Feasibility Study of Human Lumbar Motion using Variability Techniques 
 
by 
 
Yuvraj M. Pawar 
 
 
 
 
A thesis submitted to the Graduate Faculty of 
Auburn University 
in partial fulfillment of the 
requirements for the Degree of 
Master of Science 
 
Auburn, Alabama 
 May 9, 2011 
 
 
 
 
 
 
Copyright 2011 by Yuvraj M. Pawar 
 
 
Approved by 
 
P. K Raju, Chair, Thomas Walter Professor of Mechanical Engineering 
 Maruti Ram Gudavalli, Associate Professor, Palmer Center for Chiropractic research, IA 
Dan Marghitu, Professor of Mechanical Engineering 
 
 
 
 
 
 
ii 
 
 
 
Abstract 
 
 
 More than 80 percent of people suffer from low back pain at some point in their 
lifetime. It is one of the most prevalent musculoskeletal disorders in the United States. 
The pain can be short-lived or long-lasting. In either case, low back pain can make many 
daily activities difficult to perform. Even though it is one of the most studied disorders, 
the etiology of low back pain is still unknown. Kinematics and kinetics of body 
movements can be affected by low back pain and may result in spinal instability. 
Previous studies have quantified this spinal stability using linear and nonlinear motion 
analysis techniques. The purpose of this study was to obtain the variation in kinematics of 
healthy volunteer subjects using these two techniques while the volunteer subjects are 
performing the lumbar movement during flexion-extension while sitting on an unstable 
surface. In addition, a second objective was to study the variation on a cadaveric 
specimen during flexion and extension motions. 
Five healthy volunteers performed repetitive trunk flexion and extension 
movements while sitting on an unstable surface with different starting positions (forward, 
neutral, and backward) and two different movement initiation (forward and backward) 
directions. The cadaveric specimen was subjected to flexion-extension movements at 
different speeds. Traditional linear techniques such as mean, standard deviation, and 
coefficient of variation, along with nonlinear technique such as approximate entropy, 
were calculated from measured trunk kinematics to estimate the variability of the human 
motions.  
 iii 
  The average values for the motions had a range from 10.556 degrees to 26.101 
degrees. The standard deviation values had a range from 1.941 to 3.278. The coefficients 
of variation had a range from 7.258 to 26.597. The Approximate Entropy values had a 
range from 0.209 to 0.274. 
To the best of the authors? knowledge this is the first study to determine the 
variability of sitting motions on an unstable surface using approximate entropy. Linear 
and nonlinear dynamic systems analyses were successfully applied to trunk flexion-
extension data, which can serve as control data for further studies and understanding 
kinematics and kinetics of low back pain patients.    
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 iv 
 
 
 
 
Acknowledgments 
 
 
I would like to express my sincere gratitude to my research advisors, Dr. P. K. 
Raju and Dr. Maruti Ram Gudavalli, for their invaluable guidance and help during the 
course of this study. I would like to specially thank Dr. Dan Marghitu for helping me 
with nonlinear techniques and being part of my committee. 
I would like to acknowledge and extend my gratitude for the financial support 
received from Palmer Center for Chiropractic Research in Davenport, Iowa. I would like 
to thank my research group at Palmer Center for useful discussions and enjoyable 
moments in the lab. Special thanks to Dr. Ting Xia for his excellent technical assistance. 
This investigation was conducted in a facility constructed with support from Research 
Facilities Improvement Program Grant Number C06 RR15433-01 from the National 
Center for Research Resources, National Institute for Health. 
My research assistantship during my studies supported by Award Number 
U19AT004137 from the National Center for Complementary & Alternative Medicine, 
National Institute of Health, is highly appreciated. 
I would like to thank my family for unconditional love and support throughout my 
graduate studies. Finally, I am thankful to all my friends and colleagues for their 
unconditional friendship and support. 
 
 
 v 
 
 
 
Table of Contents 
 
 
Abstract .......................................................................................................................... ii 
Acknowledgments .......................................................................................................... iv 
List of Tables.................................................................................................................vii 
List of Figures ............................................................................................................. viii 
List of Abbreviations ..................................................................................................... xv 
1. Introduction ................................................................................................................. 1 
1.1 Background ............................................................................................................ 1 
1.2 Traditional Linear Methods to Quantify Variability ................................................ 4 
1.3 Nonlinear Methods to Quantify Variability ............................................................. 8 
1.4 Specific Objective ................................................................................................ 13 
2. Anatomy of Lumbar Spine ......................................................................................... 14 
2.1 Vertebral Column ................................................................................................. 14 
2.2 Vertebrae ............................................................................................................. 17 
3. Tools and Techniques ................................................................................................ 19 
3.1 State of System .................................................................................................... 19 
3.2 Time Series .......................................................................................................... 19 
3.3 State Space ........................................................................................................... 20 
3.4 Phase Space.......................................................................................................... 20 
3.5 Noise in Biomechanical Data ............................................................................... 33 
3.6 Filters ................................................................................................................... 33 
3.7 Sample Size ......................................................................................................... 40 
4. Methodology ............................................................................................................. 41 
 vi 
4.1 Instrumentations for Tests on Human Subjects ..................................................... 41 
4.2 Experimental Set-up ............................................................................................. 43 
4.3 Human Subjects ................................................................................................... 44 
4.4 Data Acquisition .................................................................................................. 44 
4.5 Data Analysis ....................................................................................................... 47 
4.6 Calculation of ApEn ............................................................................................. 51 
4.7 Instrumentation for Test on Cadaver Spine ........................................................... 52 
4.8 Potting the Specimen ............................................................................................ 56 
4.9 Digitization .......................................................................................................... 56 
4.10 Data Acquisition................................................................................................. 57 
4.11 Data Analysis ..................................................................................................... 58 
5. Results ....................................................................................................................... 60 
5.1 Human Volunteer Study ....................................................................................... 60 
5.2 Cadaver Study .................................................................................................... 115 
6. Discussions .............................................................................................................. 133 
7. Conclusions ............................................................................................................. 138 
References ................................................................................................................... 141 
Appendices .................................................................................................................. 148 
A. Technical Specifications ......................................................................................... 148 
B. MATLAB Programs ............................................................................................... 153 
C. Documentation ........................................................................................................ 157 
 
 
 
 
 
 
 vii 
 
 
 
 
List of Tables 
 
 
Table 3. 1 Linear and Nonlinear Measures ..................................................................... 30 
Table 4. 1 Subtests ......................................................................................................... 47 
Table 5. 1 Linear and Nonlinear Measures ..................................................................... 72 
Table 5. 2 Linear and Nonlinear Measures ..................................................................... 81 
Table 5. 3 Linear and Nonlinear Measures ..................................................................... 89 
Table 5. 4 Linear and Nonlinear Measures ..................................................................... 98 
Table 5. 5 Linear and Nonlinear Measures ................................................................... 106 
Table 5. 6 Linear and Nonlinear Measures ................................................................... 125 
Table A. 1 Technical Specification of Polhemus Unit .................................................. 148 
Table A. 2 Technical Specifications of Optotrak 3020 Unit ......................................... 150 
Table A. 3 Technical Specifications of System Control Unit ........................................ 151 
 
 
 
 
 
 
 
 
 
 viii 
 
 
List of Figures 
 
 
Figure 2. 1 Normal Disc ................................................................................................ 15 
Figure 2. 2 Human Lumbar Spine .................................................................................. 16 
Figure 2. 3 Vertebra ....................................................................................................... 17 
Figure 2. 4  Ligaments of Spine ..................................................................................... 18 
Figure 3. 1 Sine Wave ................................................................................................... 21 
Figure 3. 2 Phase Plane Plot........................................................................................... 21 
Figure 3. 3 3-D Phase Plane Plot .................................................................................... 22 
Figure 3. 4 Random Data ............................................................................................... 22 
Figure 3. 5 Phase Plane Plot........................................................................................... 23 
Figure 3. 6 3-D Phase Plane Plot .................................................................................... 23 
Figure 3. 7 Foam Strip ................................................................................................... 24 
Figure 3. 8 Displacement, Velocity, Acceleration Diagram ............................................ 25 
Figure 3. 9 Phase Plane Plot........................................................................................... 26 
Figure 3. 10 3-D Phase Plane Plot .................................................................................. 26 
Figure 3. 11 Displacement, Velocity, Acceleration Diagram .......................................... 27 
Figure 3. 12 Phase Plane Plot ......................................................................................... 27 
Figure 3. 13 3-D Phase Plane Plot .................................................................................. 28 
Figure 3. 14 Displacement, Velocity, Acceleration Diagram .......................................... 28 
Figure 3. 15 Phase Plane Plot ......................................................................................... 29 
Figure 3. 16 3-D Phase Plane Diagram .......................................................................... 29 
 ix 
Figure 3. 17 Mean for Foam Test ................................................................................... 31 
Figure 3. 18 Standard Deviation for Foam Test .............................................................. 31 
Figure 3. 19 Coefficient of Variation for Foam Test....................................................... 32 
Figure 3. 20 Approximate Entropy for Foam Test .......................................................... 32 
Figure 3. 21 Step Response ............................................................................................ 35 
Figure 3. 22 Frequency Response .................................................................................. 36 
Figure 3. 23 Frequency Response .................................................................................. 37 
Figure 3. 24 Filter Frequency Response ......................................................................... 38 
Figure 3. 25 Acceleration using Accelerometer and Filter .............................................. 39 
Figure 4. 1 Polhemus Unit ............................................................................................. 42 
Figure 4. 2 Subject in Test Position................................................................................ 46 
Figure 4. 3 Data Export.................................................................................................. 48 
Figure 4. 4 Optotrak SCU and Stober Units with Markers on Sensors ............................ 53 
Figure 4. 5 Optotrak 3020 Unit ...................................................................................... 54 
Figure 4. 6 Schematic Diagram of the Set-up ................................................................. 55 
Figure 4. 7 Cadaver Test Set-up and Stylus with Markers .............................................. 56 
Figure 5. 1 Displacement Time Series with Velocity and Acceleration .......................... 61 
Figure 5. 2 Phase Plane Plot........................................................................................... 61 
Figure 5. 3 3-D Phase Plane Plot .................................................................................... 62 
Figure 5. 4 Displacement Time Series with Velocity and Acceleration .......................... 62 
Figure 5. 5 Phase Plane Plot........................................................................................... 63 
Figure 5. 6 3-D Phase Plane Plot .................................................................................... 63 
Figure 5. 7 Displacement Time Series with Velocity and Acceleration .......................... 64 
 x 
Figure 5. 8 Phase Plane Plot........................................................................................... 64 
Figure 5. 9 3-D Phase Plane Plot .................................................................................... 65 
Figure 5. 10 Displacement Time Series with Velocity and Acceleration ......................... 65 
Figure 5. 11 Phase Plane Plot ......................................................................................... 66 
Figure 5. 12 3-D Phase Plane Plot .................................................................................. 66 
Figure 5. 13 Displacement Time Series with Velocity and Acceleration ......................... 67 
Figure 5. 14 Phase Plane Plot ......................................................................................... 67 
Figure 5. 15 3-D Phase Plane Plot .................................................................................. 68 
Figure 5. 16 Displacement Time Series with Velocity and Acceleration ......................... 68 
Figure 5. 17 Phase Plane Plot ......................................................................................... 69 
Figure 5. 18 3-D Phase Plane Plot .................................................................................. 69 
Figure 5. 19 Displacement Time Series with Velocity and Acceleration ......................... 70 
Figure 5. 20 Phase Plane Plot ......................................................................................... 70 
Figure 5. 21 3-D Phase Plane Plot .................................................................................. 71 
Figure 5. 22 Mean Values for Subtests ........................................................................... 75 
Figure 5. 23 Standard Deviation for Subtests ................................................................. 75 
Figure 5. 24 Coefficient of Variation for Subtests .......................................................... 76 
Figure 5. 25 Approximate Entropy for Subtests ............................................................. 76 
Figure 5. 26 Variations in Angular Velocity................................................................... 77 
Figure 5. 27 Variations in Angular Velocity................................................................... 78 
Figure 5. 28 Variations in Angular Velocity................................................................... 78 
Figure 5. 29 Variations in Angular Velocity................................................................... 79 
Figure 5. 30 Variations in Angular velocity ................................................................... 79 
 xi 
Figure 5. 31 Variations in Angular Velocity................................................................... 80 
Figure 5. 32 Variations in Angular Velocity................................................................... 80 
Figure 5. 33 Mean Values for Subtests ........................................................................... 84 
Figure 5. 34 Standard Deviation for Subtests ................................................................. 84 
Figure 5. 35 Coefficient of Variation for Subtests .......................................................... 85 
Figure 5. 36 Approximate Entropy for Subtests ............................................................. 85 
Figure 5. 37 Variations in Angular Velocity................................................................... 86 
Figure 5. 38 Variations in Angular Velocity................................................................... 86 
Figure 5. 39 Variations in Angular Velocity................................................................... 87 
Figure 5. 40 Variations in Angular Velocity................................................................... 87 
Figure 5. 41 Variations in Angular Velocity................................................................... 88 
Figure 5. 42 Variations in Angular Velocity................................................................... 88 
Figure 5. 43 Variations in Angular Velocity................................................................... 89 
Figure 5. 44 Mean Values for Subtests ........................................................................... 92 
Figure 5. 45 Standard Deviation for Subtests ................................................................. 93 
Figure 5. 46 Coefficient of Variation for Subtests .......................................................... 93 
Figure 5. 47 Approximate Entropy for Subtests ............................................................. 94 
Figure 5. 48 Variations in Angular Velocity................................................................... 94 
Figure 5. 49 Variations in Angular Velocity................................................................... 95 
Figure 5. 50 Variations in Angular Velocity................................................................... 95 
Figure 5. 51 Variations in Angular Velocity................................................................... 96 
Figure 5. 52 Variations in Angular Velocity................................................................... 96 
Figure 5. 53 Variations in Angular Velocity................................................................... 97 
 xii 
Figure 5. 54 Variations in Angular Velocity................................................................... 97 
Figure 5. 55 Mean Values for Subtests ......................................................................... 101 
Figure 5. 56 Standard Deviation for Subtests ............................................................... 101 
Figure 5. 57 Coefficient of Variation for Subtests ........................................................ 102 
Figure 5. 58 Approximate Entropy for Subtests ........................................................... 102 
Figure 5. 59 Variations in Angular Velocity................................................................. 103 
Figure 5. 60 Variations in Angular Velocity................................................................. 103 
Figure 5. 61 Variations in Angular Velocity................................................................. 104 
Figure 5. 62 Variations in Angular Velocity................................................................. 104 
Figure 5. 63 Variations in Angular Velocity................................................................. 105 
Figure 5. 64 Variations in Angular Velocity................................................................. 105 
Figure 5. 65 Variations in Angular Velocity................................................................. 106 
Figure 5. 66 Mean Values for Subtests ......................................................................... 109 
Figure 5. 67 Standard Deviation for Subtests ............................................................... 110 
Figure 5. 68 Coefficient of Variation for Subtests ........................................................ 110 
Figure 5. 69 Approximate Entropy for Subtests ........................................................... 111 
Figure 5. 70 Variations in Angular Velocity................................................................. 111 
Figure 5. 71 Variations in Angular Velocity................................................................. 112 
Figure 5. 72 Variations in Angular Velocity................................................................. 112 
Figure 5. 73 Variations in Angular Velocity................................................................. 113 
Figure 5. 74 Variations in Angular Velocity................................................................. 113 
Figure 5. 75 Variations in Angular Velocity................................................................. 114 
Figure 5. 76 Variations in Angular Velocity................................................................. 114 
 xiii 
Figure 5. 77 Displacement Time Series with Velocity and Acceleration ....................... 116 
Figure 5. 78 Phase Plane Plot ....................................................................................... 116 
Figure 5. 79 3D Phase Plane Plot ................................................................................. 117 
Figure 5. 80 Displacement Time Series with Velocity and Acceleration ....................... 117 
Figure 5. 81 Phase Plane Plot ....................................................................................... 118 
Figure 5. 82 3D Phase Plane Plot ................................................................................. 118 
Figure 5. 83 Displacement Time Series with Velocity and Acceleration ....................... 119 
Figure 5. 84 Phase Plane Plot ....................................................................................... 119 
Figure 5. 85 3-D phase Plane Plot ................................................................................ 120 
Figure 5. 86 Displacement Time Series with Velocity and Acceleration ....................... 120 
Figure 5. 87 Phase Plane Plot ....................................................................................... 121 
Figure 5. 88 3-D Phase Plane Plot ................................................................................ 121 
Figure 5. 89 Displacement Time Series with Velocity and Acceleration ....................... 122 
Figure 5. 90 Phase Plane Plot ....................................................................................... 122 
Figure 5. 91 3-D Phase Plane Plot ................................................................................ 123 
Figure 5. 92 Displacement Time Series with Velocity and Acceleration ....................... 123 
Figure 5. 93 Phase Plane Plot ....................................................................................... 124 
Figure 5. 94 3-D Phase Plane Plot ................................................................................ 124 
Figure 5. 95 Mean for Tests ......................................................................................... 127 
Figure 5. 96 Standard Deviation for Tests .................................................................... 127 
Figure 5. 97 Coefficient of Variation for Tests ............................................................. 128 
Figure 5. 98 Approximate Entropy for Tests ................................................................ 128 
Figure 5. 99 Variations in Velocity .............................................................................. 129 
 xiv 
Figure 5. 100 Variations in Velocity ............................................................................ 129 
Figure 5. 101 Variations in Velocity ............................................................................ 130 
Figure 5. 102 Variations in Velocity ............................................................................ 130 
Figure 5. 103 Variations in Velocity ............................................................................ 131 
Figure 5. 104 Variations in Velocity ............................................................................ 131 
Figure A. 1 Liberty Polhemus Unit .............................................................................. 148 
Figure A. 2 Magnetic Source Diagram ......................................................................... 149 
Figure A. 3 Sensor Diagram ........................................................................................ 150 
Figure A. 4 Optotrak Data Acquisition Unit II ............................................................. 152 
 
 
 
 
 
 
 
 
 
 
 
 xv 
 
 
List of Abbreviations 
 
 
ApEn Approximate Entropy 
CV Coefficient of Variance 
SD Standard Deviation 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1 
 
