘|𝑘−1 (4-17)
𝐻𝑘 =
𝜕
𝜕𝜉 ℎ(𝜉 )|
𝜉𝑘|𝑘
(4-18)
The error in between the predicted and updated state is given in Eq. (4-19), which is an input to
the ADWIN function described in sections 4.2.3 and 4.3 .
𝑒𝑘 = 𝜉𝑘|𝑘 − 𝜉 ̂𝑘|𝑘−1 (4-19)
82
4.2.3. Drift Detection
In this chapter a drift detector is incorporated into the main EKF loop to provide estimates of the
this concept has been previously applied to a linear estimator in [217]. The method of drift
detection that is considered in this chapter is ADWIN [218]. ADWIN is an incremental
algorithm which can accurately estimate changes in distributions of a sequence of variables.
Conceptually ADWIN operates on a principle that a window, W holds a series of past values.
During each step of the ADWIN algorithm, the length of the window is then reduced when an
abrupt change to the distribution statistics is occurring or when there is no change detected the
length of the window will remain constant. The result is an accurate detection of concept drift in
the data as well as an evaluation of variance and mean statistics.
1 function ADWIN(W,𝑒𝑘)
2 % add 𝑒𝑚 to the start of W
3 W←W∪ 𝑒𝑘
4
5 while |𝝁𝑾𝟎 − 𝜇𝑊1|≤ 𝜖
6 drop last element of W and add it
start 7 of W0
8 W1 = W
9 end
10 output W, 𝜇𝑊,𝜎𝑊2
The pseudo code in Table 4-1 shows a function representation of ADWIN for a single variable,
𝑒𝑘 and a window 𝑊 containing past values of 𝑒. Initially the current sample is added to the
beginning of W. A loop is then performed from lines 5 to 8, where the window W is split into
two windows W0 and W1, where W0 contains the last element of W. On each iteration of the
while loop, the splitting criterion 𝜖 is updated for the statistics in window 𝑊, given in Eq.
(4-20). Equation (4-20) is based on Hoeffding’s boundary [219], extended on by Bifet et al.
[218] with the summation of latter term, in order to provide a boundary that fits nearer to the
expected values of the variable.
Table 4-1 Pseudo code of ADWIN function.
83
𝜖 = √2
𝑚∙ 𝜎𝑊
2 ∙ loge (2
𝛿′) +
2
3𝑚∙ loge (
2
𝛿′)
(4-20)
The parameters required for the evaluation of the boundary 𝜖, are the harmonic mean 𝑚, given
in Eq. (21) and the confidence value per window length, given in Eq. (4-21). Where 𝑛0 is the
length of window 𝑊0, 𝑛1 is the length of window 𝑊1. 𝛿 is the confidence constant of ADWIN.
𝛿′ =
𝛿
𝑛
(4-21)
4.3. EKF-ADWIN implementation
4.3.1. EKF System Model Definition and Decomposition
For the implementation of the EKF-ADWIN estimator, a second order system is considered to
represent the Seat to Head Transmissibility (STHT) of the seated test subject. The second order
degree biodynamic transfer function, expressed in the continuous s domain, is given in Eq.
(4-22). The two unknown parameters considered by the estimator, are the natural frequency 𝜔𝑜
and the damping ratio 𝜁. The relationship between the transfer function and the input and output
is given in Eq. (4-22), where the excitation signal is denoted by 𝑈(𝑠) and the measured signal is
𝑌(𝑠), when expressed in the continuous frequency domain. The transfer function of
transmissibility can be conceptualised as a function that describes how the amplitude and phase
a vibration will change as it is transmitted from the seat to the sensor location on the human
body.
𝐻(𝑠) =
𝑌(𝑠)
𝑈(𝑠)=
2𝜁𝜔𝑜𝑠 + 𝜔𝑜2
𝑠2 + 2𝜁𝜔𝑜𝑠 + 𝜔𝑜2
(4-22)
Since the EKF provides state estimations of a state space system, the biodynamic model is
required to be expressed as a state space system. Following the steps as described in section 4.2,
initially an augmented state space model is then defined by making a substitution of the
biomechanical parameters 𝜁 and 𝜔𝑜 into, the state transition equation for the state 𝜃 is given as
Eq. (4-3) from section 4.2, resulting in a representation of the system given in Eq. (4-22) in the
forms of Eq. (4-9) and Eq. (4-10). The reason for the selection of the damping ratio and natural
frequency as the state variables of the EKF is to achieve steady state observability.
The state space model of Eq. (4-22) is decomposed into two separate state space systems with a
single parameter, in the first system the natural frequency is considered as a state variable and
84
the damping ratio is treated as a constant value that is updated. The converse is true for state
where damping ratio is a state variable and natural frequency is a constant. The purpose of
reducing the number of nonlinear states is to improve the stability of the state estimates. As
there will be two state space models, each model will have independent process noise
covariance matrices, 𝑄. A simplified flow chart of EKF-ADWIN for a single parameter, is
shown in Figure 4-1.
4.3.2. Process noise covariance update step
Although the nonlinear process and measurement model defined appears to be sufficient in its
current form for state estimation with an EKF, however in practice this system has been shown
to have a very high likelihood of diverging [220]. The assumption that is tested in this chapter is
that the mean and covariance of the distribution of the process noise in a biodynamic system is
changing over time, and by incorporating an online drift detection algorithm to detect these
changes will improve the performance of the EKF by converging to an optimal state.
A time variant error covariance process variable 𝑄𝑘 is introduced into the EKF, which is
incorporated, the definition of 𝑄𝑘 is given in Eq. (4-23) where the diagonal elements correspond
to the process covariance for each state 𝑛. The diagonal of 𝑞𝑛𝑛, denoted in Eq. (4-24) are the
outputs of the ADWIN drift detector by storing the time history of the state errors between the
Figure 4-1 Flow chart of the EKF-ADWIN process.
𝑄𝑘 = [
𝑞1 … 0⋮ ⋱ 00 0 𝑞𝑛
]
(4-23)
85
prior estimate and the updated posteriori state estimate.
The outputs of ADWIN are a mean and variance value of the state errors. Since the EKF only
takes into consideration that the mean of the process noise is zero the variance is expanded so
that Gaussian distribution with mean variance of 𝑞𝑛 can be determined.
𝑞𝑛𝑛 = 𝜇𝑘 + 𝜎𝑘2 (4-24)
4.4. Experimental Validation
A custom 6 Degrees of Freedom (6DOF) parallel robot was acquired for generating the
mechanical vibrations as described in Chapter 3. The synthesised acceleration excitation signal
was applied in the vertical axis with seated test subjects. The excitation source was set to a
sinusoid with a frequency increasing from 1 Hz to 9 Hz and with a peak amplitude increasing
from 0.1𝑚/𝑠2 to 2.5 𝑚/𝑠2. The brand and make of wireless IMUs used for the collection the
data for evaluation were Xsens MTws [198], which consist of a 3 axis accelerometer, a 3 axis
Figure 4-2 The layout of the IMUs and parralel
hexapod robot.
86
gyroscope and a 3 axis magnetometer. The layout of the wearable IMUs are shown in Figure
4-2, IMUs are affixed to the left and right shoulders, head, chest and the rear pelvis. The
definition of the complex transmissibility response is defined in Eq. (4-25), for the purpose of
simplicity only the absolute value of the vertical transmissibility response will be presented in
this chapter. The signals 𝑋(𝑗𝜔) and 𝑌(𝑗𝜔) from Eq. (4-25), are also labelled on Figure 4-2
where 𝑋(𝑗𝜔) is measured from the hexapod and 𝑌(𝑗𝜔) is measured from the IMU on the head.
For the validation of the accuracy of the estimated model, the transmissibility of both the
estimated model and the Frequency Response Function (FRF) estimation will be evaluated by
performing a 𝐻1 FRF estimation algorithm [99] where Welch’s method [221] is used to perform
the fast Fourier transform. The transmissibility response is determined by evaluating the 𝐻1
estimation of Eq. (4-25) where 𝑈(𝑗𝜔) is the input acceleration signal at the platform base and
𝑌(𝑗𝜔) is the vertical acceleration at the test subject’s head.
|𝐻(𝑠)|𝑗𝜔| = |
𝑌(𝑗𝜔)
𝑈(𝑗𝜔)|
(4-25)
4.5. Results and Discussion
The results of the evaluation of the EKF-ADWIN parameter estimator are presented on the
mesh surface in Figure 4-4. The training time axis corresponds to the time elapsed where the
model is updated at 60𝐻𝑧. The initial state plane shows the initial guess of the model for states.