 
1. Introduction 
 
 
1.1 Background 
 
Low back pain (LBP) is the second most prevalent work-related musculoskeletal 
disorder in the United States. Nearly everyone suffers from LBP at some point in his or 
her lifetime. In the United States alone, 85% of the population suffers from low back 
disorder. Of all cases of LBP, 70% are due to lumbar strain or sprain, 10% are due to age-
related degenerative changes, 4% are due to osteoporotic compression fractures, and 3% 
are due to spinal stenosis (eMedicine Mechanical Low Back Pain Hills, 2010). A 2009 
National Health Interview Survey (NHIS) reported that 28.2% of respondents had visited 
physicians within the last three months for LBP. 
The costs of LBP can be very expensive both to society and to the individual. In 
1998, the total cost attributable to LBP in the U.S. was estimated to be $26.3 billion (Lou 
et al., 2004). Moreover, indirect costs incurred due to lost work days are substantial, with 
approximately 2% of the U.S. workforce compensated for back injuries each year 
(Andersson G.B., 1999). 
The majority of LBP is non-specific and is based on conditions where no clear 
cause can be found. However, despite this lack of understanding, biomechanical or 
physiological components can reveal the initiation and resolution of the condition 
(Hoogendoorn et al., 2000; Hartvigsen et al. 2003; Mulholland 2008). Acute pain is a 
short-term low back pain that generally lasts from a few days to a few weeks. It is 
 2 
generally caused by trauma to the lower back such as a sports injury, daily work, etc., or 
by a disorder such as arthritis. Chronic pain, on other hand, is measured by duration. 
Generally, pain lasting longer than 3 months is considered chronic. It is often progressive 
and the cause can be difficult to determine. Even though acute pain is diagnosed by short 
duration, a small percentage of sufferers go on to develop more long-term conditions 
( Melloh et al., 2009). A large number of studies have focused on chronic LP, but they 
have not revealed a complete understanding of this condition (Bergman, 2007).  
In order to apprehend this condition, researchers are now focusing on the effect of 
LBP on trunk movements. Kinematic and kinetic quantities are assumed to be periodic or 
pseudo periodic based on body characteristics and personal ability to control the lumbar 
spine. With neuromuscular and musculoskeletal pathologies or injuries, these movements 
may not be periodic and may result in increased instability of the lumbar spine 
(Papadakis et al., 2009). 
Many researchers have employed linear and nonlinear techniques to gait 
parameters to develop an understanding of physiological stability. Techniques of gait 
analyses have revealed long correlations in degrees of variability, and these have been 
related to maturation and degeneration of gait stability with age and central nervous 
system pathologies (Hausdorff, 2007). Such studies have played a vital role in 
understanding the conditions of normal and unhealthy movements, and in diagnosing 
patients with motor deficiencies. Motor learning is a nonlinear process, exhibiting 
nonlinear learning curves depending on the task, conditions, and characteristics of the 
learner. Normal variations that occur in motor performance across multiple repetitions of 
 3 
a task over time are the main reason for human movement variability (Stergiou et al. 
2004). 
Many investigations have linked variability with the health of the biological 
system. Specifically, one could find thousands of research articles that have shown that 
heart rate variability is affected by changes in the autonomic nervous system?s regulation 
of the heart rate (Goldberger et al., 2002; Ho et al., 1997). However, there was not a 
complete understanding of how human movements affect these heart rate dynamics. 
Later, some studies revealed that in certain populations, gait instability is related to 
cardiovascular health (Hausdorff et al., 2003). Moreover, these studies proposed that, 
similar to heart rate studies investigating heart rate dynamics, investigating lumbar spine 
dynamics may provide insight into locomotor control and may have clinical applications. 
Traditionally, random error or noise within the system was considered the etiology of 
variability. Motor learning textbooks usually describe movement variability as error and 
skilled movement as movement with decreased variability. For example, if one is tossing 
a bean bag, sometimes someone may toss it into the hole on the wooden plank, but the 
bean bag doesn?t always go into the hole because of variability. This leads some to 
conclude that there is some problem with the motor program and that it is not executed 
correctly when the bean bag fails to go in the hole. From this viewpoint, variability is 
considered to be an error in the motor control system. 
However, there is mounting evidence that the variability in normal movement 
reveals variation not as error but as a necessary condition for function. In fact, there is 
growing understanding that the phenomena previously described as noisy are actually the 
result of nonlinear interactions and have deterministic origins (Stergiou et al. 2004). 
 4 
Hence, studying the noisy component of the measured signal can yield important 
information about the system under study. 
  Variability can be defined in behavioral, biological and statistical senses. 
Behavioral variability can be defined as differences in observed behavior when the entity 
is placed in the exactly same situation. Biological variability is the power possessed by 
living organisms, both animal and vegetable, of adapting themselves to modifications or 
changes in their environment, thus possibly giving rise to ultimate variation of structure 
or function. Statistical variability refers to measures of centrality around a mean or an 
average and includes measures such as standard deviation, range and variance. All these 
definitions contribute to an understanding of human variability (Harbourne and Stergiou, 
2009). 
1.2 Traditional Linear Methods to Quantify Variability 
 
There are numerous studies that have shown the methods and quantities to 
determine the variability in systems. Traditional and nontraditional approaches have been 
used to identify the variability in kinetic, kinematic, and temporal variables. Traditional 
methods originating from statistical dispersion theory have been effectively applied in 
many investigations, and are considered to be appropriate to quantify the total variability 
patterns in human movements. Statistical dispersion is spread in a system. Range, 
standard deviation, variance, coefficient of variance are commonly used measures of 
statistical dispersion for variability analysis. Range is the difference between the largest 
and smallest observations. Probably the most common measures of variability used by 
researchers are variance and standard deviation. Variance is the mean of the squared 
deviations of a set of measurements, while standard deviation is the square root of the 
 5 
variance.  Although variance and standard deviation are computed similarly, variance is 
used less often in variability studies because units of variance are squared, making the 
interpretation of variability somewhat harder than the interpretation of standard deviation 
(Stergiou, 2004). Coefficient of variation is a normalized measure of dispersion that is 
defined as the ratio of the standard deviation to the mean and is given as a percentage. 
Standard deviation of data must always be understood in context of the mean of the data. 
Hence, when comparing two different sets of data with different means and units, 
coefficient of variation is more useful than standard deviation.  
Of the all linear techniques used to determine variability, standard deviation and 
coefficient of variation are the most popular methods. These techniques have been used 
for a long time. Fitzgerald and colleagues (1983) presented an assessment method in 
conjunction with age-related normal values for the lumbar spinal range of motion. 172 
volunteers participated in the study, which were divided into 6 age groups. Lumbar spinal 
flexion, right and left lateral thoracolumbar spinal flexion, and spinal extension were 
observed, and the coefficient of variation was calculated for each measurement. From the 
results the author showed that the variability in the normal range of motion increased 
with the age. In a similar kind of study related to effect of age on the gait variability, 
Buzzi et al. (2003) tried to understand the difference in the variability in two different age 
groups. Subjects were asked to walk on the treadmill, and gait parameters were recorded 
and analyzed using standard deviation and coefficient of variation. The authors suggested 
that the linear measures distinguished between the groups by indicating significantly high 
variability, but that they could not distinguish between the increase in noise within the 
system and the inherent variability associated with it. In the study carried out by Jackson 
 6 
et al. (2010), the authors evaluated the normative lumbar flexion-extension values for 
different ages and racial groups. Mean and standard deviations were calculated for 
lumbar flexion-extension. It was evident from the result of standard deviation that the 
normative values of lumbar flexion and extension are different for different ages and 
racial groups.  
The lumbar spine has also been studied in terms of postural variability. 
Specifically, Herp et al. (2000) assessed the three-dimensional range of movement in 
non-pathological subjects. 100 subjects (50 males and 50 females) with no recent back 
pain history were recruited. Each subject performed lumbar flexion-extension, side 
flexion to the left and right, and axial rotation to the left and right. There was significant 
difference in range of motion observed in younger and older age groups. The results 
revealed greater flexibility in women than in men. Along the same lines, Buchnan et al. 
(2001) tried to examine the upper trunk variation in space for instabilities. Subjects stood 
on support surfaces and the variability of upper trunk anterior posterior position in space 
was computed. Each subject performed the experiment with and without open eyes. The 
results showed very little evidence of variability with open eyes, while more variability 
was observed when eyes were closed. The authors thus concluded that postural patterns 
are influenced by mechanical and neural constraints. Changes in postural patterns are 
closely associated with visual information. Some investigators (Garcia-Alsina et al., 
2005) have also concentrated on normal range, velocity, and consistency of the 
movement of active arm elevation using coefficient of variation technique; they have 
shown that analysis of arm elevation repeatability in angular position and the velocity and 
repeatability of motion can give picture of overall performance of shoulder. 
 7 
Stride-to-stride variability during gait is one of the most widely researched 
movement analyses. Hausdorff et al. (1999) characterized the development of mature 
stride dynamics. 50 boys and girls were recruited in the study and were classified in 3 age 
groups: 3?4 years old, 6?7 years old and 11?14 years old. Two force-sensitive switches 
were placed inside the subject?s right foot to record force applied to floor at the time 
when foot struck the floor. Each subject was asked to walk on a running track at his or 
her self-selected speed. The authors suggested that variations in the stride were 
significantly larger for 3?4-year-old children than that of 6?7-year-olds and 11?14-year-
olds. They also concluded that temporal structure of gait fluctuations is not fully 
developed until 7 years of age. Dingwell and Marin (2006) tried to understand the 
kinematic variability and local dynamic stability across the range of walking speed. 6 
healthy males and females participated in the study. Standard deviation was calculated 
for each of the gait time series recorded. The results showed an increase in standard 
deviation for both faster and slower preferred walking speeds, which made the authors 
conclude that variability increases at both slower and faster walking speeds.   In another 
stride variability study, Dingwell et al. (2001) characterized the local dynamic stability 
properties of locomotor behavior using standard deviation measurement and suggested 
that treadmill walking was associated with significant changes in variability and local 
stability as compared to walking over ground. Thus the motorized treadmill walking may 
produce misleading results in situations where changes in neuromuscular control are 
likely to affect the variability of locomotion. In 2004, Kurz et al. used a relative phase 
dynamics to evaluate gait in subjects who had undergone anterior cruciate ligament 
(ACL) reconstruction. The large standard deviation values indicated a considerable 
 8 
amount of variability in shank-thigh relative phase dynamics within each group during 
running conditions.  
Some researchers have also used linear techniques to study questions related to 
human cadaveric movement. Brown et al. (2002) quantified instability of the lumbar 
spine using vertebrate distractor to measure motion segment stiffness by applying defined 
load at constant rate. They used mean, standard deviation, and range of stiffness as the 
measure of variability of motion stiffness and found out that motion segment stiffness is 
dependent on disc morphology. In another human cadaveric study, Hitchon et al. (2005) 
compared the biomechanical performance of human cadaveric spines implanted with 
artificial discs at the L4-L5 levels with spines in an intact state and after anterior 
discectomy. They concluded that the artificial disc stabilized the spine after discectomy, 
comparable to that of the intact spine.  
1.3 Nonlinear Methods to Quantify Variability 
 
Physicians in the medical field use linear models for prediction and problem 
solving (Harbourne et al., 2009).  However, biological systems, including humans, are 
complex adaptive systems, characterized by multiple interconnected and interdependent 
parts (Harbourne et al., 2009). These are highly non-linear systems with inherent 
variability in all healthy organisms. There is growing understanding that linear models 
are limited in many cases and are certainly not the best models for understanding the 
nonlinear human system (Stergiou, 2004). Professionals in different medical fields such 
as biomechanics, epidemiology, etc. are now turning to nonlinear tools for solving such 
complex systems. 
 9 
Although a large number of studies use linear tools as a measure of variability, 
they may not be truly representative of the variability. They ignore the viewpoint that 
variability may have a structure that can be useful and provide information about the 
strategies used to control the body?s degree of freedom (Harbourne et al., 2003). 
Structure in human movement data may not be visibly clear and it may become difficult 
to distinguish the data sets. Hence, many researchers are now focusing on nonlinear 
techniques that can lead to a better understanding of the holistic behavior of the motor 
system. The structure of the time series can reveal important information regarding 
properties of posture that evolve during human movement. Hence, it is advantageous to 
have variability in movement. The theory of optimal variability concentrates on the 
advantages of having a balance between rigid control and randomness in movement, i.e. 
complexity (Stergiou et al., 2006). Complexity can indicate spatial variation in 
movements (Mijna Hadders-Algra, 2007). Currently the most commonly used method to 
determine the complexity is approximate entropy (ApEn) (Karmakar et al., 2006; 
Khandoker et al., 2008). Approximate entropy is a method that quantifies the regularity 
or predictability of a time series. Pincus and Goldberger (1994) and Pincus (1995) 
suggested that in order to calculate ApEn two main parameters are required: m, the 
number of observation windows to be compared and r, the tolerance factor. Generally 
m=2 for all ApEn calculations while r ranges from 0.1 to 0.25 SD of the u(i) data (Pincus 
and Goldberger 1994; Pincus 1995).  
Vast number of studies has been conducted in trying to understand deterministic 
dynamical systems through reconstructed dynamics. Techniques such as correlation 
dimension, K-S entropy, and the Lyapunov Exponent have been extensively studied. 
 10 
ApEn has been compared with correlation dimension (Grassberger and Procaccia 1983) 
and the Lyapunov Exponent (Pincus 1997). Most of these techniques effectively employ 
embedding dimensions larger than m=2 (Pincus and Goldberger, 1994). As a result these 
techniques are superior in deterministic dynamical system environment as compared to 
ApEn. However, in biological settings we encounter mixed (stochastic and deterministic) 
systems, and those parameters and techniques produce poor results (Stergiou, 2004). 
There are many advantages of ApEn over the aforementioned nonlinear techniques that 
are explained by Pincus (1997) as: 
1. ApEn is unaffected by the noise below the tolerance factor ?r?. 
2. ApEn is unaffected by outliers (such as a spike in the time series). 
3. ApEn requires a small amount data to quantify complexity. 
4. ApEn is finite for random, noisy deterministic and mixed data sets, which are 
observed in biological environments. 
5. Increased complexity in the data is associated with an increase in ApEn.  
Because of these advantages, ApEn technique is more widely used in biological 
environments than other nonlinear techniques. Pincus and Goldberger (1994) studied the 
utility of ApEn over other nonlinear tools. They studied the heart rate data of aborted 
sudden infant death syndrome (aborted-SIDS) infants compared to normal infants, which 
showed the ApEn values for aborted-SIDS infant was significantly higher than normal 
infants. The authors concluded that abnormal physiology is associated with more 
regularity while normal physiology is linked with greater complexity. In another study 
ApEn was utilized to examine the effect of age on luteinizing hormone and follicular 
 11 
stimulating hormone serum concentrations in men and women. In a similar kind of study 
Veldhuis et al. (1998) studied the orderliness of hormone release patterns. 
Several human studies have also used ApEn measures for postural analysis. In 2003 
Harbourne et al. examined the lumbar postural control during various stages of 
development in independent sitting. Five infants of age 4 were categorized into 3 groups 
based on their ability to maintain posture. The result suggested that infants dynamically 
assume the sitting posture by increasing the stability and regularity of their strategy and 
by controlling the degree of freedom, first to approximate the skill, then to explore 
adaptations to function in the environment. Justin et al. (2009) tried to understand the 
effect of speed on the trunk motion during overground walking. The acceleration time 
series of the lower trunk was used to calculate the ApEn values. The result showed lack 
of regularity and repeatability difference, indicating that trunk motion was preserved 
between same conditions. Khandoker et al. (2008) examined the variability indexes in 
minimum foot clearance that would correlate with balance impairments. The results 
showed that ApEn is effective in quantifying gait dynamics in normal and pathological 
conditions and hence could be useful for early diagnosis of at-risk gait. 
Nowadays nonlinear tools are gaining more attention as a way to examine the 
variability in human movement.  There are several reasons for selection of nonlinear 
techniques as an alternative measure for variability, which are:  
1. The use of linear tools can mask the true structure of variability in the 
movement pattern. These techniques generally average the kinematic data to 
produce a mean picture of the subject?s movement, which completely ignores 
the temporal variation.  
 12 
2. Use of linear tools assumes the variability in movement to be random and 
independent, while studies have shown that they are not random and may have 
deterministic origin which can be revealed using nonlinear tools. 
3. Deterministic movement dynamics offer a moving system the ability to be 
adaptable and flexible in a constantly changing environment. Nonlinear tools 
have the ability to understand these dynamics. 
Although nonlinear tools have aforementioned advantages, they have some 
demerits as well (Harbourne et al. 2009): 
1. Nonlinear measurement techniques require mathematical equations and 
software to evaluate time series data; as a result they must be carried out in a 
research environment. 
2. There is a lack of understanding of variability and complexity in most medical 
fields. 
3. Translation of nonlinear measures to clinical problems requires concurrent use 
of linear tools to make associations and determine clinical meaning. 
4. Most of these measures require multiple repetitions or cycles of a movement. 
The aforementioned limitations of nonlinear techniques and the simplicity of 
calculation using linear tools make linear tools the preferred methods for understanding 
variability even in most biomechanical settings today. 
Searching the literature has revealed that both linear and nonlinear techniques can 
be effectively applied for motion analysis. Linear tools are useful but may ignore the 
hidden structure which can be determined by nonlinear techniques. The main objective of 
this study, which is a part of a bigger project investigating the effectiveness of 
 13 
chiropractic adjustments on the vertebral column by understanding its postural stability 
and lumbar proprioception, is to identify the best method to quantify the variability in the 
human lumbar spine during sitting flexion-extension in healthy subjects and in human 
cadaveric spine movements. 
1.4 Specific Objective 
 
Specific goals that would be required to achieve this objective are: 
1. To measure and record human lumbar movements and conduct in-vitro testing 
on the human cadaveric spine using the six-degree-of freedom-motion 
tracking system. 
2. To denoise the displacement data and differentiate it to get velocity and 
acceleration values. 
3. To quantify the effect of movement direction on the variability of human 
lumbar spine motion. 
4. To quantify the effect of speed and incision on flexion-extension kinematics 
in the cadaveric lumbar spine. 
5. To measure change in velocity while performing flexion-extension tasks. 
 
                                                                                                                                                                                                                          
 
 
 
 
 14 
 
 
 
2. Anatomy of Lumbar Spine 
 
 
The human spine is a complex structure that comprises the spinal cord and its 
protective and supportive coverings, known as the vertebral column or backbone. The 
spinal cord is a thin tubular bundle of nervous tissue that extends from the brain and 
makes up the central nervous system. It has a hollow center that is filled with fluid. The 
delicate tissues of the spinal cord are enclosed in and protected by the vertebral column.  
2.1 Vertebral Column 
 
The vertebral column is a column that has series of 33 stacked bones known as 
vertebrae (including the 5 that are fused to form the sacrum). Each vertebra is separated 
by spongy cushioning material called intervertebral discs that help to avoid contact 
between the adjacent vertebrae (University of Maryland Medical Center). These discs 
consist of strong annulus fibers known as annulus fibrosus, which distribute pressure 
evenly across the disc, and a soft jelly-like center known as the nucleus pulposus. The 
nucleus polposus contains loose fibers suspended in mucoprotein gel. This nucleus 
polposus helps to absorb shock during daily activities and prevent intervertebral contact.  
 15 
 
Figure 2. 1 Normal Disc 
Each vertebra is connected to its adjacent vertebra by an intricate web of muscles, 
ligaments and connective tissue. The pair of vertebrae and their supporting structures 
contributes to the specific and limited range of motion. Together they create a movable 
spine structure with the spinal cord as the feedback system continuously monitoring 
motion by sending and receiving signals to and from the brain. The spine has 31 
segments. The motor nerve roots branch out at each segment from the ventral side and 
sensory nerve roots enter from the dorsal side. The ventral and dorsal roots collectively 
form paired spinal nerves on each side of the spinal cord. 
The vertebral column has several curves when viewed laterally. These curves 
represent various regions on the column: cervical, thoracic, lumbar and pelvic (Anatomy 
of Human Body, 1918). 
 16 
 
Figure 2. 2 Human Lumbar Spine 
The cervical spine: This vertebral region curves convexly from the anterior and is the 
upper part of the spine. It is made up of 7 vertebrae (C1 to C7). 
The thoracic spine: Below the cervical spine is the thoracic spine, which curves 
concavely from the front. It consists of 12 vertebrae (T1 to T12) and forms middle 
portion of the spine.  
The lumbar spine: The lower part of spine is called the lumbar spine. It is convex 
anteriorly. It is made up of 5 vertebrae (L1 to L5); however, some people may have 6 of 
them.  Since the lumbar region is a lower region and these vertebrae have to support more 
weight than the vertebrae above them, they have developed larger and stronger bodies. 
The sacrum: The sacrum is a large triangular bone at the base of the lumbar spine. Its 
narrow lower part joins the coccyx. The sides are connected to the ilium (the largest bone 
forming pelvis). 
The coccyx: It is the lowest part of the spine that is attached by ligaments to the margins 
of the sacral hiatus. 
 17 
2.2 Vertebrae 
 
Figure 2. 3 Vertebra 
An individual bone of the spine is called a vertebra. Together, the vertebrae make 
a freely mobile yet sturdy structure that enables human beings to perform various 
common tasks in daily routine. When the vertebrae are combined together the bodies 
form a strong pillar for the support of the head and trunk. The vertebral body is the main 
part, which bears about 80% of the load while standing and provides attachments for 
discs between the vertebrae. Both the vertebral body and the discs increase in size from 
the head to the sacrum. Each vertebra has two cylinder-shaped projections that stick out 
from the back of the vertebral body, providing side protection for the spinal cord and 
nerves. The lamina is the roof of the spinal canal that provides support and protection for 
the backside of the spinal cord. The bony projection that is felt on the back of any person 
is called the spinous process and is at a right angle to the midline of the lamina. These 
spinous processes are attached to each another by ligaments. The transverse processes are 
located at right angles to the junction of the pedicles and the lamina. They provide a place 
for the back muscles to attach to the spine. The spinal canal is the bony tunnel that 
surrounds the spinal cord. It is made up of vertebral bodies in front, the pedicles on the 
sides and the laminae on the back. Each vertebra has a paired joint on the left and right of 
 18 
the spinous process that allows the connection between the vertebrae above and below it 
and is called the facet joints. 
There is a complex web of infinitesimal muscle that controls the bony structure of 
the spine, known as the soft tissue. The paraspinal muscles are the muscles running along 
either side of the spine that mainly control the vertebral movement. These muscles and 
ligaments are covered by layers of skin and connective tissue (University of Maryland 
Medical Center).  
 
 
Figure 2. 4  Ligaments of Spine 
 
 
 
 
 
 
 
 
 19 
 
 
3. Tools and Techniques 
 
 
3.1 State of System 
 
Different methods of nonlinear techniques are based on the examination of 
structural characteristics of a time series that is embedded in an appropriately constructed 
state space. State space is the vector space where a dynamical system can be defined at 
any point. The characteristics of that state space then can be examined to gain insight into 
the motor control of posture. The phase space is a plot that represents the behavior of a 
dynamic system in state space. The phase plane plot is simply a representation of the state 
space.  
3.2 Time Series  
 
A time series is a sequence of data collected at regular intervals over a period of 
time. Time series data analysis can be very useful in understanding the variability of 
given parameters over a period of time. For example, if we record the distances from the 
center of bicycle to the edge of the road, we may categorize the rider?s driving as learner 
or proficient. Time series can be analyzed in two ways: Time domain and Frequency 
domain. A discrete time series is one in which the data is collected at discrete time points. 
Discrete time series refers to fixed time intervals between two adjacent observations 
(Deng et al. 1997). 
 
 
 
 20 
3.3 State Space 
 
The dynamic behavior of the system can be examined only if the structural 
characteristic of time series is studied. To understand the structural characteristic we need 
to reconstruct the state space where the behavior of the system is embedded. State space 
is a vector space where the dynamical system can be defined at any point. By examining 
the state space one can gain insight into the motor control of the posture (Stergiou, 2004).  
3.4 Phase Space 
 
Phase space is the space in which all possible states of systems are represented, 
with each possible state of the system corresponding to one unique point in the phase 
space. The phase space usually comprises all possible values of position and momentum 
variables and the plot of it is sometimes called a phase plane plot. For every behavior of 
the dynamic system, a point is plotted in multidimensional space. The trajectory of the 
plotted points is analogous to the system?s changing state.  
Figure 3.1 below shows the time series and phase plots for sine curve and random 
data. 
 
 21 
0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1
-1
- 0 . 8
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1
T i m e ( s e c )
D
is
p
la
c
e
m
e
n
t
 
 
Figure 3. 1 Sine Wave 
 
 
-1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1
- 8 0
- 6 0
- 4 0
- 2 0
0
20
40
60
80
D i s p l a c e m e n t
V
e
lo
c
it
y
 
 
Figure 3. 2 Phase Plane Plot 
 
 22 
-1
- 0 . 8
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1
- 8 0
- 6 0
- 4 0
- 2 0
0
20
40
60
80
- 4 0 0 0
- 2 0 0 0
0
2000
4000
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
 
Figure 3. 3 3-D Phase Plane Plot 
 
 
0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1
- 1 . 5
-1
- 0 . 5
0
0 . 5
1
T i m e ( s e c )
D
is
p
la
c
e
m
e
n
t
 
 
Figure 3. 4 Random Data 
 
 23 
- 1 . 5 -1 - 0 . 5 0 0 . 5 1
- 2 0 0 0
- 1 5 0 0
- 1 0 0 0
- 5 0 0
0
500
1000
1500
2000
D i s p l a c e m e n t
V
e
lo
c
it
y
 
 
Figure 3. 5 Phase Plane Plot 
 
 
- 1 . 5
-1
- 0 . 5
0
0 . 5
1
- 2 0 0 0
- 1 5 0 0
- 1 0 0 0
- 5 0 0
0
500
1000
1500
2000
-4
-2
0
2
4
6
x  1 0
6
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
 
Figure 3. 6 3-D Phase Plane Plot 
Figures 3.1 and 3.4 show time series from a simple periodic function (sine curve) 
and random data respectively. Figures 3.2 and 3.5 show phase plane plots of these time 
series. The phase plane plot of the periodic function gives a closed orbit with complete 
overlapping of the trajectories and no divergence at all. However, the phase plane plot for 
 24 
random data has no order and the trajectories have no clear patterns. Hence, one can see 
that in a biomechanical setting, if the time series is perfectly periodic and repetitive, the 
phase plane plots will show clear patterns with no divergence, while the time series with 
some variability will show some divergence. 
As mentioned previously, a phase plane plot is just a representation of state space 
and many times it is difficult to distinguish between the different time series data that we 
are comparing. In order to clarify this point, an experiment was carried out on a flexible 
foam strip (2 ft long) with 4 sensors attached at different levels spaced similarly to the 
T1, L1, L3, and S1 levels of the human spine (Figure 3.7). The motion data was recorded 
at the sampling rate of 120 fps. 
 
 
Figure 3. 7 Foam Strip 
This foam strip was then placed on a fixed bottom and was moved to and fro 
smoothly, so that it replicated motions similar to human lumbar flexion-extension 
 25 
movement. Three types of data were recorded: 1) Smooth and Controlled; 2) Smooth and 
Uncontrolled; 3) Unsmooth and Uncontrolled. 
These data were filtered using a second-order Butterworth filter with a cut-off 
frequency of 10 Hz; Sample rate = 120 fps; Nyquist frequency = sample rate/2 = 60 Hz; 
Wn= cut off freq/Nyquist freq = 10/60 = 0.166 Hz (England and Granata, 2006). 
 
 
1) Smooth and Controlled Movement 
 
 
0 5 10 15
- 2 0
- 1 5
- 1 0
-5
0
5
10
15
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( d e g )
V e l o c i t y ( d e g / s e c )
A c c e l e r a t i o n ( d e g / s e c 2 )
 
 
Figure 3. 8 Displacement, Velocity, Acceleration Diagram 
 
 26 
- 2 0 - 1 5 - 1 0 -5 0 5 10 15
-3
-2
-1
0
1
2
3
D i s p l a c e m e n t
V
e
lo
c
it
y
 
 
Figure 3. 9 Phase Plane Plot 
- 2 0
- 1 5
- 1 0
-5
0
5
10
15
-3
-2
-1
0
1
2
3
- 1 . 5
-1
- 0 . 5
0
0 . 5
1
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
 
Figure 3. 10 3-D Phase Plane Plot 
 
 
 
 
 
 
 
 
 
 
 27 
2) Smooth and Uncontrolled Movement 
 
 
0 2 4 6 8 10 12
- 5 0
- 4 0
- 3 0
- 2 0
- 1 0
0
10
20
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( d e g )
V e l o c i t y ( d e g / s e c )
A c c e l e r a t i o n ( d e g / s e c 2 )
 
 
Figure 3. 11 Displacement, Velocity, Acceleration Diagram 
 
- 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 10 20
-6
-4
-2
0
2
4
6
8
10
D i s p l a c e m e n t
V
e
lo
c
it
y
 
 
Figure 3. 12 Phase Plane Plot 
 28 
 
- 5 0
- 4 0
- 3 0
- 2 0
- 1 0
0
10
20
-6
-4
-2
0
2
4
6
8
10
-2
0
2
4
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
 
Figure 3. 13 3-D Phase Plane Plot 
 
 
 
3) Unsmooth and Uncontrolled Movement 
 
 
0 2 4 6 8 10 12 14 16 18 20
- 2 5
- 2 0
- 1 5
- 1 0
-5
0
5
10
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( d e g )
V e l o c i t y ( d e g / s e c )
A c c e l e r a t i o n ( d e g / s e c 2 )
 
 
Figure 3. 14 Displacement, Velocity, Acceleration Diagram 
 
 29 
- 2 5 - 2 0 - 1 5 - 1 0 -5 0 5 10
-3
-2
-1
0
1
2
3
4
D i s p l a c e m e n t
V
e
lo
c
it
y
 
 
Figure 3. 15 Phase Plane Plot 
 
- 2 5
- 2 0
- 1 5
- 1 0
-5
0
5
10
-3
-2
-1
0
1
2
3
4
-2
-1
0
1
2
3
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
 
Figure 3. 16 3-D Phase Plane Diagram 
As seen from the phase plane plots (Figures 3.9, 3.12, 3.15) we see that the 
smooth and controlled motions produce the trajectories that have some elegant pattern, 
while the unsmooth and uncontrolled movements showed no pattern in trajectories. The 
smooth and uncontrolled time series showed some pattern and was similar to that of the 
smooth and controlled. Hence, we can say that phase plane plots could be used to 
 30 
distinguish between different conditions. But they do not quantify variability, and provide 
a visual means to distinguish between various movements. In general, phase plots should 
just be considered as a guideline and not the method to differentiate the variability in time 
series, which could be more effectively done by the linear and nonlinear techniques 
explained above. 
 
Table 3. 1 Linear and Nonlinear Measures 
 
Test Condition Total ROM 
 Mean SD CV (%) ApEn 
Smooth and 
Controlled 
Movement 
 
27.51279 2.660723 9.670858 0.117 
Smooth and 
Uncontrolled 
Movement 
 
46.67221 15.47123 33.14869 0.2269 
Unsmooth and 
Uncontrolled 
Movement 
 
23.02578 2.624542 11.39828 0.2757 
 
 
 31 
 
Figure 3. 17 Mean for Foam Test 
 
 
Figure 3. 18 Standard Deviation for Foam Test 
 
 32 
 
Figure 3. 19 Coefficient of Variation for Foam Test 
 
Figure 3. 20 Approximate Entropy for Foam Test 
Linear and nonlinear techniques estimated and represented above in Figures 3.17?
3.20 showed that linear techniques have higher variability for smooth and uncontrolled 
movement, which is unexpected since unsmooth and uncontrolled movement is more 
random than others. On other hand, approximate entropy (ApEn) showed higher 
variability for unsmooth and uncontrolled movement after smooth and uncontrolled 
movement. The linear techniques failed to account for the underlying structure of the 
 33 
movement. ApEn showed an increasing trend as the randomness in the movement 
increased, which was in agreement with the phase plane plots.  
3.5 Noise in Biomechanical Data 
 
As described previously, the data collected above was filtered before it was used 
to plot various graphs. The filter mainly removes the unwanted features from the signals. 
These unwanted signals, or noise, are inherent in all electronic devices. Noise can be 
generated for various reasons, the most common being the random thermal motion of 
electrons inside an electrical conductor, regardless of any applied voltage and random 
fluctuations of the electric current in an electrical conductor. 
3.6 Filters 
 
The displacement data in most biomechanical studies are differentiated in order to 
get the velocity and acceleration. These signals are generally of more importance than the 
displacement data and may yield important clinical information. If a noise-contained 
signal is differentiated it results in more noisy velocity and acceleration data. To 
understand the severity of this problem, let us consider a time series x(t) consisting of a 
sum of N sinusoids of frequency ?i, amplitude of ai and phase ?i. This could be 
mathematically represented as: 
            N 
x(t) = ? aicos (?it + ?i) 
           i = 1 
The amplitude of velocity is expressed by ?i ai and that of acceleration is defined 
as ?i 2 ai (Wachowiak et al., 2000). Hence, as we can see from these expressions, if the 
displacement signal frequency has noise, then it would only be amplified by 
 34 
differentiating the displacement signal twice. In order to avoid this, signals are de-noised 
using filters. 
Filters are one of the crucial parameters of signal processing and mainly have two 
important roles: (a) separation of signals that have been combined and (b) restoration of 
signals that have been distorted in some way (Smith, 1999). Original signals may get 
adulterated with noise, interference, or other signals. They may even get distorted in 
some way. A good filter should have the ability to nullify these errors.  
Every filter has various characteristics that give complete information of that 
particular filter. These are known as the responses of the filter. They are impulse, step, 
and frequency responses. All responses contains the same information but in different 
arrangements and can be calculated directly if any one response is known (Smith, 1999).  
One of the important functions of the filter is to represent the information 
contained in the signal. This can be done in time domain or frequency domain. 
Information represented in time domain explains when something occurred and what the 
amplitude was, while the frequency domain representation is more indirect. In the 
frequency domain the information is represented by the relationship between many points 
in the signal (Smith, 1999). 
The step response explains how the information represented in time domain is 
modified by the system, while the frequency response reveals how information 
represented in the frequency domain is being changed (Smith, 1999).  
In order to measure the performance of a filter in time domain, certain parameters 
of step response should be considered. 
 
 35 
These are: 
a) Risetime 
b) Overshoots 
c) Phase Linearity  
 
Figure 3. 21 Step Response 
The filter design is poor if the designed filter has risetime as shown in Figure 
3.21.a; however, the risetime in Figure 3.21.b is steep and short, making it an ideal 
response for time domain filter.  
Figure 3.21.c shows overshoot in the step response. Overshoot is basically 
distortion of information in the signal and results in a change in the amplitude of the 
samples of the signals. Conversely, Figure 3.21.d shows the desired step response with no 
overshoot. 
Phase linearity is the symmetry between the upper and lower halves of the step 
response. Figure 3.21.e shows unsymmetrical step response which is due to the faulty 
filter design and is undesirable.  
 36 
The purpose of designing a filter in the frequency domain is to allow signals with 
certain frequencies to pass through while blocking some signals with other frequencies. 
Passbands are the frequencies that are passed untouched, while stopband refers to 
frequencies that are blocked. Between the two is the transition band. When this region is 
narrow it is called fast roll-off. The division between the passband and the transition band 
is called the cutoff frequency. 
 
 
 
Figure 3. 22 Frequency Response 
 
In order to estimate how well a filter performs in the frequency domain, three 
frequency response parameters should be considered: 
a) Roll?off 
b) Passband 
c) Stopband Attenuation 
 37 
 
Figure 3. 23 Frequency Response 
As shown in Figure 3.23, in order for the filter to be effective, it must have a fast 
roll-off. Figure 3.23.b shows faster roll-off as compared to Figure 3.23.a, which is 
desirable. The presence of a ripple in the passband adds distortion to the passband 
frequencies when moving through filter (Figure 3.23.c). In order to avoid this, no ripple 
should be present in the passband (Figure 3.23.d), and lastly, to completely block the 
stopband frequencies, the designed filter should represent good stopband attenuation in 
frequency response as shown in Figure 3.23.f. 
Normally one has to trade off between the two responses as it is not possible to 
optimize a filter for both applications. A filter designed for step response may not yield 
good results for frequency response. In order to check our designed filter in the frequency 
domain, the frequency response of the filter was estimated using MATLAB R2008a; see 
Figure 3.24 below. 
 38 
0 5 10 15 20 25 30 35 40 45
- 6 0
- 5 0
- 4 0
- 3 0
- 2 0
- 1 0
0
F r e q u e n c y  ( H z )
M
a
g
n
it
u
d
e
 (
d
B
)
M a g n it u d e  R e s p o n s e  ( d B )
 
Figure 3. 24 Filter Frequency Response 
The frequency response of the designed filter shows slow roll-off, but the 
passband is flat.  
In order to calculate dynamic stability, the present study uses a 10 Hz low pass 
second-order Butterworth filter. The filter cut-off frequency was selected to assure 
sufficient bandwidth to represent torso dynamics?i.e. approximately ten times the 
natural frequency of the torso (Moorhouse and Granata 2005; Winter et al. 1998). The 
filter was designed in MATLAB R2008a, which also determined the velocity and 
acceleration from biomechanical displacement signals (See appendix for program). 
In order to test this program, a small experiment was carried out using Optotrak 
3020 (Northern Digital Inc.; Ontario, Canada) and an accelerometer CXL10HF3 
(Crossbow Technology Inc.; California, USA). The Optotrak unit was stationed on a 
table and connected to the computer through ODAU II and a system control unit. First 
principal, motion analysis software, was used in conjunction with the Optotrak unit to 
record the data at 100 Hz. Three infrared markers were firmly attached on the surface of 
 39 
the accelerometer facing the Optotrak unit using double-sided tape. These markers were 
supplied with the strober voltage of 9V. 
On starting the recording in First principal, the accelerometer was given upward 
and downward motion for three complete cycles. The Optotrak unit recorded the motion 
data while the accelerometer simultaneously collected the acceleration data. This data 
was then exported and saved on the computer hard drive in Microsoft Excel spreadsheet 
format. The accelerometer data was collected in Volts. To convert this data into m/sec2, 
zero-G volts of 2.5 V in each particular direction and for the particular accelerometer was 
subtracted from the available data values. This data was then divided by the sensitivity of 
0.105 V/g for that particular direction and multiplied by 9.81 to get the value in m/sec2. 
Both zero-G voltage and sensitivity values were given in a calibration sheet provided for 
that particular accelerometer (See Appendix C). Results of this test are shown below 
(Figure 3.25).  
 
0 5 10 15
- 1 0
-5
0
5
10
15
T i m e ( s e c )
A
c
c
e
le
ra
ti
o
n
(m
/s
e
c
2
)
 
 
A c t u a l  A c c e l e r a t i o n  ( m / s e c 2 )
C a l c u l a t e d  A c c l e r e a t i o n  ( m / s e c 2 )
 
Figure 3. 25 Acceleration using Accelerometer and Filter 
 40 
As we can see from the graph, values from actual acceleration data taken from 
accelerometer and values of calculated acceleration computed from the MATLAB 
R2008a program are in close agreement with each other. To estimate error in numerical 
value, the Root mean square error of the two signals was calculated using MATLAB 
R2008a and was found to be 0.4724. This value is very low and suggests that the 
designed filter is efficient in performance and gives accurate velocity and acceleration 
data. 
3.7 Sample Size 
 
A biomedical study, to be successful, has to have a well-defined problem, 
appropriate population, a reliable procedure and instruments, among other resources. In 
addition to these, an adequate sample size is one of the most critical parameters to be 
considered. It must be ?big enough? that it does not waste resources on an inconclusive 
study, and ?short enough? that it can yield useful results in a timely manner. Sample size 
is of the utmost importance in experiments involving human or animal subjects for 
ethical issues. In an over-populated experiment, an unnecessary number of participants 
are exposed to potentially hazardous tests, while under-populated studies expose subjects 
to potentially hazardous tests without advancing the research knowledge (Lenth, 2001). 
Finally, the study must be of adequate size, which would be relative to the goal of the 
study. The present study is a preliminary study, and time and cost are the main 
constraints. Keeping this in mind, a sample size of five subjects was considered adequate 
for this study. 
 