The 𝐻1 FRF plane shows the transmissibility response estimated using the entire time history
acceleration input/output. The closeness of the fit between the EKF-ADWIN surface value at
the time instant the 𝐻1 estimation is evaluated at gives an indication of the accuracy of the
model, when the assumption is made that the 𝐻1 estimation is the ideal transmissibility
response.
The 2-dimensional equivalent of Figure 4-4 is given in Figure 4-5, where the 𝐻1 FRF estimation
is performed over a 30 second time series. It can be observed in Figure 4-5, that the
transmissibility response fits the actual response more accurately near the peak but less
accurately at frequencies from 0 to 3 𝐻𝑧 and at the frequencies higher than. The EKF-ADWIN
estimator only considers the model to be a grey box system model of a second order whereas a
higher order system will fit the second local peak value. Since the amplitude of the
transmissibility is less than 0.5 at frequencies of 10 𝐻𝑧 or greater in this case the accuracy of the
fit in this region is not crucial in the application example used in this chapter.
A problem encountered with the estimator is occurrence of EKF divergence when the initial
87
state value varies too far away from the real value, that is, a random initial state will result in a
divergence in the EKF. A possible solution that could be implementing in the future is the
incorporation of an ADWIN algorithm into a particle filter. Multiple state and covariance
estimate or particles could be considered, a test for divergence would be performed on the
particles. The particles which tend to diverge would then be discarded. The main benefit of the
particle filter approach would be the result in the reliable state estimations in conditions where
the initial state values are unknown.
Applying the theorem formulated in [220], the estimation asymptotic corrective function 𝑓𝑎(. . )
is applied to assess the conditions which EKF-ADWIN diverge at. The function 𝑓𝑎(. . ) evaluates
the correlation between the EKF residuals and the parameter estimates Figure 4-6 , Figure 4-7
and Figure 4-8, show a plot of the corrective asymptotic function. The function 𝑓𝑎(. . ) has been
evaluated as a function of the natural frequency in Figure 4-6 and the damping ratio in Figure
4-7. The stationary points of the curve and the regions near to zero indicate the parameter values
at which the EKF is likely to converge at. Both the standard EKF and the EKF-ADWIN have
been evaluated in Figure 4-6, to validate the improvements in the convergence of EKF-ADWIN.
Figure 4-8 shows the corrective asymptotes function while both the natural frequency and the
damping ratio are the input variables.
Figure 4-3 and show the incremental estimation of the model parameters with EKF-ADWIN
where the ARMAX model is used a performance benchmark which is fitted offline.
Figure 4-3. A comparison between the EKF-ADwin damping ratio tracking performance to and
ARMAX model parameter estimation.
88
Figure 4-4. Simulated model transmissibility during validation run.
Figure 4-5. Comparison of the model to time series 𝐻1estimation.
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2
2.5
3
Transmissibility Response
Frequency (Hz)
Tra
nsm
issib
ilit
y
H1 Estimation
EKF-ADWIN
89
Figure 4-6. Asymptote function of natural frequency for EKF-ADWIN and EKF.
Figure 4-7. Asymptote function of damping ratio for EKF-ADWIN.
90
4.6. Conclusion
This chapter presented a novel algorithm, EKF-ADWIN which has the benefits of the ability to
stably converge to a satisfactory estimation and a low computational cost. It has been shown to
provide satisfactorily accurate estimates of biodynamic model parameters. Furthermore, it has
been proven to yield real time performance in the estimation of the parameters of the
biodynamic equations of motions for a seated body. Further work in this field could investigate
further into the asymptotes and estimation failure conditions of EKF-ADWIN when compared
to existing equivalent methods solving the problem presented, such as the IEKF and the UKF.
An improvement of an estimator based on IEKF or UKF with drift detection applied to the
process error statistics could be implemented in future work. Improvements could be made to
EKF-ADWIN to improve estimation convergence with the inclusion of a step where the states
and covariance are tested for divergence or the incorporation of a Particle Filtering approach,
where multiple initial state values are considered simultaneously, and the particles that diverge
Figure 4-8. Surface plot of EKF-ADWIN Asymptotes.
91
would be discarded via the resampling process.
92
5. Approximation methods for estimation of biomechanical
models of whole-body vibration
This section presents a paradigm for biomechanical model estimation considering
a soft particle filter tuned through the process of supervised learning. A tuned
particle filter with an intelligent adaptive resampling strategy is introduced. The
results of the proposed method demonstrate its suitability to solving the problem of
incremental whole-body vibration estimation. This section presents a novel
strategy for biomechanical whole-body vibration model estimation with a
bootstrap particle filter. Currently, there is a challenge associated with issue of
utilising idealised state space estimators to incrementally predict the whole-body
vibration model. A particle filter is employed, which is an approximate method
that is independent of the process noise equation. In this chapter the heuristic
distribution of damping ratio and natural frequency values is exploited to generate
the initial state estimates. The results allow for the self-tuned estimation of
biomechanical whole-body vibration models, along 6 degrees of freedom of motion
incrementally without the use of postprocessing. This whole-body vibration model
estimator was experimentally verified with experimental data from test subjects
with a network of wearable wireless Inertial Measurement Units. The results of the
presented method demonstrate its suitability to solving the problem of incremental
whole-body vibration estimation without the need for computing a process error
model. The results demonstrate that the estimator presented in this chapter
provides stable performance along the rectilinear and rotary axes of motion and is
a noninvasive scheme for estimating the response of the seated human body
instantaneously.
5.1. Introduction
THE modelling and monitoring of the seated WBV response of the human body has become
an increasingly active topic of research interest due its usefulness in applications in workplace
health and safety . In the existing laboratory environment based research, the main sensing
technologies applied in the analysis of WBV response dynamics include the use of piezo-
electronic inertial sensors [2], Micro Electromechanical systems (MEMS) [9] and optical
systems [10].
93
A challenging aspect of estimating the biomechanical whole body vibration models, especially
in the case of using wearable sensors where motion artifacts can be introduced due errors in the
measurement and modelling due sources including the cross and skew axis misalignment of the
IMUs, thermal disturbances, magnetic disturbances, and artefacts due to the accelerometer and
gyroscope bandwidth.
A solution to estimate the states of a system is through the use of an Extended Kalman Filter
(EKF) [215] in the parameter estimator form [220]. Due to its recursive properties, the Kalman
filter and its variants have been applied in applications such as robotics, navigation technologies
and image signal processing [222]. In practice, the parameter estimator configuration of the
EKF for a biomechanical model and parameter estimator with states for any unknown sensor
locations will be unobservable. A solution to such problems, where the appropriate rules are
available, is to incorporate constraints on the estimator, where the Kalman gain is projected to
satisfy an inequality [223, 224]. The study in [16] estimates the biomechanical parameters of a
WBV model in the rectilinear axes with the use of EKF and an adaptive sliding window [217].
A soft approach to the filtering problem is the use of particle filters [209] such methods have
been applied to solve the filtering problem in a wide area of problems like object tracking [225],
localisation and mapping [226], fault detection [227] and image surveying [228]. In the
biomechanical context, particle filters have been applied to solve problems of body limb
tracking [229], seated vibration tracking using a spring mass damper ground reaction force
model [230, 231]. In this chapter we consider the particle filter variant the bootstrap filter [232],
where resampling is applied at each update step.
In this chapter a solution to the estimation problem for a biomechanical whole-body vibration
model is presented. The bootstrap filtering [232] strategy in combination with systematic
resampling [233, 234] was found to produce observations of the biomechanical response to
vibrations with a sound accuracy.
The applications of estimating the response of the human body to vibrations include the
integration into vehicle suspension control and vibration monitoring technologies. In this
chapter the observer models the transmissibility between the IMUs located at the head, left
shoulder, right shoulder, pelvis, sternum and the platform of the parallel robot. In this
configuration, we consider that the process error model of the sensors is unknown, as a result
the estimator will normally be unobservable and the EKF parameter estimator will not suffice
for the scenario investigated due to the filter’s tendency to diverge. This work has made further
improvements on the study in [16] by demonstrating that this strategy is reliable in multiple
degrees of motion and does not require tuning.
94
This chapter is arranged as follows, the background on instantaneous biomechanical model
parameter identification is presented in section II, the overview of the particle filter design is
given in section III, the experimental set up is described in IV, the discussion of the
results is given in section V. and the conclusions are given in section VI.
5.2. Whole Body Vibration Model
The biodynamic model in this section is represented as point to point broken down 2nd order
spring mass damper systems. Both systems are simplified as a 2nd order Newtonian spring mass
damper, the transmissibility transfer function of the system is given in Eq (5-1) [235] where ‘s’
denotes the Laplace transform operator variable, and U(s) and Y(s) are the driving point and
output acceleration or angular velocity signals at each sensor location. In this study we apply
this law to both the rotary and rectilinear components.
𝐻(𝑠) =
𝑌(𝑠)
𝑈(𝑠)=
2𝜁𝜔0𝑠 + 𝜔02
𝑠2 + 2𝜁𝜔0𝑠 + 𝜔02
(5-1)
The state transition equation is given in Eq.(5-2) , where uk is the input of IMU acceleration and
matrix A is given in Eq.(5-3) . denotes the state transition equation describes the relationship
between the inertial state variables and inputs between each step.
𝑓𝑘(𝑥𝑘−1, 𝑢𝑘−1) = A(𝑥𝑘−1)𝑥𝑘−1 + [
1100
] 𝑢𝑘−1
(5-2)
𝐴(𝑥𝑛,𝑘−1) = [
−2 ∙ 𝑥3,k−1 ∙ 𝑥4,k−1 −𝑥4,k−12 0 0
1 0 0 00 0 1 00 0 0 1
]
(5-3)
ℎ𝑘(𝑥𝑘) = [2 ∙ 𝑥3(𝑘) ∙ 𝑥4(𝑘) 𝑥4(𝑘)2 0 0] [
𝑥1(𝑘)
𝑥2(𝑘)𝑥3(𝑘)
𝑥4(𝑘)
]
(5-4)
95
𝑥𝑘 =
{
[
a𝑣𝜁𝜔0
] , 𝑓𝑜𝑟 𝑟𝑒𝑐𝑡𝑖𝑙𝑖𝑛𝑒𝑎𝑟 𝑠𝑡𝑎𝑡𝑒𝑠
[
�̈��̇�𝜁𝜔0
] , 𝑓𝑜𝑟 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑠𝑡𝑎𝑡𝑒𝑠
(5-5)
The observer equation is given in Eq.(5-4). The observer equation is a function of the states and
describes the transform required to compute the measured reference point. States 3 and 4 are
value of the damping ratio and natural frequency respectively. As per Eq. (5-5). States 1 and 2
denote the acceleration and velocity for the transmissibility measurements in the linear
component. When the rotational transmissibility is considered.
Table 5-1 Pseudo code for the particle filter [209].
1
2 for i = 1:Np %Number of samples
3 evaluate equation 7 to compute
4 prediction xk
5 evaluate equation 8 to compute
5 new weights
6 end
7 %Get total
8 𝑇 = ∑ 𝑤𝑘𝑖𝑁𝑝
𝑖=1
9 for
10 Normalise weights 𝑤𝑘𝑖 = 𝑇−1𝑤𝑘
𝑖
11 end
12 %Perform systematic resampling
13 *Xk = systematic resampling(xk)
The input signal from the driving point source is given in Eq.(5-6). For the rotational case the
input unit is angular velocity and in the linear component of motion the input is acceleration.
96
𝑢𝑘 = {[a 0 0 0], 𝑟𝑒𝑐𝑡𝑖𝑙𝑖𝑛𝑒𝑎𝑟 𝑠𝑡𝑎𝑡𝑒𝑠
[0 �̇� 0 0] , 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑠𝑡𝑎𝑡𝑒𝑠
(5-6)
5.3. Particle Filter Strategy
In this a chapter a bootstrap particle filter is implemented. The state transition function of the
plant is given in Eq.(5-7) where 𝑣𝑘−1 is an additive arbitrary noise term which represents the
process error. 𝑓𝑘 denotes the state transition function, 𝑥𝑘 are the state values and 𝑢𝑘 is the input
variable.
𝑥𝑘 = 𝑓𝑘(𝑥𝑘−1, 𝑢𝑘−1) + 𝑣𝑘−1 (5-7)
The measurement or observer equation is given in Eq. (5-8). Where 𝑛𝑘 is the noise term.
𝑦𝑘 = ℎ𝑘(𝑥𝑘) + 𝑛𝑘 (5-8)
The working principle of the generic particle filter involves two key initial steps of particles are
drawn from a density function given in Eq. (5-9) to generate an initial prediction of 𝑥𝑘
considering the sampled particles as the prior state value.
𝑞(𝑥𝑘|𝑥𝑘−1𝑖 , 𝑧𝑘) = 𝑝(𝑥𝑘|𝑥𝑘−1
𝑖 ) (5-9)
Equation (5-10). denotes the simplified weight update equation. The particle weights are
computed using the principle of importance density [232]. That is the weights are computed to
calculate the particles that the observer can fit closer to the measured value 𝑦𝑘 .
𝑤𝑘𝑖 ∝ 𝑤𝑘
𝑖𝑝(𝑦𝑘|𝑥𝑘𝑖 ) (5-10)
Experimentally, the damping ratio and natural frequency of the whole body response to
vibrations will be in the range given in Eq. (5-11). [39]. Considering this, we use a uniform
distribution to narrow the search space of the particle filter when performing the prediction step.
𝜆 = 𝑈(0,1), 𝜔0 = 𝑈(0,20𝜋) (5-11)
The states 1 and 2 are modelled with a gaussian noise as in Eq. (5-12).
𝑥1, 𝑥2 = 𝑁(0, 𝜎2) (5-12)
The pseudo code for the complete working process of the particle filter is presented in Table. 1.
97
In this thesis systematic sampling is utilised every iteration, to redraw the particles that are
degrading. In this study, systematic resampling was experimentally determined to produce
stable results. Systematic resampling has a time complexity of O(Np) [209]. Systematic
sampling consists of the step of initialing a value u drawn from a uniform random distribution as
per Eq. (5-13).
𝑢𝑠~𝑈 [0,
1
𝑁𝑝]
(5-13)
For each particle the value u is computed as in Eq. (5-14).
𝑢𝑖 =
𝑖 − 1
𝑁𝑝+ 𝑢𝑠
(5-14)
The update stage of the systematic resampling consists of selecting the particles are when the
following interval in Eq. (5-15) is satisfied [232].
𝑢𝑖 ∈ [∑𝑤𝑝
𝑗−1
𝑝=1
,∑𝑤𝑝
𝑗
𝑝=1
)
(5-15)
The resampling strategy is given in Table 5-2 as a programmatic representation [236].
Table 5-2 Pseudo code for systematic resampling [236].
1 j = 1
2 sumW=w(j)
3 u = U[0,1]/Np
4 for i = 1..Np
5 while sumW < u
6 j = j + 1
7 sumW = sumW +w(j);
8 end
9 *x(i) = x(j)
10 u = u+1/Np
11 end
98
5.4. Experiment
For obtaining the validation and testing data, 6 degrees of freedom of inertial motion data were
obtained from several seated test subjects on a parallel hexapod robot. The parallel robot
consisted of electromechanical actuators that can generate vibrations of up to 10 HZ with a
displacement of 10mm. For testing and validating the particle filter. Figure 5-1 shows the
location of the IMUs used for the collection of test and validation data sets. The data sets
including the IMU on the seat and the chassis of the parallel robot are considered, the
biomechanical parameters and the WBV response are computed using the data from the IMUs
embedded in the seat-cushion, platform base and the locations of the test subject as shown in
Figure 5-1 including, the head sternum, left shoulder right shoulder and the pelvis. The parallel
robot that is used is shown in Chapter 3 in Figure 3-10 and with the complete experimental
setup with the IMUs configured as shown in Figure 5-1.
A sample of the excitation motion provided at the motion platform is given in Figure 5-2. For
each of the test subjects a swept source in multiple combinations of rotation and linear vibration
Figure 5-1. The configuration of the IMUs.