 41 
 
 
 
4. Methodology 
 
 
In this approach of estimating the variability of the human spine motion, two 
methods were employed. In the first approach the subject performed flexion-extension of 
the spine while sitting on an unstable seat. The second set of data recorded flexion-
extension motion of a human cadaveric spine that was motor controlled. Both of these 
methods were designed and developed by the investigators at Palmer Center for 
Chiropractic Research, Davenport, Iowa and made available to conduct this study.  
As with most Biomechanical studies, this study involved instrumentation that 
accurately recorded the spine?s movement. Tests were carried out on human subjects and 
a cadaveric spine. Each of these tests had specific goals and procedures as described 
below. 
4.1 Instrumentations for Tests on Human Subjects 
 
  A six-degree electromagnetic tracker, Liberty 24/8 (Polhemus, Vermont, USA) 
unit in conjunction with Motion Monitor software was used to record human lumbar 
movement in human subjects. 4 sensor channels were attached to the ports available on 
the chassis of Liberty, the other ends of which were attached to the T1, L1, L3, and S1 
vertebral levels of the subject?s spinous processes. The data was recorded at a maximum 
speed of 120 updates per second. 
 42 
 
Figure 4. 1 Polhemus Unit 
The Liberty unit consists mainly of the following: 
a) System Electronic Unit (SEU): 
System electronic units interface with the host computer via USB port and consist 
of the hardware and software necessary to generate and sense the magnetic field and 
estimate the position and orientation. 
b) Source: 
The source is a plastic-enclosed cube which contains electromagnetic coils that 
generate the magnetic field around it. It acts as the system?s reference frame for sensor 
measurement. 
c) Sensors 
The sensor is a small plastic-coated cube composed of electromagnetic coils that 
detects the magnetic field generated by the source. It precisely measures the position and 
orientation as it moves. The sensor is a passive device (i.e. no active voltage is applied to 
it). 
The Liberty unit was used with Motion Monitor software to record the human 
lumbar motion and export the data. The software allowed independent user-defined 
profiles for setting system parameters such as coordinate rotations, output format, etc. 
 43 
The user can select positions in Cartesian coordinates, Euler angles, and orientations in 
direction cosines or quaternion?s. It can be used to set tracker parameters such as 
sampling frequency, recording time, calibration, etc.  
The general workings of the Liberty unit are described as follows: 
? The source generates an electromagnetic field around it. 
? This electromagnetic field is sensed by an activated sensor. 
? Upon starting the recording using Motion Monitor, the motion is recorded at a 
sampling rate of 120 Hz and the sensors output is given as input to the host 
computer through the SEU. 
? This data is then saved on the computer RAM for further processing and display. 
4.2 Experimental Set-up 
 
The study required subjects to perform continuously repeated trunk flexion 
extension-movements while sitting on an unstable seat. On its bottom, the rectangular 
cushioned seat (15 ? 9 inches) had semi-circular wedges on either side that restricted its 
movement in the subject?s anterior-posterior direction. The seat was kept on wooden 
structure, the height of which could be adjusted. A safety railing was provided on either 
side of the wooden structure to provide safety in situations in which subjects might lose 
their balance or feel discomfort during the test. The source unit was stationed on the 
posterior side of the seated subject at a distance of 10 inches. In order to avoid any 
interference with the electromagnetic field developed by the source, the whole structure 
was made out of wood.  
During the test, subjects were asked to sit on the unstable seat with their hips and 
knees flexed at an angle of 90 degrees. To help shorter subjects maintain the 90-degree 
 44 
knee angle, a wooden board (36 ? 24 ? 1 inches) was placed underneath their feet. The 
wooden board was placed on a rubber mat so as to avoid any slippery action.  
4.3 Human Subjects 
 
5 subjects with no history of low back pain were recruited and performed 
repetitive trunk flexion-extension tasks. All 5 subjects were male. The criteria were as 
follows: 
Inclusion Criteria: 
a) Men and women aged 18 to 65 years, with no history of LBP. 
b) Subjects would be examined by a professional clinician to evaluate whether they were 
healthy enough to perform biomedical tests. 
 
Exclusion Criteria: 
a) Co-morbid pathology such as bone and joint pathology, extreme obesity, etc. 
b) Poor general health condition. 
c) History of spine surgery.  
d) Pregnancy. 
Each eligible participant signed an institutional review board (IRB) approved 
informed consent form. The consent form was approved by both the Auburn University 
IRB and the Palmer College of Chiropractic IRB.  
4.4 Data Acquisition 
 
Each of the selected participants was given a randomized unique 5-digit code and 
was identified with that particular code. Before performing the actual biomechanical tests 
each participant had to answer different written questionnaires. A brief video 
 45 
demonstrating the biomechanical test was shown to each participant. A quick double-
sided tape skin test was performed on each participant in order to check for any allergic 
reactions. Upon successfully completing the aforementioned tasks, the height and weight 
of each participant was measured and entered into the computer system. All the relevant 
information (such as height, code, and weight) for each participant was written on the 
biomechanics test log form. (See appendix C for details.) Upon entering the 
biomechanics laboratory, the lumbar repositioning (flexion-extension test) was briefly 
explained to each participant. Before starting the test each participant was asked whether 
there were any removable or non-removable metal objects on his body. If they had any 
non-removable metal such as a surgical plate or rod, a note was made on that particular 
subject?s test log form. However, if participants had any removable metal on their bodies, 
they were asked to remove it (if possible), as metal parts may cause interference with the 
magnetic field generated by the source of the Liberty unit. The participant was then asked 
to sit straight on the unstable seat, making 90-degree angles at the knees and hips. If the 
participant was short a wooden board (Riser) was placed underneath the participant?s feet 
in order to maintain the 90-degree knee angle. Two pieces of white tape were placed at 
the toes of each participant and tendon to center of the unstable seat distance was 
recorded on the test log form so that if the participants took a break in between the 
subtests and wished to stand, they could return to same sitting position after the break. 
The test setup began with palpating the spinous processes of participants in order 
to identify the vertebral levels (T1, L1, L3, and S1). Meanwhile the sensors were 
activated. These levels were then noted on the test log form and marked with a water-
soluble skin marker. An alcohol cleaning pad was used to clean these levels so that the 
 46 
markers would firmly stick on the spinous processes using double-sided tape. These 
sensors were then covered with a piece of tape in order to prevent the double-sided tape 
from sticking to participant?s clothes. 
At this point each participant was asked to sit upright wearing the blindfold and 
cross his arm across the chest, resting on the shoulder (As shown in figure 4.2). This 
position was called neutral position. Every participant performed seven subtests. Each 
subtest had a different starting position, which was called the target position, and a 
different starting direction. The aim for the participant, starting each subtest at a different 
starting position (target position) and in different starting directions, was to return to the 
target position after rocking the pelvis forward and backward for ten complete cycles. 
The target position could be either neutral, between neutral and extreme forward, or 
between neutral and extreme backward. 
 
Figure 4. 2 Subject in Test Position 
 47 
Participants were asked to move at a comfortable speed and comfortable range 
and stop the test at their discretion. This represented their natural movement. This self-
selected speed eliminates any potential risk to the low back due to predetermined speed. 
In order to avoid fatigue, every participant was given two mandatory short breaks after 
the second and fifth subtests. Each subtest, depending on the permutation and 
combination of the starting position and starting direction, was addressed with different 
alphabetical notations as:  
Table 4. 1 Subtests 
Test  Starting Position  Starting Direction 
RA (NF) Neutral Forward 
RB (FF) Forward Forward  
RC (FB) Forward  Backward 
RD (BF) Backward  Forward 
RE (BB) Backward Backward 
RF (NF) Neutral  Forward 
RG (NB) Neutral  Backward 
 
4.5 Data Analysis 
 
The main aim of data analysis was to extract useful information from the recorded 
signal that can be processed further to understand the results. The motion monitor 
program was mainly used for this purpose. It provided an interface to observe the real-
time data, and to record and manage the data. A preference file with certain parameters 
was created and saved in order to set up the test. It mainly required two parameters as 
input (i.e. height and weight of the subject), after which the test could be recorded. The 
 48 
recorded data displayed the graphs of parameters such as flexion-extension, lateral 
flexion, rotation, etc. Each graph showed cursor value and the minimum and maximum of 
that graph in the upper right corner, which is advantageous in analyzing the real-time 
data. The recorded data was stored on the computer hard drive with a specific participant 
code. All saved data was exported by loading the same preference file which would 
extract particular information from that data (Figure 4.3).  
 
Figure 4. 3 Data Export 
A time series of 10 cycles was analyzed. This allowed for the evaluation of both 
linear and nonlinear methods. The continuous 10 cycles allowed examination of an 
average of 3000 data points for each subtest. This number was considered adequate for 
the nonlinear analysis performed in this study (Stergiou, 2004; Pincus 1994; Harbourne, 
et. al 2003). It was assumed that because the same instrument was used for all subjects, 
the level of noise from the measurement would be consistent in all cases. Therefore, any 
differences would be due to the changes within the system (Kaplan et. al, 1995).  
As mentioned before, to quantify the variability during lumbar movement in terms 
of nonlinear dynamics, the Approximate Entropy (ApEn) method was used. Entropy is a 
statistical concept, which was first introduced by Shanon and Weaver in 1963 as a 
 49 
measure of uncertainty or variability. Similarly, ApEn is a specific method to determine 
complexity that can quantify the regularity or predictability of a time series (Pincus, 
1994). Predictability and regularity are inversely proportional to complexity. The more 
predictable and regular the time series, the less would be the complexity, and vice versa. 
Approximate Entropy measures the logarithmic probability that a series of data points a 
certain distance apart will exhibit similar relative characteristics on the next incremental 
comparison within the state space (Pincus, 1994). Data points that exhibit greater 
possibilities of remaining the same distance apart upon comparison will result in lower 
ApEn values, while those with large differences in distance between them will result in 
higher ApEn values. 
In order to mathematically define ApEn, we need to form a time series of data u 
(1), u (2) ????. u (N). These are N raw data values from measurements taken at 
equally spaced points in time. We then fix m, an integer, and r, a positive real number. 
The input parameter m is the length of compared runs, and r is the tolerance that specifies 
a filtering level. The first step is to form a sequence of vectors x(1), x(2)??.. x(N - m + 
1) in Rm, real m-dimensional space, defined by x(i ) = [u(i)?.. u(I + m - 1)]. The second 
step is to use the sequence x(1), x(2)?.. x(N - m + 1) to construct for each I, 1 ? i ? N - m 
+ 1, Cim(r) = (number of x(j) such that d[x(i),x(j)]? r)/(N - m+1). We must define 
d[x(i),x(j)] for vectors x(i) and x(j). We follow the Takens modification of formula by 
defining d[x,x*]= max ?u(a) - u*(a)?, where the u(a) are the m scalar components of x. d 
represents the distance between the vectors x(i) and x(j), given by the maximum 
difference in their respective scalar components. Next we define ?m(r) = (N - m + 1)-1 
 50 
?i=1N-m+1 In Cim(r), where In is natural logarithm. Lastly we define Approximate Entropy 
as: 
ApEn(m,r,N) = ?m(r) - ?m+1(r) 
As seen above, calculation of ApEn requires selection of two parameters: m, the 
number of observation windows to be compared, and r, the tolerance factor. In order to 
compare the results, these parameters, along with the data length, must be kept the same 
for all calculations (Pincus and Goldberger, 1994). 
Typically m = 2 or 3; r depends greatly on the application (Pincus et. al, 1991). 
This choice of m is made to ensure that the conditional probabilities, defined in the 
equation below for fixed m and r, are reasonably estimated from the N input data points. 
Theoretic calculations indicate that reasonable estimates of these probabilities, for fixed 
m and r chosen as discussed below, are achieved with between 10m and 30m points, 
analogous to findings of Wolf et al. (Pincus et. al, 1991). 
An important observation that we can deduce from the ApEn definition is:  
-ApEn= ?m+1(r) - ?m(r) = average over I of In [conditional probability that Iu(j + m) - u(i 
+ m)I ? r, given that Iu(j + k) - u(i + k)I ? r for k = 0,1,?.., m-1].?????. ( A) 
The number of input points for ApEn computations ranges typically from 50 to 
5,000 points (Stergiou, 2003; Pincus, 1994; Pincus et. al, 1991). Using fewer than 50 data 
points yields less meaningful results, especially for m = 2 or 3, while using more than 
5,000 points will result in unacceptably long computational time (Pincus et. al, 1991). 
For noiseless, theoretically described systems, such as Henon maps and logistic 
maps, it has been shown that if entropy (A) ? entropy (B), then ApEn (m, r) (A) ? ApEn 
(m, r) (B) and vice versa. Moreover, for both theoretical and experimental systems, if 
 51 
ApEn (m1, r1) (A) ? ApEn (m1, r1) (B), then ApEn (m2, r2) (A) ? ApEn (m2, r2) (B) and 
vice versa. This ability of ApEn to preserve the order is a relative property and is an 
important utility of ApEn (Pincus et al, 1991). Considering this, one should not conclude 
that for the same systems, ApEn (m1, r1) (A) ? ApEn (m2, r2) (B), as ApEn values differ 
with different m and r values. The strength of ApEn is its ability to compare systems. 
As explained above there are two critical parameters (m and r) that need to be set 
in order to achieve reasonable results while using ApEn. Different m and r values would 
result in different results. ApEn (2, 0.1) may be different form ApEn (3, 0.01) values. 
This leads to the question of which one should be chosen. ?r? is effectively a filter level 
and in order to eliminate the effect of noise in the ApEn calculation, ?r? must be chosen 
such that its value is above most of the noise. In order to achieve reasonable results the 
magnitude of noise should rarely reach ?r?. 
Another key factor in choosing the value of r is that it should be large enough to 
achieve numerically stable conditional probability estimates in equation (A) above 
(Pincus et. al, 1991). If the ?r? value is small, one gets unstable conditional probability 
estimates, while larger ?r? values results detailed system information to be lost due to 
filter coarseness. In the current study a value of 2 was used for m and r was 0.2 (Pincus 
1990; Pincus 1994; Stergiou, 2004). 
4.6 Calculation of ApEn 
 
Consider a time series SN, consisting of N number of sample size. To compute 
ApEn we must choose two input parameters, m and r. We denote a pattern of m time 
series, beginning at measurement i within SN, by the vector pm(i). Two patterns,  pm(i) and 
 52 
pm(j) , are said to be similar if the difference the difference between any pair of 
corresponding measurements in patterns is less than r, i.e, if  
|N(i+k)-N(j+k)|< r, for k=0 to m 
Now consider the set Pm of all patterns of length m (i.e., pm(1),pm(2),? pm(N-
m+1)), within SN. So we may define Cim(r) = nim(r)/ (N-m+1) where nim(r) is the number 
of patterns in Pm that are similar to pm(i)(provided similarity criterion ?r?). The quantity 
Cm(r) is the fraction of length m that is identical to the pattern of the same length that 
begins at interval i. We can calculate Cim(r) for each pattern in Pm, and we define Cm(r) as 
the mean of these Cim(r) values. The quantity Cm(r) expresses the prevalence of repetitive 
patterns of length m in SN. Finally, we define approximate entropy of SN, for patterns of 
length m and similarity criterion r, as  
ApEn(m,r)=In[Cm(r)/Cm+1(r)] 
Thus, if we find similar patterns in a time series, ApEn estimates the logarithmic 
likelihood that the next intervals after each of the patterns will differ. Smaller values of 
ApEn imply greater likelihood that similar patterns of measurements will be followed by 
additional similar measurements. If the time series is highly irregular, the occurrence of 
similar patterns will not be predictive for the following measurements, and ApEn will be 
relatively large. 
4.7 Instrumentation for Test on Cadaver Spine 
 
In this approach of testing a cadaver spine, a six-degree-of-freedom motion 
tracking device, Optotrak 3020 (Northern Digital Inc. Onterio, Canada) along with 7 
sensors were used. Each sensor was an L-shaped acrylic bracket with 3 infrared markers 
(Northern Digital Inc. Onterio, Canada) that emits infrared signals. 6 out of 7 sensors 
 53 
were used to attach to different levels of the cadaver specimen (T12-L5). The seventh 
sensor was attached to a specially designed stylus which served as a digitizer (Figure 
4.4). Each infrared marker was attached to a stober unit. Every stober unit could 
accommodate only 2 sensors at a time. Hence, a total of 4 stober units were used. Each 
stober unit was daisy-chained to the preceding stober unit in order to provide additional 
ports. Every stober unit incorporates an LED that indicates power and status information. 
These stober units were supplied with a stober voltage of 6.4V through the System 
Control Unit (SCU).  
 
Figure 4. 4 Optotrak SCU and Stober Units with Markers on Sensors 
The System Control Unit provides interference between the Optotrak 3020, the 
stober unit and the markers. It allows quick and easy triggering and synchronization of 
third-party devices. One port of the SCU was connected to the Optotrak Data Acquisition 
Unit II (ODAU II); it is used for true?synchronous capture of data from third-party 
hardware devices such as force plates, pressure transducers, etc. Another port was 
connected to the Optotrak 3020 unit that detects the perturbation of infrared markers on 
triggering. A computer is connected to the SCU through the PCI card. The PCI card 
provides the port in order to connect the SCU to the computer.  
 54 
 
Figure 4. 5 Optotrak 3020 Unit 
In our effort to understand the inter-vertebral variability due only to injuries made 
in the cadaver spine, we wanted to keep other parameters such as motion control, distance 
of movement, etc. fixed. If we had controlled the cadaver motion by hand, there would 
have been variability due to injuries as well as due to hand movement.  Hence, we used a 
servo motor (CMC Cleveland, OH, USA) to control the flexion-extension movement of 
the cadaver spine. This servo motor was triggered by another computer using the 
Labview program (National Instrument, Austin Texas). The Labview program provided 
an interface to write the program in order to control the motion of servo motor. The servo 
motor had a plunger that could move forward and backward by the displacement and 
time/cycle given as input to the Labview program. A specially designed U-joint yoke was 
attached to the tip of a plunger. The U-joint yoke provided a flexible connection between 
the cadaver specimen and the motor plunger. A steel cup of a 6-inch diameter was firmly 
fixed to the leveled table through a force transducer, which could measure the forces and 
moments with respect to global axes. This cup could accommodate the PVC cup that was 
fixed to the sacrum of the cadaver spine that provided the stability and support to the 
 55 
cadaver spine. Eight Philips-head screws were used to restrict the motion between the 
PVC cup and the steel cup. 
On triggering the servo motor using the computer, the servo motor plunger 
pushed and pulled the specimen with a defined distance for a complete 10 cycles. 
Simultaneously a pulse was sent out to the SCU, which then triggered the Optotrak 3020 
unit. On activating, the Optotrak 3020 system precisely detected the markers? movement 
and sent those signals to the computer, which then saved them on the computer RAM. 
Motion monitor (Innovative Sports Training, Inc. IL) software was used to display, 
record, play back and manage the data. 
 