99
was applied to obtain the evaluation and testing data.
5.5. Results and Discussion
The Summary of the overall tracking performance is given in Table 5-3 and Table 5-4 ,for the
rectilinear and rotary components respectively. The units are expressed as a Mean Squared Error
of between the estimation and the measured reference data set. The cases where the goodness of
fit value is infinite are due to the particle filter failing to track the WBV response to the
vibration source. A suggested reason for the filter failing at the pelvis location is due to the
pelvis being located nearer to the seat the differential between the source is smaller resulting in
a smaller signal to noise ratio. It can be observed that the filter failed at tracking the roll rate at
the head. The shoulders gave the most stable results, with all 6DOF of motion exhibiting stable
tracking performance.
Figure 5-2. a Sample of the excitation motion source measured at the motion platform.
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A sample of the estimated state values is given in Figure 5-3 for the rectilinear
transmissibility between the seat and the sternum. The first and second state are the estimated
velocity and acceleration. The lambda and 𝜔0, curves on Figure 5-3 illustrate the tracking of the
damping ratio and natural frequency.
Sensor Location Goodness of Fit Tracking MSE (rotary
component)
X Y Z
Pelvis 0.0112 ∞ 0.0245
Sternum ∞ 0.0362 0.1811
Right shoulder 0.0187 0.0048 0.0219
Left Shoulder 0.0130 0.0060 3.49e-05
Figure 5-3. A sample of the state space estimation data at the sternum.
Table 5-3. Summary of the tracking goodness of fit performance for each location for the rotary
components of motion.
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Head ∞ 9.8114e-05 0.0058
A sample of the rectilinear acceleration tracking at the head from the seat is given in Figure 5-4.
The results at the head could be observed to be the least consistent out of all locations. This has
been speculated to be due to the higher degree of maneuverability of the head resulting in a
higher probability of motion artefacts being introduced to the signals.
A sample of the angular velocity tracking at the sternum sensor is given in Figure 5-5. The
estimated value in this case lags and overestimates the actual angular velocity response, by
approximately 10 samples and 5 degrees/second.
Figure 5-4. A sample of the acceleration IMU tracking at the head.
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Sensor Location / Axis Goodness of Fit MSE (Tracking rectilinear components)
X Y Z
Pelvis ∞ ∞ ∞
sternum ∞ ∞ 12.1 × 10−3
Right Shoulder 11.5 × 10−3 11.8 × 10−3 4.5 × 10−3
Left Shoulder 2. 8 × 10−3 1.1 × 10−3 18.4 × 10−3
Head 0.6 × 10−3 0.758 × 10−3 53.5 × 10−3
Table 5-4. Summary of the tracking goodness of fit performance for each location for the
rectilinear components of motion.
103
5.6. Conclusion
This chapter presented a particle filtering strategy that has been used for identifying the WBV
biomechanical model and its response to a vibration source. Of note the filter produced
repeatable results for the shoulder sensor locations. Future work may focus on dealing with
uncertainties due to contaminating motion artefacts and the development of an improved
algorithm that performs in the case of a low SNR and an investigation into the particle filter
modes of failure. Such strategies that can be incorporated include the use of motion
classification to identify when the measured signals are mixed with motion artefacts.
Figure 5-5. A sample of the angular velocity IMU tracking at the sternum.
104
6. Whole Body Vibration Response Identification Through
Unsupervised Learning for Sensor Dimension Reduction
This chapter introduces a transmissibility estimation model that estimates the
biomechanical vibration frequency response model to point locations when a
sensor has been removed. The identification of human biomechanical models and
parameters in real time is problem of significant interest and useful for real world
applications, in the field of ergonomic monitoring and automotive suspension
control technologies. This chapter presents a method of biomechanical model
identification based on the principle that the responses at each sensor can be
categorised into multiple classes using density-based clustering analysis. From the
identified clusters a set of polygons are extracted to define the limits to which the
biomechanical model will be bounded. The definition of the boundaries is then
utilized as a constraint by an Extended Kalman Filter state space estimator. The
clusters that are allocated to the sampled inertial data reflect the regions where the
human biomechanical frequency responses to whole body vibrations vary between
the softening and stiffening regions or the condition where vibration exposure is
negligible. The results demonstrate that the estimator presented in this chapter
provides stable performance under the condition when the number of sensors are
reduced.
6.1. Introduction
A challenging aspect of time series clustering is the ability to extract meaningful information
from datasets with contaminating noise, especially in the case of using wearable sensors where
motion artifacts can be introduced due to the fastenings and misalignment of the IMUs. Datasets
that contain time series have been clustered in the literature using model based clustering [237],
feature clustering or directly clustering of the time series datasets [238]. Existing research has
applied Expectation Maximization (EM) [239] clustering to ARMA models [237]. For structural
vibration frequency response datasets, clustering analysis has been undertaken with fuzzy
clustering [240, 241], with the objective of identifying the cluster centers for the identification
of harmonics in structures. In this study, density based clustering is considered due to its ability
in finding clusters within datasets that have attributes, which are non-uniformly distributed, and
105
contain noise [204]. The clustering algorithm used in this chapter is the Density-Based Spatial
Clustering of Applications with Noise (DBSCAN)[203]. DBSCAN searches for clusters based
on the criteria that a set of attributes coincide within a set radius of their neighboring attributes
in the common class. That is, they are reachable within a specified distance parameter within a
defined number of minimum points. Other existing notable density based clustering algorithms
include OPTICS, Generalized DBSCAN and DENCLUE [242].
The damping ratio and natural frequency parameters are estimated using an Extended Kalman
Filter (EKF) [215] in the parameter estimator form [220]. Due to its recursive properties, the
Kalman filter and its variants have been applied in applications such as robotics, navigation
technologies and image signal processing [222]. In practice, the parameter estimator
configuration of the EKF for a biomechanical model and parameter estimator with states for any
unknown sensor locations will be normally numerically unstable. A solution to such problems,
where the appropriate rules are available, is to incorporate constraints on the estimator, where
the Kalman gain is projected to satisfy an inequality [223, 224].
The focus of this chapter is the fusion of information from three domains, including the inertial
measurement, the plant model and a set of inequalities obtained from an offline cluster analysis.
The applications of estimating the response of the human body to vibrations include the
integration into vehicle suspension control and vibration monitoring technologies. In this
chapter the observer model represents the relationship between the state variables and the output
from sensors located in the seat-cushion and the cabin floor of the vehicle.
This section is arranged as follows; the problem definition associated with clustering
biomechanical vibration data are discussed in Section 6.2, the description of the inequality state
space estimator is described in Section 6.3, the discussion of the results is presented in Section
6.4 and the conclusion is given in Section 6.5.
6.2. Identification of clusters
The concept tested in this chapter is that the parameters of the biomechanical human body
model response to whole body vibration are bounded by a set of rules trained by an
unsupervised learning algorithm. The geometry of which is loosely distributed around a curve
resembling an open loop control system locus graph [243], as given in Figure 6-1.
Initially, six degrees of freedom of inertial motion data are obtained from several seated test
subjects on a parallel robot, for the purpose of synthesizing the boundary polygons from the
experimental data. The final step in this chapter is the evaluation of the boundaries on an EKF
parameter estimator for the purpose of validation. Figure 6-2 shows the experimental
106
configuration of the IMUs in each of the two steps, (a) is the subset of the training data to train
the model using a clusterer and (b) indicates the validation stage where the biomechanical
dynamics at the head and torso are to be estimated. For validation, the data sets including the
IMU embedded in the seat-cushion and the chassis of the parallel robot are considered, whilst
for the estimation phase, the biomechanical parameters are computed using the data from the
IMU embedded in the seat-cushion and platform base.
An Autoregressive–Moving-Average Model with Exogenous noise (ARMAX) model [244, 245]
is developed from the inertial time series dataset. The ARMAX parameters are then defined to
be the attributes in the training data set for the clustering algorithm. Figure 6-1 the DBSCAN
cluster assignments to the poles of the ARMAX polynomials with an exogenous noise, on a
complex plane graph, with the clusters identified as noise being omitted. Each attribute
corresponds to a sample point from the training dataset where the ARMAX models are
generated using the inertial motion data measurements taken from the seat to the head of each
test subject. The clusters identify groups of pole locations where the frequency response
dynamics of the biomechanical body model varies or is diverted between the softening and
stiffening regions [5] depending on the composition of the excitation source and the hidden
intra-subject variability.