 
Figure 4. 6 Schematic Diagram of the Set-up 
 
 
 56 
 
4.8 Potting the Specimen 
 
In order to carry out a test on cadaver specimen, it needs to be prepared so that the 
specimen can accommodate the steel cup firmly fixed at the sacrum level. Two different-
sized PVC cups of diameters 5 inches and 4 inches were fixed using (8d) screws at the 
T12 and S1 levels, respectively. In order to get permanent fixed joint, additional filling 
material (Bondo body filler) was used to fill the cup. On solidification, the screws and the 
filling material provided a firm joint between the cups and the cadaver spine. 
4.9 Digitization 
 
As mentioned before, there are several vertebral levels in the lumbar spine. Each 
of these vertebrae is interconnected to each other through a facet joint that provides 
flexibility between these vertebrae. As a result each vertebra moves differently with 
respect to the others and the global coordinate system. In order to estimate the motion of 
different vertebrae, we need to align each intervertebral local coordinate axis with the 
global coordinate axis. This is achieved through the process of digitization.  
 
Figure 4. 7 Cadaver Test Set-up and Stylus with Markers 
 57 
For digitization 3 small Philips headed screws (2d) were drilled at the anterior and 
2 sides, making 90 degree angles with each other at each vertebral level of the cadaver 
spine (T12-L5). These screws were placed anteriorly, right laterally, and left laterally to 
determine the local coordinate axes for that particular vertebral level. The specimen was 
fixed to the steel cup so that it was vertical. A stylus with a copper ball (diameter = 3mm) 
was used as digitizer. It was used to record the position of each of those small screws 
during the initial set-up for the Motion Monitor software. Each position data was 
recorded for a period of 2 seconds. A total of 18 data samples were collected for each 
small screw. 
  Alongside the small screws on the anterior/posterior sides of the specimen another 
screw (6d) was drilled at each vertebral level on the specimen. These screws facilitated 
the attachment of each sensor at different vertebral levels. In order to fix the yoke to the 
spine, these screws were drilled on the posterior side for the upper vertebral levels (T12-
L2). For the lower levels (L3-L5), these screws were fixed on the anterior side of the 
cadaver spine. 
4.10 Data Acquisition  
 
The cadaver specimen with two PVC cups attached at each end was mounted 
vertically on the steel cup using four 20d-sized screws. A yoke extending from the servo 
motor plunger was attached at the L1 level of the cadaver, so that it could restrict its 
motion in the anterior-posterior direction. A sensor with 3 markers each was fixed to each 
screw at different vertebral levels on the cadaveric spine. 
The cadaveric specimen was tested for the effects of speed and injuries to the 
intervertebral segments. Upon running the Labview program, which also triggered the 
 58 
Optotrak unit, the servo motor reciprocated the plunger in the anterior-posterior direction 
for 10 complete cycles. This resulted in intervertebral movements in the specimen which 
was then recorded by the Optotrak unit. The same procedure was repeated at a higher 
plunger speed and was recorded. On completion of this test, the specimen was then cut 
about an inch deep on the right side of the disc at the L4-L5 level using an incision blade. 
The complete aforementioned procedure was then repeated twice at 2 different speeds. 
The test progressed with making another inch-deep cut at the L4-L5 level on the right 
hand capsule of the cadaveric specimen. Triggering the servo motor resulted in 
intervertebral movement in the cadaveric spine which was recorded simultaneously by 
the Optotrak 3020 unit. 
4.11 Data Analysis 
 
Motion Monitor software was used for recording and extracting important 
information from the data. The software provided a platform to set up the experiment, a 
global coordinate system, and markers, as well as selecting the frame rate. It allowed a 
real-time visual feedback. It provided an option to trigger with an external pulse from 
ODAU to record the signal without actually hitting the record button.  
The experiment was first set up using the digitizing process, by recording the 
signal for two seconds at each vertebral level. Following digitization, the data was 
recorded for a period of 20 seconds at a frequency of 100 Hz, which was triggered 
externally. A preference file with certain parameters was created and saved. After loading 
the preference file, the recorded data displayed the graphs of flexion-extension. The data 
was then saved to the hard drive of the computer in .exp format. This saved data, when 
exported loading the same preference file, extracted only the intended information 
 59 
described by the preference file, such as the flexion- extension data. This file was in 
ASCII format which was then converted to the Excel file and loaded in the MATLAB 
R2008 to process it further for linear and nonlinear methods.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 60 
 
 
 
5. Results 
 
 
Linear tools such as mean, standard deviation, and coefficient of variation, and 
nonlinear techniques such as approximate entropy were calculated to estimate the 
neuromuscular control of stability during repeated trunk flexion-extension movements in 
human subjects and the cadaveric spine. Phase plane plots for flexion-extension time 
series data were also calculated. 
5.1 Human Volunteer Study 
 
  Subjects included 5 males with no history of lower back pain and with average 
mean age = 31.5 years, SD = 12.32 years; mean weight = 64.47 kg, SD = 7.184 kg; and 
mean height = 67 inches, SD = 2.371 inches. All subjects were physically examined by a 
professional clinician before entering the study. Each participant performed seven 
subtests with different starting positions and directions. The phase plane plots for each 
subtest are shown below. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 61 
Phase Plane Plots for Subject #1 
 
RA 
 
0 5 10 15 20 25 30
- 3 0 0
- 2 0 0
- 1 0 0
0
100
200
300
400
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( d e g )
V e l o c i t y ( d e g / s e c )
A c c e l e r a t i o n ( d e g / s e c 2 )
 
 
Figure 5. 1 Displacement Time Series with Velocity and Acceleration 
- 2 5 - 2 0 - 1 5 - 1 0 -5 0 5 10 15 20
- 8 0
- 6 0
- 4 0
- 2 0
0
20
40
60
80
D i s p l a c e m e n t
V
e
lo
c
it
y
 
Figure 5. 2 Phase Plane Plot 
 62 
- 2 5
- 2 0
- 1 5
- 1 0
-5
0
5
10
15
20
- 8 0
- 6 0
- 4 0
- 2 0
0
20
40
60
80
- 4 0 0
- 2 0 0
0
200
400
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
Figure 5. 3 3-D Phase Plane Plot 
Figure 5.1 shows the displacement time series with velocity and acceleration for 
the lumbar flexion-extension movement, with neutral as starting position and forward as 
starting movement. As seen from figure 5.2 and 5.3, the phase plane plots show some 
elegant patterns with some divergence in the trajectories. Hence, the structure can be 
thought of having less variability.  
RB 
0 5 10 15 20 25 30 35
- 3 0 0
- 2 0 0
- 1 0 0
0
100
200
300
400
500
600
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( d e g )
V e l o c i t y ( d e g / s e c )
A c c e l e r a t i o n ( d e g / s e c 2 )
 
Figure 5. 4 Displacement Time Series with Velocity and Acceleration 
 63 
- 3 0 - 2 5 - 2 0 - 1 5 - 1 0 -5 0 5 10 15 20
- 8 0
- 6 0
- 4 0
- 2 0
0
20
40
60
80
D i s p l a c e m e n t
V
e
lo
c
it
y
 
Figure 5. 5 Phase Plane Plot 
- 3 0
- 2 5
- 2 0
- 1 5
- 1 0
-5
0
5
10
15
20
- 8 0
- 6 0
- 4 0
- 2 0
0
20
40
60
80
- 5 0 0
0
500
1000
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
Figure 5. 6 3-D Phase Plane Plot 
Figure 5.4 shows a time series with forward as starting position and direction of 
first motion. Compared to phase plots of RA (Neutral-Forward) movement, these phase 
plane plots show more divergence in their trajectories. It can be concluded that this may 
be due to more variability. 
 
 
 64 
RC 
0 5 10 15 20 25 30 35
- 4 0 0
- 3 0 0
- 2 0 0
- 1 0 0
0
100
200
300
400
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( d e g )
V e l o c i t y ( d e g / s e c )
A c c e l e r a t i o n ( d e g / s e c 2 )
 
Figure 5. 7 Displacement Time Series with Velocity and Acceleration 
- 2 0 - 1 5 - 1 0 -5 0 5 10 15 20
- 6 0
- 4 0
- 2 0
0
20
40
60
80
D i s p l a c e m e n t
V
e
lo
c
it
y
 
Figure 5. 8 Phase Plane Plot 
 65 
- 2 0
- 1 5
- 1 0
-5
0
5
10
15
20
- 6 0
- 4 0
- 2 0
0
20
40
60
80
- 5 0 0
0
500
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
Figure 5. 9 3-D Phase Plane Plot 
The time series for forward target position and backward starting movement is 
shown in figure 5.7. The phase plane plots for the RC subtest has more periodic pattern 
and less divergence in the trajectories. It can be said that its variability is near to the 
variability of the RA subtest. 
RD 
0 5 10 15 20 25 30
- 4 0 0
- 3 0 0
- 2 0 0
- 1 0 0
0
100
200
300
400
500
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( d e g )
V e l o c i t y ( d e g / s e c )
A c c e l e r a t i o n ( d e g / s e c 2 )
 
Figure 5. 10 Displacement Time Series with Velocity and Acceleration 
 66 
- 3 0 - 2 5 - 2 0 - 1 5 - 1 0 -5 0 5 10 15 20
- 8 0
- 6 0
- 4 0
- 2 0
0
20
40
60
80
100
D i s p l a c e m e n t
V
e
lo
c
it
y
 
Figure 5. 11 Phase Plane Plot 
- 3 0
- 2 5
- 2 0
- 1 5
- 1 0
-5
0
5
10
15
20
- 8 0
- 6 0
- 4 0
- 2 0
0
20
40
60
80
100
- 5 0 0
0
500
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
Figure 5. 12 3-D Phase Plane Plot 
The divergence in the trajectories is evidence of variability in the structure of the 
time series. It can be seen that the trajectories separate out with greater distance on the 
left side of the phase plane plot (Figure 5.8). These phase plane plots are plotted for 
backward target position and first motion as forward. 
 
 
 67 
RE 
0 5 10 15 20 25 30 35
- 4 0 0
- 3 0 0
- 2 0 0
- 1 0 0
0
100
200
300
400
500
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( d e g )
V e l o c i t y ( d e g / s e c )
A c c e l e r a t i o n ( d e g / s e c 2 )
 
Figure 5. 13 Displacement Time Series with Velocity and Acceleration 
- 4 0 - 3 0 - 2 0 - 1 0 0 10 20
- 1 5 0
- 1 0 0
- 5 0
0
50
100
D i s p l a c e m e n t
V
e
lo
c
it
y
 
Figure 5. 14 Phase Plane Plot 
 68 
- 4 0
- 3 0
- 2 0
- 1 0
0
10
20
- 1 5 0
- 1 0 0
- 5 0
0
50
100
- 5 0 0
0
500
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
Figure 5. 15 3-D Phase Plane Plot 
Figure 5.10 shows the time series for backward target position with first motion as 
backward. As seen in the phase plots for RD, RE phase plots show similar patterns, and it 
could be said that the variability of the two is similar. 
RF 
0 5 10 15 20 25 30
- 4 0 0
- 3 0 0
- 2 0 0
- 1 0 0
0
100
200
300
400
500
600
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( d e g )
V e l o c i t y ( d e g / s e c )
A c c e l e r a t i o n ( d e g / s e c 2 )
 
Figure 5. 16 Displacement Time Series with Velocity and Acceleration 
 69 
- 2 5 - 2 0 - 1 5 - 1 0 -5 0 5 10 15 20
- 8 0
- 6 0
- 4 0
- 2 0
0
20
40
60
80
100
D i s p l a c e m e n t
V
e
lo
c
it
y
 
Figure 5. 17 Phase Plane Plot 
- 2 5
- 2 0
- 1 5
- 1 0
-5
0
5
10
15
20
- 8 0
- 6 0
- 4 0
- 2 0
0
20
40
60
80
100
- 4 0 0
- 2 0 0
0
200
400
600
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
Figure 5. 18 3-D Phase Plane Plot 
Figure 5.17 and 5.18 show phase plane plots for a time series with neutral as 
target position and forward motion as first motion. These plots show a maximum 
separation of the trajectories, and it can be claimed that this time series has maximum 
variability. 
 
 
 70 
RG 
0 5 10 15 20 25 30
- 3 0 0
- 2 0 0
- 1 0 0
0
100
200
300
400
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( d e g )
V e l o c i t y ( d e g / s e c )
A c c e l e r a t i o n ( d e g / s e c 2 )
 
Figure 5. 19 Displacement Time Series with Velocity and Acceleration 
 
- 2 0 - 1 5 - 1 0 -5 0 5 10 15
- 6 0
- 4 0
- 2 0
0
20
40
60
80
D i s p l a c e m e n t
V
e
lo
c
it
y
 
Figure 5. 20 Phase Plane Plot 
 71 
- 2 0
- 1 5
- 1 0
-5
0
5
10
15
- 6 0
- 4 0
- 2 0
0
20
40
60
80
- 5 0 0
0
500
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
Figure 5. 21 3-D Phase Plane Plot 
Figure 5.19 shows a time series for neutral target position with initial backward 
movement. The phase plane plots show uniform separation of the trajectories, and it can 
be said that there is a moderate amount of variability in this time series. 
Although interpretations are made by observing the phase plane plots, they are 
very subjective. Therefore, there is always an element of doubt when coming to 
conclusions based on these observations. As said earlier, phase plane plots are just 
representations of the state space and should be used as a guideline and not as a method 
to distinguish the time series. 
In order to get more assured results, linear and nonlinear techniques were used. 
Mean, standard deviation, coefficient of variation, and approximate entropy were 
computed using MATLAB R2008a for each of the time series data (See Appendix B for 
programs).  
Variation in velocity during flexion?extension tasks in human subjects and 
cadaveric specimens may help us to understand the effect of LBP on lumbar spine 
movements and help us to distinguish between healthy subjects and LBP subjects. Hence, 
 72 
change in velocity over directional movement of lumbar spine for complete flexion-
extension task was estimated in this study. Each cycle was divided into four sections 
depending on lumbar spine movement with respect to neutral position. Mean, SD, and 
CV was then estimated for complete ten cycles. 
Table 5.1 below shows the values for each of the subtests performed on a subject. 
 
Table 5. 1 Linear and Nonlinear Measures 
Subject # 1 
Test 
Condition  Mean SD CV (%) ApEn 
RA (NF) 
Total ROM 25.16346 2.938947 11.67942 
0.32 
Angular 
Velocity 
Forward 
(V1) 
-33.9267 6.729789 -19.8362 
Angular  
Velocity 
Backward 
(V2) 
48.71503 6.514936 13.37357 
Angular  
Velocity 
Backward 
(V3) 
24.15682 4.532175 18.76147 
Angular 
Velocity 
Forward 
(V4) 
-20.9836 5.04977 -24.0654 
RB (FF) 
Total ROM 28.80245 5.18246 17.99312 
0.2463 
Angular 
Velocity 
Forward 
(V1) 
-30.3168 6.789041 -22.3937 
Angular 
Velocity 
Backward 
(V2) 
39.69756 9.539105 24.02945 
Angular 
Velocity 
Backward 
35.19676 10.7602 30.57157 
 73 
(V3) 
Angular 
Velocity 
Forward 
(V4) 
-26.6625 5.309553 -19.9139 
RC (FB) 
Total ROM 16.91608 1.606685 9.497971 
0.2441 
Angular 
Velocity 
Backward 
(V1) 
40.28309 4.4422 11.02746 
Angular 
Velocity 
Forward 
(V2) 
-29.5164 3.606675 -12.2192 
Velocity 
Forward 
(V3) 
-23.352 7.452419 -31.9134 
Angular 
Velocity 
Backward 
(V4) 
19.57475 6.829137 34.88749 
RD (BF) 
Total ROM 10.76923 3.038641 28.21595 
0.2222 
Angular 
Velocity 
Forward 
(V1) 
-38.9616 6.185729 -15.8765 
Angular 
Velocity 
Backward 
(V2) 
51.20373 8.025324 15.67332 
Angular 
Velocity 
Backward 
(V3) 
11.37887 6.493213 57.06378 
Angular 
Velocity 
Forward 
(V4) 
-8.43334 4.009456 -47.5429 
RE (BB) 
Total ROM 19.11888 5.711879 29.87559 
0.1945 
Angular 
Velocity 
Backward 
(V1) 
16.24144 5.849587 36.01644 
Angular 
Velocity 
Forward 
-15.6025 5.188226 -33.2525 
 74 
(V2) 
Angular 
Velocity 
Forward 
(V3) 
-50.5192 14.54363 -28.7883 
Angular 
Velocity 
Backward 
(V4) 
46.69309 11.17078 23.92385 
RF (NF) 
Total ROM 28.01136 5.323007 19.00303 
0.1917 
Angular 
Velocity 
Forward 
(V1) 
-31.0571 7.949072 -25.595 
Angular 
Velocity 
Backward 
(V2) 
40.10805 10.43883 26.02678 
Angular 
Velocity 
Backward 
(V3) 
26.76865 7.03382 26.27634 
Angular 
Velocity 
Forward 
(V4) 
-20.4708 7.899119 -38.5872 
RG (NB) 
Total ROM 22.16171 2.642313 11.92287 
0.3065 
Angular 
Velocity 
Backward 
(V1) 
28.66511 7.5263 26.25596 
Angular 
Velocity 
Forward 
(V2) 
-20.5705 4.063352 -19.7533 
Angular 
Velocity 
Forward 
(V3) 
-26.257 5.744034 -21.8762 
Angular 
Velocity 
Backward 
(V4) 
22.17435 5.369799 24.21626 
 
 
 75 
 
Figure 5. 22 Mean Values for Subtests 
As seen in figure 5.22, the mean values for total range of motion increased for the 
first two subtests and then reduced to the minimum value for the RD subtest. The mean 
again rose for the RF subtest and eventually decreased, revealing an M-shaped function 
of variability.   
 
Figure 5. 23 Standard Deviation for Subtests 
 76 
In the case of standard deviation, the graph shows a rise and decline to the 
minimum for the RC subtest, followed by another rise and decline in values. The shape 
was similar to the graph of mean values, but the variability was lowest for RC. 
 
Figure 5. 24 Coefficient of Variation for Subtests 
The coefficient of variation also shows a similar pattern, with the RC subtest 
having minimum variability. 
 
Figure 5. 25 Approximate Entropy for Subtests 
 77 
On the other hand, ApEn measurements show a completely different pattern. The 
values show a gradual decline before a rise for final subtest, showing that complexity 
decreased as the test progressed. 
 
 
 
Figure 5. 26 Variations in Angular Velocity 
Figure 5.26 shows variation in angular velocity of lumbar spine movement. The 
mean values show that the subject moved quickly while moving from neutral to 
backward, while CV values show angular velocity while moving from backward to 
neutral. SD values were almost uniform for all cases. 
 
 78 
 
Figure 5. 27 Variations in Angular Velocity 
As before, the mean of angular velocity is maximum for backward movement 
from front position to neutral position, while CV shows more angular velocity for 
neutral-to-backward movement. 
 
Figure 5. 28 Variations in Angular Velocity 
Figure 5.28 show the angular velocity was greater for neutral-to-backward 
movement, while CV was greater for front position to neutral position, with SD 
remaining more or less uniform. 
 79 
 
Figure 5. 29 Variations in Angular Velocity 
In figure 5.29, the mean value of angular velocity was maximum for front 
position to backward position, while CV was greater for neutral to backward position. SD 
was higher for forward to backward movement. 
 
Figure 5. 30 Variations in Angular velocity 
The mean value for this subtest shows a higher value of angular velocity while 
moving from neutral to forward position, while CV and SD are highest for neutral-to-
backward movement and neutral-to-forward movement respectively. 
 80 
 
Figure 5. 31 Variations in Angular Velocity 
Figure 5.31 shows higher value of mean of angular velocity while moving from 
forward to neutral position, with CV at its maximum for backward to neutral position. SD 
values remained uniform. 
 