Figure 6-1. Complex graph of ARMAX model DBSCAN cluster assignments.
107
Figure 6-3 shows a similar scatter plot of the data set used in Figure 6-1 except using the EM
cluster algorithm [239] and using the Weka package [201]. Graphically, the different results in
these two algorithms indicate that density-based clustering identifies the arcs of model pole
locations whereas EM clustering has a greater overlap between these arcs. Both the EM and
DBSCAN clusters identify identical noisy attributes.
Figure 6-2. (a) Analysis for clustering (b) Implementation and testing of the state estimator
108
Figure 6-4 and Figure 6-5 indicate the down sampled scatter plots of the cluster assignments
and the polygons for the real and imaginary plots for the ARMAX model. The real-imaginary
polygons were considered between the measured and unmeasured parameters due to the dataset
having a higher degree of dispersion between the clusters. The attribute on the vertical axis is
the constraint for the measured state of the EKF parameter estimator. The polygon covers the
common region where the two state constraints are coupled. The samples indicated by diagonal
crosses are the noise samples that are not associated with any of the DBSCAN clusters, that is,
they are not reachable by any common linkage as per the criteria of DBSCAN. DBSCAN
clustering labels the data points as either noise or a nominal class.
Figure 6-3. Complex graph of ARMAX model EM cluster assignments
109
Figure 6-4. ARMAX Cluster assigments real imaginary graph, DBSCAN1 and DBSCAN2
Denote the cluster class label
110
Likewise, the results of clustering for the natural frequency and damping ratio are displayed in
Figure 6-7 and Figure 6-8 respectively. Figure 6-9 illustrates a sample of time series
acceleration data, where the solid line is the input excitation, and the markers indicate the class
and value of acceleration.
6.3. Constrained State Estimation
The biomechanical model in this chapter is represented with two main systems, the unmeasured
and the measured systems. Both systems are considered to be a 2nd order Newtonian spring mass
damper, the transmissibility transfer function of the system is given in Eq.(5-1) [235] from
Chapter 5, where ‘s’ is the Laplace transform operator variable, and 𝑈(𝑠) and 𝑌(𝑠) are the
driving point and output acceleration signals.
Figure 6-5. ARMAX Cluster assigments real imaginary graph, DBSCAN1 and DBSCAN2
Denote the cluster class label
111
For the purpose of identifying the EKF parameter estimator matrices [220], the spring-mass-
damper systems are represented as state space models, where the first two states are the inertial
parameters of the systems and the latter two are the damping ratio and natural frequency, as
given in Eq. (6-1), where 𝐿 is the number state transition equations, in the case of this chapter
two, the first equation for the vehicle body to seat cushion, the ‘measured plant’ and the second
is from the vehicle body to the head, the ‘unknown plant’. The sample number is denoted by 𝑘.
𝜉𝑘 = [
𝜉1(𝑘)𝜉2(𝑘)𝜉3(𝑘)𝜉4(𝑘)
] = [
𝑥1𝑥2ζ𝜔
]
(6-1)
Figure 6-6. The measured and unkown plant.
112
Figure 6-7. Sample cluster assignments natural frequency
Figure 6-8. Cluster assignments for damping
113
The state transition equation is given in Eq.
(6-2), where 𝑢𝑘 is the input of IMU acceleration and matrix A is given in Eq.
(6-3). The state transition equation describes the relationship between the state variables and
inputs between each sample.
𝑔𝐿(𝜉𝑘, 𝑢𝑘) = 𝐴𝑛(𝜉𝑘)𝜉𝑘 + [
1000
] 𝑢𝑘
(6-2)
𝐴𝐿(𝜉𝑘) = [
−2 ∙ 𝜉3(𝑘) ∙ 𝜉4(𝑘) −𝜉4(𝑘)2 0 0
1 0 0 00 0 1 00 0 0 1
]
(6-3)
The observer equation is given in Eq.(6-4), where matrix C is given in Eq.(6-5). The observer
equation is also dependent on a function of the states.
ℎ𝐿(𝜉𝑘) = 𝐶(𝜉𝑘) [
𝜉1(𝑘)𝜉2(𝑘)𝜉3(𝑘)𝜉4(𝑘)
]
(6-4)
𝐶(𝜉𝑘) = [2 ∙ 𝜉3(𝑘) ∙ 𝜉4(𝑘) 𝜉4(𝑘)2 0 0] (6-5)
𝑑𝑓(𝜉 ) =
[
𝜉(3)𝜉(4)
𝑅𝑒(−𝜉(3) ∙ 𝜉(4) + 𝜉(4)√1 − 𝜉(3)2)
𝐼𝑚(−𝜉(3) ∙ 𝜉(4) + 𝜉(4)√1 − 𝜉(3)2)
𝑅𝑒(−𝜉(3) ∙ 𝜉(4) − 𝜉(4)√1 − 𝜉(3)2)
𝑅𝑒(−𝜉(3) ∙ 𝜉(4) − 𝜉(4)√1 − 𝜉(3)2)]
(6-6)
114
Figure 6-10 illustrates a schematic overview of this approach. The human seat body model is
evaluated by the EKF as a separate state space model from the cabin floor to the seat-cushion.
Figure 6-10. flow chart of the estimation process.
Figure 6-9 Cluster assignment against a time series of acceleration data
115
From each pair of attributes of the DBSCAN clusters a polygon is extracted. For each polygon
one attribute will correspond to the constraint on the observed model that is the one measured
from the vehicle floor to the cushion. The second attribute will be the constraint for the
unmeasured model from the floor to the head. If the sample is within a noise cluster, then the
estimator is reinitialized.
The method of constraining the state estimates in this chapter is the use of gain projection [223,
224]. At each Kalman gain update step of the EKF, the constraint function is evaluated to
determine if the inequality holds true. If the inequality has not been satisfied, then the EKF gain
update is projected such that the estimated state satisfies the inequality. Equation (6-6) is the
constraint function, which denotes the relation between the states and the constraint boundaries
denoted by 𝑙𝑏 and 𝑢𝑏 as given in Eq. (6-7). To project the Kalman gain, the states are
extrapolated.
After each iteration of the EKF, the state dependent boundary function for each row of the
function 𝑑𝑓(𝜉 ) is evaluated considering the inequality given in Eq. (6-7). Since the boundary
defined in Eq. (6-6) is a function of the EKF states it is required to perform a linearisation to
compute the approximation. The constraint function is linearized by computing the Jacobian
partial derivative matrices where the state values are the inputs as given in Eq. (6-10) [246],
where D is the linearized constraint function and d is the desired constraint value. The state
update equation with the linearized constraints applied is given in Eq. (6-9). The pseudo code
for this operation is given in Table 6-1.
Table 6-1 Pseudo code the constrained estimator
1 if sample resides in a non-noise 2
2 cluster
3 % check if the eq.8 holds true
4 % where the limits are the 5intersections between
6 % the state value and the associated 7polygon
8 for all ‘measured’ states
9 define ub as max point of polygon
10 define lb as min point of polygon
116
𝑙𝑏 < 𝑑𝑓(𝜉 ) < 𝑢𝑏 (6-7)
𝑑𝑓(𝜉𝑘|𝑘) = 𝑑 (6-8)
𝜉𝑘|𝑘 = 𝜉𝑘|𝑘 − 𝐷𝑇(𝐷𝐷𝑇)−1(𝐷𝜉𝑘|𝑘 − 𝑑) (6-9)
11 end
12
13 if Eq.8 is false
14 Evaluate Eq.10
15 where each row of d is the cluster
16 boundary that was exceeded
17
18 else
19 row of d is equated to Eq.7
20 end
21
22 for ‘unmeasured’ states
23 find the intersections between d and
24 the unmeasured axis of each polygon
25 define ub and lb as the maximum and
26 minimum intersection points
27 repeat lines 13 to 20 for the
28 unmeasured attributes
29 end
117
𝐷 = 𝐻𝑘
=𝜕
𝜕𝜉 𝑑𝑓(𝜉 )|
𝜉𝑘|𝑘
(6-10)
Figure 6-11. Transmissibility verification of the constrained EKF, where FRF is the estimated
frequency response function
118
Sensor Location Goodness of Fit MSE (Tracking
rectilinear component)
Head
Seat Cushion 0.2273
6.4. Discussion of results
The results from this study present an application of density-based clustering to achieve
recursive constrained state space model parameter estimation of a biomechanical model.
Through the results obtained in simulation, the averaged transmissibility response of the seat to
head biomechanical system is verified against the time series frequency response function, post
processed using the H1 estimation method [99] and Welch’s power spectra [221]. From Figure
6-11 a comparison can, be made between the estimation from the constrained EKF and the
transmissibility that is directly measured from the actual test data set, the mean estimation
overestimates the peak transmissibility magnitude by 0.5 units by a frequency of 0.2Hz. The
performance of the estimation is considerably less precise than the results reported in [16], with
Figure 6-12, The comparsion of the estimation vs the validation data.
Table 6-2. A summary of the tracking accuracy of the estimator.
119
an online feedback IMU at the head. However, this system does not require a direct
measurement from the IMU located at the head and is demonstrated to perform when sensors
are removed. WBV inertial motion data sets have been observed to have a large noise to
nominal class ratio and contain time series drifts.
6.5. Conclusion
This chapter presented a frequency response function estimator for identifying the WBV
biomechanical model of the human body that has been subjected to inter and intra subject
variabilities in the case that the number of IMUs are removed, with the assistance of density-
based clustering analysis. The key advantages demonstrated with the use of polygons identified
with DBSCAN, are to constrain the EKF parameter estimator. It has been shown that density-
based clustering can identify meaningful modalities of WBV frequency response through
identifying a frequency response function of a seated biomechanical body. This was confirmed
through the implementation of the EKF parameter estimator evaluated on a separate validation
set of biomechanical motion data. Possible limitation of this approach is that the clustering
algorithm must be evaluated with data for the worst, average, and best-case scenarios to be
presented in the training dataset. Without the presence of these training data samples in the
extreme regions it is not possible to identify an associated cluster. Further future work will
include the development of a more effective clustering technique specifically for addressing the
issues associated with clustering time series of inertial motion datasets.
120
7. A State Based Seated Whole-Body Biodynamic Model
Abstract- This chapter presents a representation of whole-body vibration models with a
hierarchical model obtained through unsupervised learning. A state transitional model
representing the jerking and rocking motions between each linkage of the biomechanical
model of the human body is presented. In contrast to the previous chapters, a model that
considers the instantaneous transfer function of the biodynamic model of the human body
response to vibration was considered. In this chapter a model is taken n into account
inputs a sequential process before asserting a decision on the biodynamic model. A state
transition is proposed to represent the whole-body vibration model. The advantages of this
method include the ability to forecast trends and sequential patterns in the whole-body
vibration response. And the model can consider cases where the biodynamic response is
time invariant. The overarching objective is to model sequences of motions and activities
that are not covered by the analytical kinematic methods.
7.1. Introduction
This chapter presents a method of whole-body vibration modelling, that is composed of a
sequence of states that can be inferred through unsupervised learning. In the literature a domain
of statistical clustering analysis algorithms exists and are utilised to identify finite models from
unlabeled datasets. The process of unsupervised learning for clustering is experimented in this
chapter to extract motions that provide information beyond the physical dynamics, for example
the ability to identify and predict a sequence of rocking or jerking motions of a seated subject.
As previously reviewed in in Chapter 6, the EM approach to clustering mixture models requires
tuning of parameters for initialisation. An approach that overcomes this is the use of a class of
finite clustering, where various parameters like the number of mixtures are learnt in an
unsupervised manner. Minimum message length clustering MML [247] is an invariant Bayesian
point clustering method used for statistical model selection where the simplest model is
preferred when the comparison model has a similar goodness of fit. MML clustering is a model
based clustered that has proven results in accurately clustering large volume data sets [248].
More recent extensions of MML clustering have explored the use of integrating the estimation
and model selection step and has demonstrated results of a lower mean clustering error rate but
has a higher standard deviation error that MML clustering [249]. A notable automatic clustering
algorithm is Autoclass [250],[251] where the classes are represented in terms of probability
density functions as well as a set of maximum posteriori probability parameters.
121
Dirichlet process mixture models have been utilized widely for clustering tasks. Such examples
include: Acoustic waveform classification performed with a hierarchical Dirichlet process [252]
and robot trajectory learning[253].
The Dirichlet mixture model Hidden Markov model has the flexibility of nonparametric
learning of systems with an unknown number of classes [254].
In the context that is related to the works of biomechanical modelling. Unsupervised clustering
using GMM has been applied to various topics of research. Such examples include, for the use
of motion activity classification and segmentation [255], dynamic posture analysis [256],
fatigue estimation [257].
For the HMM sequential models have been formulated to solve biomechanical problems model
[258], motion recognition of [255] for fatigue estimation through motion classification [257]. To
the candidate’s knowledge there is not any works on the Dirichlet process GMM modelling for
explicit biodynamic whole-body vibration models.
7.2. Problem Background Design
The solution presented in this chapter is multi-faceted to solve the problem of formulating a
sequential biodynamic whole-body vibration model. The first step is to formulate a clustering
strategy that is suitable for meaningfully categorising the unlabeled and inertial motion data that
can extract a sequence meaningful motions for the biodynamic model.
7.2.1. Mixture Models
The Gaussian mixture models (GMM) are used in this this as a clustering strategy for the
inertial motion data. GMM are a linear superposition of K multivariate Gaussian components
[259]. The Definition of the gaussian mixture model is given in Eq. (7-1), where function N is
the function for the multivariate gaussian distributions. Each component has an independent
covariance and mean denoted by Σk Denotes the covariance and μ respectively.
p(x) = ∑𝜋𝑘𝑁(x|μk, Σk)
𝐾
𝑘=1
(7-1)
In this chapter we use an implementation of MML clustering to train the GMM [249].
In this chapter a mixture constructed using the Dirichlet process. This method is considered for
selecting a finite number of clusters for inertial motion data due to the Dirichlet process being a
multinomial distribution, is tested an alternative to the GMM.
122
Equation (7-2) defines the predictive density of the Dirichlet process [260] Dirichlet process is
given in Eq . (7-2). G0 is the base random measure taken from the Dirichlet distribution [261],
alpha is the scaling parameter of the Dirichlet distribution scaling, ƞ denotes the random
variable drawn from the Dirichlet Process and x is the samples for the mixture model.
𝑝(𝑥|𝑥1, … , 𝑥𝑁 , 𝛼, 𝐺0) = ∫𝑝(𝑥|ƞ)p(ƞ|x1, … , xN, α, G0)dƞ (7-2)
It is aimed to find a solution to (7-2), however there is not a general closed form solution [260].
The Gibbs sampling strategy is utilised to provide the Dirichlet Mixture models [262].
7.2.2. Hidden Markov Model Training
The Hidden Markov model is a probabilistic model that considers the probability of an event is
dependent on the state of the previous event . By constructing an HMM using the mixture
models from the previous section, a state transition matrix can be constructed. An overview of
the main steps to compute the HMM state transition matrices are given as follows [259].
The Equation for the conditional probability of the state transition matrix is given in (7-3),
where A is the state transition probability, and z is the variable representing the current node in
the graph. K is the total number of states.
𝑝(𝑧𝑛|𝑧𝑛−1,𝐴) =∏∏𝐴𝑗𝑘𝑧𝑛−1,𝑧𝑛𝑘
𝐾
𝑗=1
𝐾
𝑘=1
(7-3)
As the HMM requires a state to have a previous state the, a special case for the definition of the
initial conditional probability is given in Eq. (7-4), pi will denote the array of initial
probabilities.
p(z1|π) =∏πkz1k
K
k=1
(7-4)
Equation (7-5) is the expression for the initial array of probabilities, where γ denotes the
posterior distribution of z [259].
𝜋𝑘 =
γ(z1k)
∑ γ(z1j)𝐾𝑗=1
(7-5)
123
Equation (7-6) defines the state transition matrix that is used to provide the probabilities of
transition between each node of the HMM graph. The value of the row wise sum of elements in
A must equal 1. The function ξ denotes the denote the joint posteriori distribution of 𝑧𝑛−1,𝑗 and
𝑧𝑛𝑙.