Figure 5. 32 Variations in Angular Velocity 
Mean, SD, and CV for this subtest correlate well and show similar results, with 
angular velocity being at its maximum for neutral-to-backward movement. 
 
 81 
Table 5. 2 Linear and Nonlinear Measures 
Subject # 2 
Similar measures were computed for remaining subjects. 
Test 
Condition  Mean SD CV (%) ApEn 
RA (NF) 
Total ROM 12.41608 1.227959 9.89007 
0.3247 
Angular 
Velocity 
Forward 
(V1) 
-11.9891 2.426826 -20.242 
Angular 
Velocity 
Backward 
(V2) 
10.40492 2.123236 20.40609 
Angular 
Velocity 
Backward 
(V3) 
11.39617 4.007765 35.16764 
Angular 
Velocity 
Forward 
(V4) 
-11.3373 4.197697 -37.0255 
RB (FF) 
Total ROM 15.23287 1.667311 10.94548 
0.2834 
Angular 
Velocity 
Forward 
(V1) 
-18.4673 2.835653 -15.355 
Angular 
Velocity 
Backward 
(V2) 
15.06725 2.066963 13.71825 
Angular 
Velocity 
Backward 
(V3) 
16.61443 2.739847 16.49077 
Angular 
Velocity 
Forward 
(V4) 
-15.2235 1.680097 -11.0362 
RC (FB) Total ROM 12.19406 2.106081 17.27137 0.3517 
 82 
Angular 
Velocity 
Backward 
(V1) 
13.54575 3.753753 27.71166 
Angular 
Velocity 
Forward 
(V2) 
-14.8997 4.951029 -33.2291 
Angular 
Velocity 
Forward 
(V3) 
-12.2544 2.962727 -24.1769 
Angular 
Velocity 
Backward 
(V4) 
11.12784 2.537487 22.80304 
RD (BF) 
Total ROM 8.15035 2.757983 33.83883 
0.2487 
Angular 
Velocity 
Forward 
(V1) 
-10.7115 5.781007 -53.9701 
Angular 
Velocity 
Backward 
(V2) 
9.599615 3.189955 33.23003 
Angular 
Velocity 
Backward 
(V3) 
14.89396 2.437355 16.36472 
Angular 
Velocity 
Forward 
(V4) 
-10.5298 3.052364 -28.988 
RE (BB) 
Total ROM 11.14685 2.094973 18.7943 
0.3273 
Angular 
Velocity 
Backward 
(V1) 
18.80549 6.628579 35.24809 
Angular 
Velocity 
Forward 
(V2) 
-16.6217 3.477871 -20.9237 
Angular 
Velocity -13.5795 5.260933 -38.7417 
 83 
Forward 
(V3) 
Angular 
Velocity 
Backward 
(V4) 
9.607045 2.551823 26.56199 
RF (NF) 
Total ROM 8.933566 2.349299 26.29744 
0.275 
Angular 
Velocity 
Forward 
(V1) 
-8.87902 3.418681 -38.5029 
Angular 
Velocity 
Backward 
(V2) 
7.304628 3.012098 41.23547 
Angular 
Velocity 
Backward 
(V3) 
18.10217 4.15362 22.94542 
Angular 
Velocity 
Forward 
(V4) 
-14.273 2.823111 -19.7794 
RG (NB) 
Total ROM 11.49755 0.928005 8.07133 
0.2634 
Angular 
Velocity 
Backward 
(V1) 
19.35916 2.527288 13.05474 
Angular 
Velocity 
Forward 
(V2) 
-19.9396 1.871287 -9.38476 
Angular 
Velocity 
Forward 
(V3) 
-8.97626 3.015788 -33.5974 
Angular 
Velocity 
Backward 
(V4) 
6.650135 2.724628 40.97101 
 
 84 
 
Figure 5. 33 Mean Values for Subtests 
The mean value shows a rise and then reaches its minimum value at RD. The 
mean values rise again for RE and remain almost uniform. A similar pattern was 
observed for the previous participant. 
 
Figure 5. 34 Standard Deviation for Subtests 
The standard deviation graph shows a gradual increase in values with the 
maximum for the RD subtest, which shows an increase in variability followed by a 
decline with the minimum at final subtest RG. 
 85 
 
Figure 5. 35 Coefficient of Variation for Subtests 
The coefficient of variance values show a similar pattern of variability as that of 
standard deviation. 
 
Figure 5. 36 Approximate Entropy for Subtests 
The values show complexity, rising and falling alternatively. The result can be 
correlated with the previous ApEn graph. 
 86 
 
Figure 5. 37 Variations in Angular Velocity 
The mean and SD of angular velocity is more or less uniform for all cases in this 
subtest, with CV for angular velocity being at its maximum for backward to neutral 
position. 
 
Figure 5. 38 Variations in Angular Velocity 
The mean, SD, and CV of angular velocity for this subtest is more or less similar 
throughout and shows a strong correlation with each other. Mean was maximum for 
 87 
neutral to forward movement, while SD and CV remained high for neutral to backward 
motion. 
 
Figure 5. 39 Variations in Angular Velocity 
In figure 5.39, the mean and CV of angular velocity is higher for backward-to-
neutral movement, while the SD values remain constant throughout. 
 
Figure 5. 40 Variations in Angular Velocity 
The mean and SD of angular velocity for this subtest is almost uniform while CV 
shows its maximum value for neutral-to-forward movement. 
 88 
 
Figure 5. 41 Variations in Angular Velocity 
Figure 5.41 shows a maximum value of mean and SD of angular velocity for 
neutral-to-backward movement of lumbar spine, while CV was highest for neutral-to-
forward motion. 
 
Figure 5. 42 Variations in Angular Velocity 
In this case, the mean of angular velocity was highest for neutral-to-backward 
movement while CV showed higher values for forward-to-neutral movement. 
 89 
 
Figure 5. 43 Variations in Angular Velocity 
Figure 5.43 showed higher values of mean for backward-to-neutral movement, 
while CV was higher for forward-to-neutral movement. 
 
Table 5. 3 Linear and Nonlinear Measures 
Subject # 3 
Test 
Condition  Mean SD CV (%) ApEn 
RA (NF) 
Total ROM 36.73951 3.056539 8.319488 
0.2378 
Angular 
Velocity 
Forward 
(V1) 
-29.0413 5.056237 -17.4105 
Angular 
Velocity 
Backward 
(V2) 
48.12021 10.75091 22.34178 
Angular 
Velocity 
Backward 
(V3) 
52.99896 28.6905 54.13409 
Angular 
Velocity 
Forward 
-50.9074 28.60734 -56.1949 
 90 
(V4) 
RB (FF) 
Total ROM 28.51049 3.063703 10.74588 
0.3383 
Angular 
Velocity 
Forward 
(V1) 
-34.9918 12.57378 -35.9335 
Angular 
Velocity 
Backward 
(V2) 
43.68193 8.517696 19.49936 
Angular 
Velocity 
Backward 
(V3) 
71.21064 10.73873 15.08024 
Angular 
Velocity 
Forward 
(V4) 
-63.4849 8.360227 -13.1688 
RC (FB) 
Total ROM 34.42308 3.107815 9.028289 
0.2714 
Angular 
Velocity 
Backward 
(V1) 
56.40662 13.6145 24.13636 
Angular 
Velocity 
Forward 
(V2) 
-59.9368 10.26253 -17.1222 
Angular 
Velocity 
Forward 
(V3) 
-32.4561 10.47805 -32.2837 
Angular 
Velocity 
Backward 
(V4) 
41.95977 8.2506 19.66312 
RD (BF) 
Total ROM 14.45804 2.228288 15.4121 
0.236 
Angular 
Velocity 
Forward 
(V1) 
-49.8195 10.30889 -20.6925 
Angular 
Velocity 
Backward 
68.94928 7.31262 10.6058 
 91 
(V2) 
Angular 
Velocity 
Backward 
(V3) 
37.75827 5.450713 14.43581 
Angular 
Velocity 
Forward 
(V4) 
-45.4241 6.416788 -14.1264 
RE (BB) 
Total ROM 12.70629 2.736122 21.53359 
0.2031 
Angular 
Velocity 
Backward 
(V1) 
34.86804 11.47596 32.91256 
Angular 
Velocity 
Forward 
(V2) 
-41.7084 10.18281 -24.4143 
Angular 
Velocity 
Forward 
(V3) 
-46.3261 4.326236 -9.33866 
Angular 
Velocity 
Backward 
(V4) 
64.24768 6.847152 10.65743 
RF (NF) 
Total ROM 37.49126 1.898663 5.064281 
0.2017 
Angular 
Velocity 
Forward 
(V1) 
-32.046 5.813095 -18.1399 
Angular 
Velocity 
Backward 
(V2) 
50.79108 6.88803 13.5615 
Angular 
Velocity 
Backward 
(V3) 
57.08177 8.580514 15.03197 
Angular 
Velocity 
Forward 
(V4) 
-61.5926 5.16197 -8.38083 
 92 
RG (NB) 
Total ROM 35.8951 4.15122 11.56486 
0.1993 
Angular 
Velocity 
Backward 
(V1) 
59.91014 18.08741 30.19089 
Angular 
Velocity 
Forward 
(V2) 
-60.4231 9.658142 -15.9842 
Angular 
Velocity 
Forward 
(V3) 
-39.6265 8.803663 -22.2166 
Angular 
Velocity 
Backward 
(V4) 
57.10554 9.258534 16.21302 
 
 
Figure 5. 44 Mean Values for Subtests 
The mean values remained almost uniform for the first three subtests and decrease 
sharply at RD, followed by sharp rise, showing that variability in motion was maximum 
for initial and final subtests. 
 93 
 
Figure 5. 45 Standard Deviation for Subtests 
The standard deviation graph shows uniform values for most of the subtests, with 
a sharp rise for the final subtest. 
 
Figure 5. 46 Coefficient of Variation for Subtests 
This time, the values of the coefficient of variation do not correlate with standard 
deviation values and show a gradual rise followed by a sharp decline for the final 
subtests, describing a decrease in variability for initial and final subtests. 
 94 
 
Figure 5. 47 Approximate Entropy for Subtests 
The approximate entropy continues to show a similar pattern, with a gradual 
decrease in complexity with the subtests. 
 
Figure 5. 48 Variations in Angular Velocity 
In this subtest the mean, SD, and CV are higher for backward-to-neutral 
movement. 
 95 
 
Figure 5. 49 Variations in Angular Velocity 
In figure 5.49 the mean of angular velocity depicts higher values for neutral-to-
backward movement, while CV is maximum for neutral-to-forward motion. SD of 
angular velocity remains uniform.  
 
Figure 5. 50 Variations in Angular Velocity 
The mean values of angular velocity for the FB subtest show higher angular 
velocity for backward-to-neutral movement, while CV was maximum for neutral-to-
forward direction. 
 96 
 
Figure 5. 51 Variations in Angular Velocity 
Figure 5.51 shows higher values of mean of angular velocity for forward-to-
neutral lumbar spine movement, while SD and CV of velocity are higher for neutral-to-
forward movement. 
 
Figure 5. 52 Variations in Angular Velocity 
Figure 5.52 shows mean value of angular velocity at maximum for forward-to-
neutral movement, while SD and CV show higher values for neutral-to-backward 
movement. 
 97 
 
Figure 5. 53 Variations in Angular Velocity 
For the NF subtest, the mean of angular velocity was maximum for backward-to-
neutral position, while CV shows higher values for neutral-to-forward movement. 
 
Figure 5. 54 Variations in Angular Velocity 
The mean values for this subtest are higher for neutral-to-backward as well as 
backward-to-neutral movements. SD and CV of angular velocity had higher values for 
neutral-to-backward movement. 
 
 98 
  Table 5. 4 Linear and Nonlinear Measures 
 
Subject # 4 
Test 
Condition  Mean SD CV (%) ApEn 
RA (NF) 
Total ROM 33.77797 2.284872 6.764385 
0.153 
Angular 
Velocity 
Forward 
(V1) 
-14.2121 2.774334 -19.5209 
Angular 
Velocity 
Backward 
(V2) 
16.93159 3.2964 19.46894 
Angular 
Velocity 
Backward 
(V3) 
34.69094 7.296463 21.03277 
Angular 
Velocity 
Forward 
(V4) 
-22.6464 2.868871 -12.6681 
RB (FF) 
Total ROM 16.28671 2.457953 15.09177 
0.2392 
Angular 
Velocity 
Forward 
(V1) 
-15.0804 7.142536 -47.3629 
Angular 
Velocity 
Backward 
(V2) 
16.03047 3.765113 23.48722 
Angular 
Velocity 
Backward 
(V3) 
23.12471 2.786331 12.04915 
Angular 
Velocity 
Forward 
(V4) 
-20.8657 4.401871 -21.0962 
RC (FB) Total ROM 39.02972 3.831347 9.816486 0.2334 
 99 
Angular 
Velocity 
Backward 
(V1) 
42.80203 7.865588 18.37667 
Angular 
Velocity 
Forward 
(V2) 
-45.5397 7.544675 -16.5672 
Angular 
Velocity 
Forward 
(V3) 
-14.8892 9.539037 -64.0668 
Angular 
Velocity 
Backward 
(V4) 
15.36966 5.489797 35.7184 
RD (BF) 
Total ROM 19.38199 2.771309 14.29837 
0.2073 
Angular 
Velocity 
Forward 
(V1) 
-19.5606 4.270326 -21.8313 
Angular 
Velocity 
Backward 
(V2) 
19.69281 4.273012 21.69833 
Angular 
Velocity 
Backward 
(V3) 
23.82195 4.271951 17.93283 
Angular 
Velocity 
Forward 
(V4) 
-23.1147 5.567579 -24.0867 
RE (BB) 
Total ROM 25.45804 2.998514 11.77826 
0.2065 
Angular 
Velocity 
Backward 
(V1) 
18.51214 7.067774 38.17912 
Angular 
Velocity 
Forward 
(V2) 
-26.0891 7.496292 -28.7335 
Velocity 
Forward -27.9967 6.604632 -23.5908 
 100 
(V3) 
    
Angular 
Velocity 
Backward 
(V4) 
29.43636 8.561432 29.08455 
RF (NF) 
Total ROM 30.36189 4.518289 14.88145 
0.1873 
Angular 
Velocity 
Forward 
(V1) 
-17.9121 8.960335 -50.0238 
Angular 
Velocity 
Backward 
(V2) 
13.66101 6.092742 44.59949 
Angular 
Velocity 
Backward 
(V3) 
42.74039 7.127892 16.67718 
Angular 
Velocity 
Forward 
(V4) 
-31.3397 4.781819 -15.258 
RG (NB) 
Total ROM 35.74301 2.16374 6.053603 
0.1909 
Angular 
Velocity 
Backward 
(V1) 
51.36497 12.12502 23.60562 
Angular 
Velocity 
Forward 
(V2) 
-34.3373 3.970481 -11.5632 
Angular 
Velocity 
Forward 
(V3) 
-22.5267 5.884504 -26.1223 
Angular 
Velocity 
Backward 
(V4) 
18.51593 5.072447 27.39504 
 101 
 
Figure 5. 55 Mean Values for Subtests 
The mean values show a gradual increase, with increase in variability at first and 
second subtests. 
 
Figure 5. 56 Standard Deviation for Subtests 
This standard deviation graph shows consistent variability, with a sudden rise for 
couple of tests, eventually depicting the M-shaped pattern also seen for first two subjects. 
 102 
 
Figure 5. 57 Coefficient of Variation for Subtests 
The variability shown by the coefficient of variation graph was higher for 
alternate subtests. 
 
Figure 5. 58 Approximate Entropy for Subtests 
As seen before, ApEn values decreased gradually after a low complexity for first 
subtest. 
 103 
 
Figure 5. 59 Variations in Angular Velocity 
The mean, SD, and CV values of angular velocity were maximum for neutral-to-
backward movement. 
 
Figure 5. 60 Variations in Angular Velocity 
This subtest shows higher values for CV and SD of angular velocity for neutral-
to-forward direction, while the mean was maximum for neutral-to-backward direction. 
 104 
 
Figure 5. 61 Variations in Angular Velocity 
The mean of angular velocity in figure 5.61 shows a higher value for neutral-to-
backward and backward-to-neutral movements. The CV values are maximum for neutral-
to-forward direction, with SD being uniform. 
 
Figure 5. 62 Variations in Angular Velocity 
Figure 5.62 shows mean, SD, and CV values of angular velocity are more or less 
uniform for all four sections of the cycle, with mean being maximum for neutral-to-
backward movement and CV higher for backward-to-neutral movement. 
 105 
 
Figure 5. 63 Variations in Angular Velocity 
For the BB subtest, mean of angular velocity has maximum value for forward-to-
neutral movement, while CV is higher for neutral-to-backward movement. 
 
Figure 5. 64 Variations in Angular Velocity 
Figure 5.64 shows higher mean values for neutral-to-backward movement, while 
CV is maximum for neutral-to-forward movement. 
 106 
 
Figure 5. 65 Variations in Angular Velocity 
The mean of the angular velocity shows higher values while moving backward 
from neutral position, while CV is maximum while moving from forward-to-neutral 
direction. 
Table 5. 5 Linear and Nonlinear Measures 
Subject # 5 
Test 
Condition  Mean SD CV (%) ApEn 
RA (NF) 
Total ROM 5.83479 1.606169 27.52745 
0.1493 
Angular 
Velocity 
Forward 
(V1) 
-5.42485 2.209598 -40.731 
Angular 
Velocity 
Backward 
(V2) 
6.615675 1.904245 28.78384 
Angular 
Velocity 
Backward 
(V3) 
6.08009 3.136002 51.57821 
Angular 
Velocity 
Forward 
-5.24073 2.785412 -53.1494 
 107 
(V4) 
RB (FF) 
Total ROM 5.512587 1.054312 19.12554 
0.2044 
Angular 
Velocity 
Forward 
(V1) 
-6.68542 2.715881 -40.624 
Angular 
Velocity 
Backward 
(V2) 
9.493815 3.447403 36.31209 
Angular 
Velocity 
Backward 
(V3) 
8.985679 3.201779 35.63202 
Angular 
Velocity 
Forward 
(V4) 
-8.01148 3.324271 -41.4938 
RC (FB) 
Total ROM 7.188811 1.195438 16.62915 
0.2036 
Velocity 
Backward 
(V1) 
9.869443 1.546233 15.66687 
Angular 
Velocity 
Forward 
(V2) 
-8.25132 1.583974 -19.1966 
Angular 
Velocity 
Forward 
(V3) 
-6.35202 1.81681 -28.6021 
Angular 
Velocity 
Backward 
(V4) 
10.37606 4.416373 42.56312 
RD (BF) 
Total ROM 4.867832 1.589348 32.65001 
0.1958 
Angular 
Velocity 
Forward 
(V1) 
-7.41343 1.088973 -14.6892 
Angular 
Velocity 
Backward 
(V2) 
9.248163 1.925909 20.82478 
 108 
Angular 
Velocity 
Backward 
(V3) 
8.447628 3.017001 35.71418 
Angular 
Velocity 
Forward 
(V4) 
-7.79239 2.896464 -37.1704 
RE (BB) 
Total ROM 7.31049 1.404553 19.21284 
0.225 
Angular 
Velocity 
Backward 
(V1) 
8.004571 2.694866 33.66659 
Angular 
Velocity 
Forward 
(V2) 
-8.95576 2.372645 -26.4929 
Angular 
Velocity 
Forward 
(V3) 
-7.82318 1.506173 -19.2527 
Angular 
Velocity 
Backward 
(V4) 
8.243921 2.697588 32.72215 
RF (NF) 
Total ROM 7.907692 0.825406 10.43801 
0.2072 
Angular 
Velocity 
Forward 
(V1) 
-7.92154 1.936026 -24.44 
Angular 
Velocity 
Backward 
(V2) 
10.00056 1.840471 18.40367 
Angular 
Velocity 
Backward 
(V3) 
8.662963 1.654945 19.10368 
Angular 
Velocity 
Forward 
(V4) 
-7.15391 1.371799 -19.1755 
RG (NB) Total ROM 9.94965 0.756835 7.606651 0.2026 
 109 
Angular 
Velocity 
Backward 
(V1) 
10.96658 1.490266 13.58916 
Angular 
Velocity 
Forward 
(V2) 
-9.68718 1.520841 -15.6995 
Angular 
Velocity 
Forward 
(V3) 
-8.42253 2.65833 -31.5621 
Angular 
Velocity 
Backward 
(V4) 
8.325599 1.89399 22.74899 
 
 
 
Figure 5. 66 Mean Values for Subtests 
The graph shows a gradual increase in variability as the tests progress, with 
maximum variability at the final subtest. 
 110 
 
Figure 5. 67 Standard Deviation for Subtests 
The standard deviation graph shows decline in variability after high variability for 
the first test. The variability increases for RD, followed by a gradual decline before 
reaching minimum at the last subtest. 
 