Ajk =
∑ 𝜉(𝑧𝑛−1,𝑗, 𝑧𝑛𝑘)Nn=2
∑ ∑ 𝜉(𝑧𝑛−1,𝑗, 𝑧𝑛𝑙)𝑁𝑛=2
𝐾𝑙=1
(7-6)
7.2.3. Biomechanical Modelling
Initially, the angular and rectilinear jerk of motion at the locations of the seat cushion sternum
and head are computed using numerical differentiation of the angular velocity and acceleration
data. This is used for the validation of this model. Figure 7-1 shows a sample of the computed
jerk from the IMUs on the parallel robot and the head.
The inertial motion data including the angular velocity and accelerations of the IMUs at the
head sternum and seat location are sampled for the training data of the mixture models and the
HMM. The results of the clustering analysis using the Dirichlet process mixture model is given
Figure 7-1 A sample of the jerk measurement ground truth dataset.
124
in Figure 7-2 with the distribution kernels shown plot along the horizontal and vertical axes. By
cross referencing the cluster assignments, the result from the jerk motion, a ground truth
reference is provided for evaluating the accuracy of the model inference. The plot in Figure 7-2
shows a sample of the acceleration at the seat and the head to illustrate the probability
distributions of each class. Figure 7-3 shows the times series plot of the corresponding cluster
assignments on the training dataset. The cluster assignments successfully identify the phases of
kinematic motion, that is it shows the states when the head is in phase with the seat and when
they are out of phase.
Figure 7-2 the assigned mixture models.
125
Figure 7-3 A sample of the acceleration waveform with the superimposed cluster inference.
126
7.1. Results and Discussion
This section presents the sign of a model that that responds to the human bodies’ responses of
vibrations in a sequential manner. Using the HMM the state transition matrices are computed
using the full training set described in chapter 3. The probabilities of the state transition are
given in Table 7-1 computed using the EM algorithm to solve Equations (7-5) and (7-6) [265].
Figure 7-5 shows the constructed HMM directed graph and the state transitions. The node of the
graph is labelled by their respective cluster assignment as listed in Figure 7-2. The class names
are abbreviated to preserve the brevity of the figure. Referring to the direction of the jerking
motion, PS denotes positive seat, NS denotes negative seat, PH denotes positive head and NH
denotes negative head. The idle state is the condition when there is a negligible vibrational
excitation source. The edges of the directed graph are labelled to show the probability of the
transition to the following state.
The results of the inference accuracy on the evaluation data set are given in Table 7-2. The main
outcome observed was that the multinomial Dirichlet process had a higher mixture model
inference accuracy than the continuous gaussian distribution. This experiment shows that the
Figure 7-4 a sample of the Time Series angular velocity with the superimposed cluster
assignments
127
Dirichlet mixture model provides a strong goodness of fit solution to whole body vibration
model when predicting sequential motion patterns.
𝐴𝑖1 𝐴𝑖2 𝐴𝑖3 𝐴𝑖3 𝐴𝑖5 𝐴𝑖6 𝐴𝑖7
𝐴1𝑗 10.018 11.668 2.619 2.720 15.264 3.203 54.507
𝐴2𝑗 7.7249 5.482 0.407 2.300 45.207 9.782 29.098
𝐴3𝑗 1.475 0.218 92.387
7.83 ×10−5 2.688 2.550 0.682
𝐴4𝑗 0.419 1.743
21 ×10−6 93.399 0.823 1.265 2.350
𝐴5𝑗 10.368 52.105 0.151 0.209 27.114 3.241 6.811
𝐴6𝑗 0.583 41.702 2.209 2.998 6.044 10.853 35.610
𝐴7𝑗 6.434 15.490 0.688 3.738 38.176 3.927 31.547
Figure 7-5 the trained HMM for the sequence of seat to head states.
Table 7-1 State edge probabilities.
128
Model Evaluation Accuracy
(Percent %)
GMM 73.79
Dirchlet MM 78.83
7.2. Conclusion
This chapter introduced a concept of a sequential model that is proven to accurately identify the
categorical phases of motion under seated whole-body vibration. The results show that the
Dirichlet mixture model can infer at a 78.83% accuracy. Furthermore, the model provides a
novel solution to forecasting the likelihood of patterns of motions of test subjects under whole
body vibration. Further work could be done to extend the HMM model to include motor
movements such as arm reflexes, the arm sensors were excluded from this study due to the
increased complexity in finding a solution.
The practical use of the result from this study is the ability to use the HMM to forecast the lag
time between the seat and the head. The ability to predict the lag time provides the possibility to
predict the damping and natural frequency of the human body.
Table 7-2 Summary of the inference results of each models on the evalutation data set
129
8. Artificial Neural Network Approach to Cross Coupled Whole-
Body Vibration Modelling
Abstract- In this section a model is presented using a residual neural network
spectrogram regression model. The hypothesis tested is that the model will be able to learn
the unknown variables of a biodynamic system that are extremely difficult to model
analytically. The strategy presented in this chapter consists of a Residual Neural Network
that has been trained for spectrogram regression of biodynamic data. Which consists of a
multiple image input layer of the short time Fourier transform of the inertial motion data.
The network is trained to make forecasts of biomechanical model parameters based on the
patterns in the spectrogram of the input inertial motion sensors. The model is bench
marked against a linear transfer function as a litmus test and a convolutional neural
network. The model is a proposed as a new approach to solve the problem of modelling
cross coupling of the transmissions of whole-body vibrations and provides a precedent for
applying deep learning to induce a new biomechanical model of an unknown system.
8.1. Problem Background
The effects of cross coupling in biomechanical whole-body vibration analysis have been
introduced in Chapter 2 of this thesis. The problem this chapter addresses is the viable
estimation of a biomechanical model using a black box artificial neural network approach.
Deep learning [266] has been widely applied to solve problems in the area of image signal
processing [267]. Deep learning is a sub field of machine learning that incorporates algorithms
in machine learning that often uses artificial neural networks to aim to learn in multiple layers.
The layers correspond to different concepts and abstraction [268]. Residual neural networks
[269] were introduced the goal of solving image classification problems without over fitting,
that is the increase in layers translates to an increase in the gain between the testing and training
accuracy of the model. Deep residual neural networks have been applied to both classification
and regression problems [270].
Biomechanical systems are analogous to acoustic systems which have seen a wide application
of deep learning in the for classification and regression tasks. For example, to date it has been
reported that the ResNN provides results in spectrogram classification. An example of such
includes the use of a Residual Neural network to classify bird calls [271], acoustic range
estimation [272], acoustic activity recognition [273], audio spectrogram classification [274] and
large scale CNN audio classification [275] and Surface estimation [276].
130
In the field of seated whole-body vibration modelling, as previously described in Chapter 2,
related works on using artificial neural networks include the use of multilayer perceptron [144,
277, 278]. The task that these models are used for is to forecast the time series inertial motion at
various locations of the human body whereas the problem solved in this chapter is the prediction
of model parameters at various locations by passing the input of the spectrogram response at an
input location.
8.2. ANN Whole-Body Vibration Model Design
This section describes the design process of the ANN used in this chapter. Through the literature
survey recent works have proven results of the accuracy of deep residual neural networks for
classifying spectrogram patterns. As discussed in the preceding section it is tested in thus
chapter that the similar structured neural networks used for classification of spectrograms of
acoustic and visual images will perform for biodynamic modelling tasks. This is due to the
analogy taken between nature of acoustic and visual systems with biomechanical systems
express multiple modes, interference and share much of the similar physical phenomena.
Figure 8-1 shows the high-level block diagram of the model used. For both training and
inference, initially the inertial motion data is processed to get the spectrogram using a STFT.
From each axis of motion, a 32 by 32 matrix is constructed and then passed to the input layer of
the CNN or residual neural network. Both architectures are considered and are trained using the
identical training data set.
8.2.1. Spectrogram Construction and Preprocessing
Initially the training dataset was prepared for using the logarithmic magnitude STFT with 32
points and a sliding Hanning window at 60 Hz [105], these parameters where selected of this
size mainly as a balance between complexity and resolution. A sample of the training data
(presented in a higher resolution for improved readability) is given in Figure 8-2. The STFT is
taken for the 3 exes of each IMU located on the seat cushion and the sternum to provide the
training data. A total of 2046 spectrograms were used for the training of the neural networks.