Figure 5. 68 Coefficient of Variation for Subtests 
The coefficient of variation graph shows a gradual decrease in variation for first 
half of the subtests, followed by a sudden rise at RD. After that, there is decline in 
variability.  
 111 
 
Figure 5. 69 Approximate Entropy for Subtests 
This graph depicts an initial increase in ApEn values, after which the values are 
fairly uniform throughout. 
 
Figure 5. 70 Variations in Angular Velocity 
The mean and SD for angular velocity for the NF subtest shows uniform values 
throughout, while CV is higher for neutral-to-backward and backward-to-neutral 
movements. 
 112 
 
Figure 5. 71 Variations in Angular Velocity 
For subtest FF, the mean, SD, and CV are uniform for all four sections of the 
cycle and show correlation among them. 
 
Figure 5. 72 Variations in Angular Velocity 
Figure 5.72 shows higher mean values of angular velocity for movement in 
neutral-to-backward and forward-to-neutral directions, while CV is maximum for 
forward-to-neutral movement. 
 113 
 
Figure 5. 73 Variations in Angular Velocity 
Figure 5.73 shows higher mean of angular velocity values for backward 
movement, but the CV of angular velocity shows higher values for neutral-to-backward 
and backward-to-neutral directions. 
 
Figure 5. 74 Variations in Angular Velocity 
For the BB subtest, the mean and SD of angular velocity is uniform, while the CV 
is higher for neutral-to-backward and forward-to-backward movements. 
 114 
 
Figure 5. 75 Variations in Angular Velocity 
Figure 5.75 depicts uniform values for mean, SD, and CV of angular velocity. 
 
 
Figure 5. 76 Variations in Angular Velocity 
Figure 5.76 shows higher mean of angular velocity values for neutral-to-backward 
movement, while CV is higher for neutral-to-forward direction. 
As seen from the above results, the linear techniques do not show any particular 
pattern and it is difficult to conclude anything from these results. However, ApEn have 
 115 
similar pattern in all the subtests for all participants and mostly shows decreases in 
complexity as the tests progressed. 
Variations in angular velocity graphs do not show any peculiar trends as well. 
Although they show variation in angular velocity from subtest to subtest, they fail to 
show any particular pattern in healthy subjects that can be attributed to them.  
5.2 Cadaver Study 
 
Variability in a cadaveric specimen was also analyzed. The effect of speed and 
injury was analyzed for intervertebral segments. A second-order Butterworth filter with a 
cut-off frequency of 10 Hz (Sample rate = 100; Nyquist frequency = sample rate/2 = 50 
Hz; Wn = cut off freq/Nyquist freq = 10/50 = 0.2 Hz), similar to the one used for the 
subject test, was used. But the velocity and acceleration graph showed noise. 
To reduce the effect of noise, the velocity and acceleration data was filtered, but 
the results did not show any improvement. Hence, in order to nullify this error, a filter 
with a cut-off frequency of 1Hz was used to estimate velocity and acceleration data. 
The phase plane plot for each cadaveric test is shown below. 
 
 
 
 
 
 
 
 
 116 
Cadaver without Cut at Slower Speed: 
 
 
0 10 20 30 40 50 60 70 80 90
- 1 0
0
10
20
30
40
50
60
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( m m )
V e l o c i t y ( m m / s e c )
A c c e l e r a t i o n ( m m / s e c 2 )
 
Figure 5. 77 Displacement Time Series with Velocity and Acceleration 
52 53 54 55 56 57 58 59 60
-4
-3
-2
-1
0
1
2
3
4
D i s p l a c e m e n t
V
e
lo
c
it
y
 
Figure 5. 78 Phase Plane Plot 
 
 
 117 
52
53
54
55
56
57
58
59
60
-4
-3
-2
-1
0
1
2
3
4
-4
-2
0
2
4
6
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
Figure 5. 79 3-D Phase Plane Plot 
As seen from figures 5.43 and 5.44, the plot shows an elegant pattern with 
minimal separation in the trajectories. Hence, it can be concluded that this is a highly 
periodic system.  
Cadaver without Cut at Higher Speed: 
 
 
0 5 10 15 20 25 30 35 40 45 50
- 1 0
0
10
20
30
40
50
60
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( m m )
V e l o c i t y ( m m / s e c )
A c c e l e r a t i o n ( m m / s e c 2 )
 
Figure 5. 80 Displacement Time Series with Velocity and Acceleration 
 
 118 
52 53 54 55 56 57 58 59 60
-6
-4
-2
0
2
4
6
D i s p l a c e m e n t
V
e
lo
c
it
y
 
 
Figure 5. 81 Phase Plane Plot 
 
52
53
54
55
56
57
58
59
60
-6
-4
-2
0
2
4
6
- 1 0
-5
0
5
10
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
 
Figure 5. 82 3-D Phase Plane Plot 
As seen from phase plots in figures 5.46 and 5.47, the trajectories do not separate 
out much and are similar to those in the first case. 
 
 
 
 
 
 
 119 
Cadaver Disc Cut at Slower Speed: 
 
 
0 10 20 30 40 50 60 70 80 90
- 1 0
0
10
20
30
40
50
60
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( m m )
V e l o c i t y ( m m / s e c )
A c c e l e r a t i o n ( m m / s e c 2 )
 
 
Figure 5. 83 Displacement Time Series with Velocity and Acceleration 
 
 
 
 
 
Figure 5. 84 Phase Plane Plot 
 
52 53 54 55 56 57 58 59 60
-4
-3
-2
-1
0
1
2
3
4
D i s p l a c e m e n t
V
e
lo
c
it
y
 120 
52
53
54
55
56
57
58
59
60
-4
-3
-2
-1
0
1
2
3
4
-4
-2
0
2
4
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
Figure 5. 85 3-D phase Plane Plot 
The phase plane plots show peculiar patterns without much divergence in 
trajectories. 
Cadaver Disc Cut at Higher Speed: 
0 5 10 15 20 25 30 35 40 45 50
- 2 0
- 1 0
0
10
20
30
40
50
60
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( m m )
V e l o c i t y ( m m / s e c )
A c c e l e r a t i o n ( m m / s e c 2 )
 
Figure 5. 86 Displacement Time Series with Velocity and Acceleration 
 
 121 
52 53 54 55 56 57 58 59 60
-8
-6
-4
-2
0
2
4
6
8
D i s p l a c e m e n t
V
e
lo
c
it
y
 
 
Figure 5. 87 Phase Plane Plot 
 
52
53
54
55
56
57
58
59
60
-8
-6
-4
-2
0
2
4
6
8
- 1 5
- 1 0
-5
0
5
10
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
 
Figure 5. 88 3-D Phase Plane Plot 
Similar to the slower speed phase plane plots, the higher speed phase plane plots 
depict less separation of trajectories and more of the system?s periodicity. 
 
 
 
 122 
Cadaver Right Capsule Cut at Slower Speed: 
0 10 20 30 40 50 60 70 80 90
- 1 0
0
10
20
30
40
50
60
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( m m )
V e l o c i t y ( m m / s e c )
A c c e l e r a t i o n ( m m / s e c 2 )
 
Figure 5. 89 Displacement Time Series with Velocity and Acceleration 
51 52 53 54 55 56 57 58 59 60
-4
-3
-2
-1
0
1
2
3
4
5
D i s p l a c e m e n t
V
e
lo
c
it
y
 
Figure 5. 90 Phase Plane Plot 
 
 123 
51
52
53
54
55
56
57
58
59
60
-4
-3
-2
-1
0
1
2
3
4
5
-4
-2
0
2
4
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
Figure 5. 91 3-D Phase Plane Plot 
Figures 5.55 and 5.56 show phase plane plots for intervertebral spine movement 
when injured on the right capsule at L4-L5 level. The plot shows some elegant pattern 
depicting the periodicity of the system. 
Cadaver Right Capsule Cut at Higher Speed: 
0 5 10 15 20 25 30 35 40 45 50
- 2 0
- 1 0
0
10
20
30
40
50
60
T i m e ( s e c )
D
is
p
,V
e
l,
A
c
c
 
 
D i s p l a c e m e n t ( m m )
V e l o c i t y ( m m / s e c )
A c c e l e r a t i o n ( m m / s e c 2 )
 
Figure 5. 92 Displacement Time Series with Velocity and Acceleration 
 
 
 124 
51 52 53 54 55 56 57 58 59 60
-8
-6
-4
-2
0
2
4
6
8
D i s p l a c e m e n t
V
e
lo
c
it
y
 
 
Figure 5. 93 Phase Plane Plot 
 
51
52
53
54
55
56
57
58
59
60
-8
-6
-4
-2
0
2
4
6
8
- 1 5
- 1 0
-5
0
5
10
D i s p l a c e m e n t
V e l o c i t y
A
c
c
e
le
ra
ti
o
n
 
 
Figure 5. 94 3-D Phase Plane Plot 
 
Phase plane plots of this test show similar patterns as seen in the above test. In 
this case, the speed and injuries to the cadaveric spine had a minimal effect on 
intersegmental stability. 
Mean, standard deviation, coefficient of variation, and approximate entropy were 
computed for each of the time series data. Below, Table 5.1 shows the values for each of 
the tests performed on a cadaveric specimen. 
 125 
 
Table 5. 6 Linear and Nonlinear Measures 
 
Test 
Condition  Mean SD CV (%) ApEn 
Cadaver 
without Cut 
Slower 
Speed 
Total ROM 
6.468531 
 
0.039696 
 
0.613672 
 
0.0477 
 
Velocity 
Forward 
(V1) 
-1.76277 0.062718 -3.55792 
Velocity 
Backward 
(V2) 
1.773309 0.064234 3.622261 
Velocity 
Backward 
(V3) 
2.152 0.114584 5.32452 
Velocity 
Forward 
(V4) 
-2.14217 0.133844 -6.24804 
Cadaver 
without Cut 
Higher 
Speed 
Total ROM 
6.483916 
 
0.036232 
 
0.558797 
 
0.065 
 
Velocity 
Forward 
(V1) 
-3.53525 0.113045 -3.19766 
Velocity 
Backward 
(V2) 
3.669537 0.141752 3.862946 
Velocity 
Backward 
(V3) 
4.204689 0.151203 3.596049 
Velocity 
Forward 
(V4) 
-4.03208 0.372195 -9.23084 
Cadaver 
Disc Cut at 
Slower 
Speed 
Total ROM 
7.79021 
 
0.050265 
 
0.645238 
 
0.0641 
 
Velocity 
Forward 
(V1) 
-2.11924 0.081573 -3.84915 
Velocity 
Backward 
(V2) 
2.154336 0.066777 3.099663 
 126 
Velocity 
Backward 
(V3) 
2.633066 0.20561 7.808761 
Velocity 
Forward 
(V4) 
-2.4881 0.228225 -9.17269 
Cadaver 
Disc Cut at 
Higher 
Speed 
Total ROM 
7.744406 
 
0.067084 
 
0.866221 
 
0.0615 
 
Velocity 
Forward 
(V1) 
-4.34084 0.102533 -2.36206 
Velocity 
Backward 
(V2) 
4.23037 0.165887 3.921328 
Velocity 
Backward 
(V3) 
4.897316 0.265935 5.43021 
Velocity 
Forward 
(V4) 
-5.16134 0.282337 -5.47022 
Cadaver 
Right 
Capsule Cut 
at Slower 
Speed 
Total ROM 
8.206294 
 
0.054791 
 
0.66767 
 
0.061 
 
Velocity 
Forward 
(V1) 
-2.20426 0.085847 -3.89461 
Velocity 
Backward 
(V2) 
2.352996 0.108934 4.629577 
Velocity 
Backward 
(V3) 
2.686655 0.047786 1.778625 
Velocity 
Forward 
(V4) 
-2.63781 0.211405 -8.01443 
Cadaver 
Right 
Capsule Cut 
at Higher 
Speed 
Total ROM 
8.25 
 
0.045477 
 
0.551236 
 
0.0771 
 
Velocity 
Forward 
(V1) 
-4.45809 0.146835 -3.29367 
Velocity 
Backward 
(V2) 
4.692214 0.160693 3.424669 
Velocity 5.24866 0.239875 4.570218 
 127 
Backward 
(V3) 
Velocity 
Forward 
(V4) 
-5.22231 0.377887 -7.23602 
 
 
Figure 5. 95 Mean for Tests 
As seen in Figure 5.95, the mean values show uniform values throughout the tests 
irrespective of speed and injuries made to the specimen. 
 
Figure 5. 96 Standard Deviation for Tests 
 128 
Although SD shows some increase in value for the disc cut and tested at higher 
speed, it remains more or less uniform throughout the tests. 
 
Figure 5. 97 Coefficient of Variation for Tests 
Similarly to the SD graph, CV shows maximum variability for the cadaver tested 
at the higher speed and injured at the disc. However, the values are more or less uniform 
for all other tests. 
 
Figure 5. 98 Approximate Entropy for Tests 
 129 
As seen from the linear techniques, ApEn shows a similar trend. It shows a small 
rise and then the values remain constant thereafter. Its values are more or less uniform 
across the tests irrespective of speed and injuries made to the specimen. 
 
Figure 5. 99 Variations in Velocity 
The mean and the SD of velocity remain uniform across four sections, while CV 
is maximum for backward-to-neutral movement. 
  
Figure 5. 100 Variations in Velocity 
 
 130 
Similarly to the previous test, the mean and SD of velocity remains uniform 
throughout the tests, while CV is maximum for backward-to-neutral movement. 
 
Figure 5. 101 Variations in Velocity 
Figure 5.101 shows uniform values for mean and SD throughout the test, while 
the CV is maximum for backward-to-neutral motion of L4-L5 level. 
 
Figure 5. 102 Variations in Velocity 
As above, the mean and SD are uniform throughout the tests, while CV of 
velocity is higher for backward-to-neutral movement. 
 131 
 
Figure 5. 103 Variations in Velocity 
Even after injuring the specimen at several positions, the mean and SD of velocity 
remain uniform across the tests, while CV is greater for backward-to-neutral position. 
 
Figure 5. 104 Variations in Velocity 
Figure 5.104 shows similar trends as before. The mean and SD are uniform 
throughout, while CV is maximum for backward-to-neutral movement. 
 132 
For this particular cadaveric test, the linear and nonlinear techniques show good 
correlation with each other. Their values are more or less uniform throughout the test 
irrespective of the injuries made and the speed of the flexion-extension. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 133 
 
 
 
6. Discussions 
 
 
Musculoskeletal pathologies and injuries may result in non-periodic kinematic 
and kinetic quantities due to increase in instability. Studies show that both linear and 
nonlinear techniques have been applied toward understanding this condition in various 
movement analyses. Although spinal stability has been estimated using linear and 
nonlinear techniques, few previous studies aimed to compare these quantities for the 
control of stability during dynamic trunk movements in human subjects and cadaveric 
specimens. The present study is one of the few studies that were conducted using linear 
and nonlinear techniques.   
The phase plane plot is a tool that provides a perspective on some of the 
phenomena that can be found in nonlinear systems. There are many phenomena that are 
peculiar to nonlinear systems, and these plots can be helpful to understanding and 
visualizing what might happen in a nonlinear system. In this study, these plots provide 
information regarding how flexion-extension time series data were organized within a 
state space. Results show that they can distinguish between the periodic and chaotic 
movements, and can be used as a reference to determine variability involved in human 
movements. Although these plots are useful, they do not quantify the variability and fail 
to reveal the underlying dynamic structure of the system. It is difficult to distinguish 
between two subtests with almost the same variability, and the interpretation of results is 
very subjective. These findings are in agreement with previous studies (Yamada, 1995; 
 134 
Buzzi et al., 2003) that used phase plane plots to analyze human movements. Those 
studies concluded that although phase plane plots were effective in distinguishing the 
human motion under analysis, linear and nonlinear techniques were important to 
completely understand and quantify these motions. 
In order to quantify motor variability during lumbar flexion-extension movement, 
stability was examined. Stability is defined as the sensitivity of the system to small 
perturbations (Dingwell and Cusumano, 2000). If motor variability is related to stability, 
then the increased variability should result in increased instability (Buzzi et al, 2003). 
Stability can be estimated using linear and nonlinear tools. 
In case of human subject testing, the traditional linear measures of variability, 
which generally provide estimates of the average magnitude of variations across the 
subtests, demonstrate significant differences among the subtests. However, from these 
results we cannot conclude that the starting direction and position had any effect on the 
stability of the lumbar spine. The linear methods for a particular subject subtest do not 
correlate well with each other. For example, the mean graph diagram for the first subject 
shows minimum variability for the RD subtest, while the variability was lower for the RC 
subtest in case of standard deviation and coefficient of variation. Moreover, from these 
results we cannot distinguish between inherent variability and increases in noise within 
the system. This observation is also supported by previous studies in human movement 
analysis in which linear measure failed to reveal the underlying structure of the human 
motion under analysis. Specifically, Dingwell et al. (2001) found that traditional 
measures of variability poorly predicted the stability in human movement analysis as 
compared to nonlinear techniques. Buzzi et al. (2003) concluded that traditional linear 
 135 
tools cannot distinguish between inherent variability and increases in noise within the 
system. 
Estimating the nonlinear dynamics of the system may help us to understand 
changes in actual motor control mechanisms that result from change in starting position 
and direction. The nonlinear analysis depicts an increase in musculoskeletal stability as 
the complexity of the system decreases with the subtests. The ApEn values, irrespective 
of starting directions and position, increase for first couple of subtests and then show a 
gradual decline for most of the subjects. This may be due to the adaptation of the human 
body system to the flexion-extension tasks as the test progressed. During the final stages 
of the test, subjects may have discovered strategies for stable trunk movement.  
Furthermore, the ApEn and linear techniques do not correlate. This conclusion is also 
supported by previous studies using linear and nonlinear techniques to quantify 
variability. Specifically, Vaillancourt and colleagues (2001) used standard deviation and 
ApEn to quantify regularity of tremor in healthy and Parkinson?s disease patients. The 
result showed standard deviation values were very similar across the healthy subjects and 
the Parkinson?s subjects. However, the ApEn values differed significantly and the authors 
were able to distinguish between the two groups based on the complexity in the system. 
In another study (Slifkin and Newell, 2000), determining the effect of increases in 
continuous isometric force production on the variability of motor responses, the results 
showed standard deviation values increased exponentially as a function of force level 
production. However, the ApEn values showed completely different results, revealing an 
inverted U-shaped function. A similar kind of study (Stergiou, 2004), used tibial 
accelerometer data from 17 healthy subjects who walked on motorized treadmills at 5 
 136 
different speeds. The author found that the standard deviation values increased with 
speed, while the ApEn values showed a U-shaped function. In another study by 
Karmarkar et al. (2006), the authors tried to determine the relationship between ApEn 
and standard deviation. The results showed no correlation between ApEn and standard 
deviation for minimum toe clearance data of healthy elderly and healthy young group. On 
the basis of these studies and the present study findings it would be fair to conclude that 
ApEn and linear methods do not correlate well and extreme care should be taken when 
interpreting the results.  
An in vitro biomechanical model of the lumbar spine was analyzed for variability. 
The cadaveric lumbar spinal unit was loaded in flexion and extension, and the anatomic 
elements were transected from the posterior direction. The phase plane graphs were 
plotted for different tests and reveal similar patterns and periodicity. The trajectories of 
the plot show minimal separation, showing that all the tests exhibited similar variability. 
The incisions at various positions and speeds had very littlt effect on intervertebral 
movement of cadaveric spine. These results are also conformed from the linear and 
nonlinear analysis, as their values are similar throughout the tests. Several factors may 
contribute to this lack of variation in observed intervertebral variability. First, the use of 
an electric motor to control the specimen?s movement may have resulted in the specimen 
to move with strict constrain. Second, the incision made on the capsule and disc may 
have minimum effect on the intervertebral stability of the cadaveric spine. The results 
show a slight increase in variability values as the cadaver specimen was transected at 
different positions and flexed at different speeds, but they are statistically insignificant. 
This study shows a positive correlation between linear and nonlinear techniques. 
 137 
Angular velocity and acceleration charts for human subjects and cadavers were 
also estimated. These charts are used to understand the variations in speed as the subject 
or cadaveric specimen flexed and extended the lumbar spine. The angular velocity charts 
for the cadaveric specimen show a more or less uniform pattern. On other hand, all of the 
remaining charts show different patterns from which we cannot get any conclusive 
results. 
In order to remove the unwanted contamination from the original signal, filter is 
used. The present study used a second-order Butterworth filter with a cut-off frequency of 
10 Hz. Applying this filter to data from human subjects yielded the desired results, but 
the same filter failed to remove noise in the signal when applied to data from the 
cadaveric specimen. This may be due to the signal?s low power. In order for filter to be 
more effective, the cut-off frequency was changed to 1 Hz.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 138 
  