Figure 8-1 The high level Signal archetcure of the model for biodynamic parameter prediction
131
The output layer of each model is a logistic regression layer that is trained against the ground
truth time invariant biomechanical parameters of a 1DOF and 2DOF model [279].
Figure 8-2. An example of STFT Surface plot for data sampled from the chest connected IMU.
8.2.2. Deep Neural Network Architecture
For this study both Residual neural network and CNN are considered for a comparison. The
structure of both networks used in this study are given in Figure 8-3. Each network consists of a
dropout layer and a regression layer connected to the output.
Both networks consist of a dropout layer [280]. The dropout layer is used to set connections
between layers to zero at random. The desired goal of the dropout layer is to add a perturbation
to the system to avoid the training process from stalling on localised minimums. Both the CNN
and Residual Neural network had the rectified linear unit selected as the activation function
[281].
The convolution unit for the residual neural network is given in Figure 8-4. The main concept of
the of the convolution unit of a residual neural network, is to use staggered units with skip
convolution layers, as per b) in Figure 8-4 and other consist of the feedforward loop with no
convolution layer.
132
The network structure for the CNN convolutional unit is given in Figure 8-5. As per Figure 8-4,
Figure 8-5 both networks utilise convolutions layers which involve the application of a matrix
of running convolutional filters to the input, with a multiplication by a weight and an addition of
a bias term [282].
Batch normalisation layers are applied to both convolutional units to improve the speed and
stability during the training process [283]. The batch normalisation layer normalizes the inputs
to this layer by shifting and rescaling such that the mean and variance of the inputs are constant.
8.2.1. Network Training
The parameters used for the training of both networks used are given in Table 8-1. In order to
provide a valid comparison both networks used the same number of convolutional units.
Through review of the literature and by trial and error with inertial motion data the training
parameters were selected.
Parameter Value
Dropout Rate 0.1
Dropout Frequency 20
Initial learning rate 0.005
Number of Convolutional Units 48
Training method stochastic gradient descent with
momentum
Activation Function Rectified Linear Unit
In this section the ResNN and CNN models a 2DOF and a 1DOF whole body vibration model,
by estimating the damping ratios and natural frequencies of the lumped elements. The equations
for the 2DOF whole body vibration model are given in Chapter 2 from Eq. (2-39) to Eq. (2-43).
For the ground truth a recursive ARMX model was used to estimate the ideal natural frequency
and damping ratio values using postprocessing [245].
Table 8-1. The training parameters used for both architectures
133
Figure 8-3 the overall structure of the artifical neural network model.
134
Figure 8-4. The convolution units of each artifical nueral network structure tested in this
chapter.
Figure 8-5 the Convolutional Unit used for the CNN.
135
8.3. Results and Discussion
The results of the RMSE and the SD performance of the testing data set are given in Table 8-2
and. Where the minimum errors are highlighted in bold. As a benchmark to prove the results are
valid, a transfer function for a 1DOF and 2DOF model was compared that had been estimated
from the same training data set using the Instrumental Variable Method [284]. The residual
neural network training error curve is shown in Figure 8-6, which demonstrates the validation
error rate is trending with the training data error rate. The network loss curve is shown in Figure
8-7. The network loss used in this case if the half mean squared.
Overall, the results demonstrate that both structures have very similar performances, the residual
neural network performs higher than the CNN at predicting damping ratio whereas the CNN has
slightly outperformed for the prediction of natural frequency. Both models outperform the linear
transfer function. This indicates the model is fitting to a higher order and considering hidden
variables like cross coupling and intra subject variabilities that are difficult to model using
analytical methods.
Figure 8-6 Whole body vibration model Residual nueral network training.
136
Figure 8-7 Whole Body vibration residual network loss.
137
Residual
CNN
CNN Linear
Transfer
Function
Validation
MSE
Validation
RMSE
Validation
MSE
Validation
RMSE
Validation
MSE
Validation
RMSE
1DOF Damping
Ratio
0.1207 0.4057 0.2223 0.4714 1.4995 1.2245
1DOF Natural
frequency
0.3092 0.5561 0.1862 0.4315 1.3362 1.1559
2DOF Damping
Ratio
0.1980 0.4446 0.2277 0.4772 1.4995 1.2245
2DOF Natural
frequency
0.1946 0.4411 0.1644 0.4053 0.9329 0.9418
Table 8-2 A summary of the various error rates per archetecture. Minimum values are
highlighted in bold.
138
Residual CNN CNN
Validation SD Validation SD
1DOF Damping
Ratio
0.8517 0.2101
1DOF Natural
frequency
0.6866 0.4034
2DOF Damping
Ratio
0.4039 0.1973
2DOF Natural
frequency
0.1916 0.3713
8.4. Conclusion
This chapter has explored a new concept of the introduction of deep biodynamic whole-body
vibration models. The results demonstrate that the Neural network produces viable results when
provided a spectrogram and has the advantages of being able to be trained given experimentally
recorded data. It is concluded that the difference in performance between the residual and
standard CNN architectures are negligible for the class of modelling performed in this
experiment.
Table 8-3 Comparison of the standard deviations of each model
139
9. Concluding Discussions
This thesis has explored a set of new solutions to existing problems in whole-body vibration
modelling considering the recent advances in technology allow us to readily sample rich
datasets from an array of sensors. The benefits provided by the contributions of the work in this
thesis spans across multiple disciplines including ergonomics, vehicular control system design
and workplace health and safety. The overarching contribution of the work is in the proposal of
biodynamic models capable of estimating the model parameters in real time for the use in
automotive control systems and vehicle suspension monitoring systems. Currently the
overwhelming majority of works in the literature, are focused on the post analysis and
modelling
9.1. Conclusions on the Real Time Estimation of Biodynamic Models
In Chapters 4, 5 and 6 a hybrid approaches were proposed to solve the whole-body vibration
model parameter estimator filtering problem. Each approach provides a tool to solve the
problem each with its own costs and benefits.
Chapter 4 presented a biodynamic model based on an EKF with drift detection, this model has a
low order of complexity and if able to estimate the biodynamic response and parameters
instantaneously. The limitation is that only the 1DOF model produced stable results, and it
becomes increasingly difficult to achieve stability under a low signal to noise ratio.
Chapter 5 presented an approximation strategy to solve the biodynamic model using a
bootstrapping particle filter, the benefit of this method is that it can solve the same problem as in
Chapter 4 except in the omnidirectional case and performs under low signal to noise ratios,
However the computational order of complexity of the chapter 4 is the order of n3 where the
particle filer will be at least a cubic scaled by the particle parameters.
Chapter 6 presented a hybrid algorithm combining unsupervised learning to approximate the
reduction of sensors using a cluster analysis. All these methods have their respective balances in
complexity and performance.
9.2. Conclusions on the use of Machine Learnt Models
Chapter 7 and Chapter 8 define biodynamic models that are machine learnt. The advantages
proven in this chapter are they can fit models to hidden intra subject variabilities that are not
140
captured by the analytical methods.
Chapter 7 presents HMM based model that predicts motions such as rocking and jerk motions.
The contribution is that this is the first whole body vibration model based on an HMM
approach. The practical contributions of this method presented is the ability to predict the
likelihood of unsafe motions and the ability to predict the lagging time of the whole-body
vibration response.
Chapter 8 presents a deep biodynamic model that is trained from experimental data to learn a
1DOF and 2DOF model. The contribution is that this method can learn the hidden parameters
that are difficult to model without overfitting, like cross coupling and higher DOF motions.
9.3. Future Work Recommendation
The state space estimation of a biodynamic model is a challenging problem to solve. Future
work could include the proposal of a unified approach could be considered extending or creating
an ensemble of estimators from the works from Chapter 4, 5 and 6.
On the topic of the machine learnt biodynamic models there is potential for much work on
integrating the machine learnt models into a vehicular control system to validate the practical
usefulness of the work. For the work in chapter 8, the work on deep biodynamic models could
be improved through performance optimisation in time and space complexity to justify the case
for replacing the analytical methods in vehicle seating suspension control systems.
141
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11. Appendices
11.1. Appendix 1
11.1.1. Human Ethics Research Council Approval Letters
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