 
 
7. Conclusions 
 
 
The results of this study were examined from a dynamic perspective. Linear and 
nonlinear techniques were used to analyze and interpret the variability in human trunk 
movements. Results demonstrate that ApEn values computed in this study fundamentally 
quantify different aspects of locomotor behavior than traditional measures of variability. 
Mean of variability was minimum for RD subtest for all five subjects. All graphs showed 
initial rise then a gradual decline followed by rise revealing ?M? shape; more variability 
at the beginning and end part of the test. The standard deviation in human study was 
lower for RG subtest. SD showed higher variability in motion after and before initial and 
final subtests respectively. This may be due to fatigue each participant experienced as the 
test progressed. Similar to SD, CV values showed lower variability for first and last 
subtests in most of the cases. RD subtest was higher in most subjects, while RG showed 
lower values. On other hand in case of angular velocity graph for subtest (NF), mean of 
variability was higher while moving backwards. SD remained uniform for most subjects 
and CV was higher for second part of test (moving neutral to backward and neutral to 
forward). Subtest (FF) showed greater mean going backwards while SD was constant 
throughout. CV showed minimum variability while moving backward to neutral position. 
In case of subtest (FB) the mean and SD was higher for moving neutral to backward in 
most cases. CV had higher values for second part of test (neutral to forward and forward 
to neutral position). (BF) subtest showed mean was higher while moving from forward to 
 139 
neutral in most cases. SD was higher initially and decreased gradually. CV on other hand 
was higher for neutral to backward motion and backward to neutral motion. For (BB) 
subtest mean was seen to be higher for second part of the test (i.e. movement from neutral 
to forward and forward to neutral). In contrast to that, SD and CV were seen to be on 
higher side for neutral to backwards and backwards to neutral movement. (NF) subtest 
had greater mean of variability for neutral to backward and backward to neutral 
movement. SD was uniform for most of the subtests. CV was higher for neutral to 
forward and forward to neutral movement. For (NB) subtest mean and SD were higher 
moving neutral to backward for most subjects. CV was higher for neutral to backward 
movement and backward to neutral movement. 
The mean of variability for cadaver study indicated higher variability when the 
disc was cut. The variability increased marginally as the specimen was injured on 
capsule. SD and CV on other hand, showed higher variability for cadaver disc cut at 
higher speed and decreased gradually. These techniques failed to give good results as 
they did not show increase in variability as one would expect after injuring the specimen. 
Thus it is seen SD and CV was not the technique to understand variability in cadaveric 
specimen. ApEn on other hand showed increased in variability as the specimen was 
injured and pushed and pulled at higher speed. The angular velocity was higher for the 
neutral to backward and backward to neutral movements in most of the cases. 
As seen from above linear and nonlinear analysis, each of them had different 
aspect of variability. Linear techniques were good in most cases but did not show good 
results in other case, while ApEn on other hand showed better results in both human and 
cadaveric experiments. 
 140 
They concentrated on understanding how variations in these movement patterns 
change over time. It is proposed that because these measures specifically quantify the 
variability of the locomotor system, they are more relevant for examining the 
neuromuscular control during flexion-extension tests. Based on the results of this 
investigation, both linear and nonlinear measures can quantify variability during human 
bodily movement, but it?s the nonlinear measures that provide insightful understanding of 
the system.  
To the best of the author?s knowledge this is the first study to determine the 
variability of sitting motions on an unstable surface using approximate entropy. Linear 
and nonlinear dynamic systems analyses were successfully applied to trunk flexion-
extension data, which can serve as control data for further studies and understanding 
kinematics and kinetics of low back pain patients. This study is restricted to collecting 
and describing normative data on 5 healthy volunteer subjects. Such information can be 
useful in establishing the baseline data on the stability of the lumbar spine on healthy 
subjects. We need to gather more data on several healthy volunteer subjects with different 
age, gender, race, and ethnicity to establish a database on the variability of healthy 
population. Further research should incorporate other nonlinear measures to understand 
the influence of direction, speed, etc., during flexion-extension tests on the human lumbar 
spine. Future research should undertake in quantifying the variability in low back pain 
patients and changes in these as a function of low back pain treatments.  
 
 
 
 
 
 141 
 
 
 
 
References 
 
 
1. eMedicine Mechanical Low Back Pain Hills, Everett C. MD Penn State Hershey 
Rehabilitation Hospital Nov 2009. 
http://emedicine.medscape.com/article/310353-overview 
2. Luo, X., Pietrobon, R., Sun, S.X., Liu, G.G. & Hey, L. 2004 ?Estimates and 
patterns of direct health care expenditures among individuals with back pain in 
the United States.? Spine, 29 (1):79-86. 
3. Andersson, G.B. 1999. ?Epidemiological features of chronic low-back pain.? 
Lancet, 354 (9178):581-5. 
4. Hoogendoorn, W.E., van Poppel, M.N., Bongers, P.M., Koes, B.W., & Bouter, 
L.M. 2000. ?Systematic review of psychosocial factors at work and private life as 
risk factors for back pain.? Spine, 25 (16):2114-25. 
5. Hartvigsen J., Lings S., Leboeuf-Yde C., & Bakketeig L. 2003. ?Psychosocial 
factors at work in relation to low back pain and consequences of low back pain; a 
systematic, critical review of prospective cohort studies? Occupational 
Environmental Medicine, 61 (1): e2. 
6. Mulholland, R.C. 2008. ?The myth of lumbar instability: the importance of 
abnormal loading as a cause of low back pain.? European Spine Journal, 17:619?
625. 
7. Melloh M., Elfering A., Egli Presland C., Roeder C., Barz T., Rolli Salathe 
C., Tamcan O., Mueller U., & Theis J.C. 2009. ?Identification of prognostic 
 142 
factors for chronicity in patients with low back pain: a review of screening 
instruments.? International Orthopaedics, 33 (2): 301?313. 
8. Bergman, S. 2007. ?Management of musculoskeletal pain.? Best Practice & 
Research Clinical Rheumatology, 21 (1): 153-166. 
9. Papadakis, N.C., Christakis, D.G., Tzagarakis, G.N., Chlouverakis, G.I., 
Kampanis, N.A., Stergiopoulos, K.N., & Katonis, P.G. 2009. ?Gait variability 
measurements in lumbar spinal stenosis patients: part A. Comparison with healthy 
subjects.? Physiological Measurement, 30: 1171?1186. 
10. Hausdorff, J.M. 2007. ?Gait dynamics, fractals, and falls: Finding meaning in the 
stride-to-stride fluctuations of human walking.? Human Movement Science, 
26:555?589. 
11. Goldberger, A.L., Amaral, L., Hausdorff, J.M., Ivanov, P., Peng, C.K. & Stanley, 
H.E. 2002 ?Fractal dynamics in physiology: Alterations with disease and aging.? 
Proceedings of the National Academy of Sciences, 99:2466?2472. 
12. Ho, K.K., Moody, G.B., Peng, C.K., Mietus, J.E., Larson, M.G. & Levy, D. 1997. 
?Predicting survival in heart failure case and control subjects by use of fully 
automated methods for deriving nonlinear and conventional indices of heart rate 
dynamics.? Circulation, 96:842?848. 
13. Hausdorff, J.M., Herman, T., Baltadjieva, R., Gurevich, T., & Giladi, N. 2003. 
?Balance and gait in older adults with systemic hypertension.? American Journal 
of Cardiology, 91:643?645. 
14. Stergiou, N. 2004. ?Innovative Analysis of Human Movement.? Human Kinetics. 
 143 
15. Fitzgerald, G.K., Wynveen, K.J., Rheault, W. & Rothchild, B. 1983. ?Objective 
assessment with establishment of normal values for lumbar spinal range of 
motion.? Physical Therapy, 63:1776-1781. 
16. Buzzi, U.H., Stergiou, N., Kurz, M.J., Hageman, P.A., & Heidel, J. 2003.  
?Nonlinear dynamics indicates aging affects variability during gait.? Clinical 
Biomechanics, 18:435?443. 
17. Trudelle-Jackson, E., Fleisher, L.A., Borman, N., Morrow, J.R., & Frierson, G.M. 
2010. ?Lumbar spine flexion and extension extremes of motion in women of 
different age and racial groups? Spine, 35 (16):1539?1544. 
18. Van Herp, G., Rowe, P., Salter, P. & Paul, J.P. 2000. ?Three-dimensional lumbar 
spinal kinematics: A study of range of movement in 100 healthy subjects aged 20 
to 60+ years.? Rheumatology, 39:1337-1340. 
19. Hausdorff, J.M., Zemany, L., Peng, C.K., & Goldeberger, A.L. 1999. ?Maturation 
of gait dynamics: stride-to-stride variability and its temporal organization in 
children.? Journal of Applied Physiology, 36(3): 1040?1047. 
20. Dingwell, J.B., Cusumano, J.P., Cavanagh, P.R., & Sternad, D. 2001 ?Local 
dynamic stability versus kinematic variability of continuous overground and 
treadmill walking.? Journal of Biomechanical Engineering, 123:27-32. 
21. Garcia-Alsina, J., Garcia Almazan, C., Moranta Mesquida, J., & Pleguezuelos 
Cobo, E. 2005. ?Angular position, range of motion and velocity of arm elevation: 
A study of consistency of performance.? Clinical Biomechanics, 20: 932?938. 
22. Brown, M. D., Holmes, D.C., & Heiner, A.D. 2002.  ?Measurement of cadaver 
lumbar spine motion segment stiffness.? Spine, 27(9): 918 ? 922. 
 144 
23. Hitchon, P.W., Eichholz, K., Barry, C., Rubenbauer, P., Ingalhalikar, A., 
Nakamura, S., Follett, K., Lim, T.H., & Torner, J. 2005. ?Biomechanical studies 
of an artificial disc implant in the human cadaveric spine.? Journal of 
Neurosurgery: Spine, 2:339?343. 
24. Stergiou, N., Harbourne, R.T., Cavanaugh, J.T. 2006. ?Optimal movement 
variability: A new theoretical perspective for neurologic physical therapy.? 
Journal of Neurologic Physical Therapy, 30 (3): 120-129. 
25. Hadders-Algra, M. 2007. ?Putative neural substrate of normal and abnormal 
general movements.? Neuroscience and Biobehavioral Reviews, 31:1181?1190. 
26. Karmakar, C.K., Khandoker, A.H., Begg, R.K., Palaniswami, M., & Taylor, S. 
2006. ?Understanding ageing effects by approximate entropy analysis of gait 
variability.? Conf Proc IEEE Eng Med Biol Soc. 2007:1965-1968. 
27. Khandoker, A.H., Palaniswami, M., Begg, R.K. 2008. ?A comparative study on 
approximate entropy measure and poincar? plot indexes of minimum foot 
clearance variability in the elderly during walking.? Journal of NeuroEngineering 
and Rehabilitation, 5:4. 
28. Pincus, S. 1994 ?Approximate entropy (ApEn) as a complexity measure.? Chaos, 
5(1):110-117. 
29. Pincus, S. & Golberger, A.L. 1994. ?Physiological time-series analysis: What 
does regularity quantify?? American Journal of Physiology, 266(4):H1643-1656. 
30. Grassberger, P. & Procaccia, I. 1983. ?Characterization of strange attractors.? 
Physical Review Letters, 50(5):346-349. 
 145 
31. Harbourne, R.T. & Stergiou, N. 2009. ?Movement variability and use of nonlinear 
tools: Principles to Guide Physical Therapist Practice.? Physical Therapy, 89(3): 
267?282. 
32. Deng, K., Moore, A.W., & Nechyba, M.C. 1997. ?Learning to recognize time 
series: Combining ARMA models with Memory-based Learning.? Computational 
Intelligence in Robotics and Automation 246?250. 
33.  Smith, S.W. 1999. ?The Scientist and Engineer?s Guide to Digital Signal 
Processing.?  http://dspguide.com/pdfbook.htm. 
34. Wachowiak M.P., Rash G.S., Quesada P.M., & Desoky A.H. 2000. ?Wavelet 
based noise removal for biomechanical signals: a comparative study.?  IEEE 
Transactions on Biomedical Engineering, 47(3):360-368. 
35. Winter, D.A., Patla, A.E., Prince, F., Ishac, M., Gielo-Perczak, K. 1998. 
?Stiffness control of balance in quiet standing.? Journal of Neurophysiology, 
80:1211-1221. 
36. Moorhouse, K.M., & Granata, K.P. 2005. ?Trunk stiffness and dynamics during 
active extension exertions.? Journal of Biomechanics, 38:2000?2007. 
37. Lenth, R.V. 2001. ?Some practical guidelines for effective sample-size 
determination.? The American Statistician, 55:187-193. 
38. Sekiya, N., Nagasaki, H., Ito, H., & Furuna, T. 1997. ?Optimal walking in terms 
of variability in step length.? Journal of Orthopedic and Sports Physical Therapy, 
26(5):266-272. 
 146 
39. Buzzi, U.H., Stergiou, N., Kurz, M.J., Hageman, P.A., & Heidel, J. 2003. 
?Nonlinear dynamics indicates aging affects variability during gait.? Clinical 
Biomechanics, 18:435?443. 
40. Slifkin, A.B., & Newell, K.M. 2000. ?Variability and noise in continuous force 
production.? Journal of Motor Behavior, 32(2): 141?150. 
41. Vaillancourt, D.E., Slifkin, A.B., & Newell, K.M. 2001. ?Regularity of force 
tremor in Parkinson's disease.? Clinical Neurophysiology, 112:1594?1603. 
42. Harbourne, R.T. & Stergiou, N. 2009. ?Movement variability and the use of 
nonlinear tools: Principles to guide physical therapist practice.? Physical 
Therapy, 89(3): 267?282. 
43. Granata, K.P. & England, S.A. 2006. ?Stability of dynamic trunk movement.? 
Spine, 31(10):E271-276. 
44. Pincus, S.M., Gladstone, I.M., Ehrenkranz, R.A. 1991. ?A regularity statistic for 
medical data analysis.? Journal of Clinical Monitoring and Computing, 7:335?
345. 
45. University of Maryland Spine Center http://www.umm.edu/spinecenter/education/ 
46. Henry Gray. 1918. ?Anatomy of Human Body? http://www.bartleby.com/107/ 
47. John J. Buchanan Fay B. Horak. 2001 ?Transitions in a postural task: do the 
recruitment and suppression of degrees of freedom stabilize posture?? Exp Brain 
Res 139:482?494 
48. Jonathan B. Dingwell, Laura C. Marin. 2006. ?Kinematic variability and local 
dynamic stability of upper body motions when walking at different speeds.? 
Journal of Biomechanics 39:444?452 
 147 
49. Regina T. Harbourne, Nicholas Stergiou. 2003. ?Nonlinear Analysis of the 
Development of sitting Postural Control.? Dev Psychobiol. 42(4):368-77 
50. Pincus SM. 1990. ?Approximate entropy as a measure of system complexity.? 
Proc. Nati. Acad. Sci. 88: 2297-2301  
51. Pincus SM. 1995. ?Approximate entropy (ApEn) as a complexity measure.? 
Chaos 5(1):110-117 
52. Johannes D Veldhuis and Steven M Pincus. 1998. ?Orderliness of hormone 
release patterns: a complementary measure to conventional pulsatile and circadian 
analyses.? European Journal of Endocrinology 138: 358?362 
53. Justin J Kavanagh. 2009. ?Lower trunk motion and speed-dependence during 
walking.? Journal of NeuroEngineering and Rehabilitation 6: 9 
54. Kaplan, D.T., Glass, L., 1995. ?Understanding Nonlinear Dynamics.? Springer-
Verlag, New York, NY. 
55. OrthoIndy, (image on internet) 
Available from: http://www.indyspinemd.com/Normal/index.asp 
56. Inition, (image on internet)  
Available from: http://www.inition.co.uk 
57. Liberty user manual, 2008 (image on internet) 
Available from: 
http://www.polhemus.com/polhemus_editor/assets/LIBERTY%20Rev%20F%20
URM03PH156.pdf 
 
 
 148 
 
 
 
Appendices 
 
 
A. Technical Specifications 
A.1 Liberty Polhemus Specifications 
 
Figure A. 1 Liberty Polhemus Unit 
Table A. 1 Technical Specification of Polhemus Unit 
 
Degrees-of-Freedom 6DOF 
Number of Sensors 1-16 
Update Rate 240 Hz per sensor) 
Static Accuracy Position 0.03in 
Static Accuracy Orientation 0.15? RMS 
Latency 3.5ms 
Resolution Position at 30cm range 0.00015in 0.0004cm 
Resolution Orientation 0.0012? 
Range from Standard TX2 Source Up to 5 feet or 1.52 meters 
 149 
Extended Range Source Up to 15 feet or 4.6 meters 
Interface RS-232 or USB (both included) 
Host OS compatability GUI/SDK 2000/XP 
 
A.2 Magnetic Source   
The source is the device which produces the electromagnetic field and is normally 
the reference for the position and orientation measurements of the sensors. It is usually 
mounted in a fixed position to a non-metallic surface or stand that is located in close 
proximity to the sensors. The standard source is the 2? cube. 
 
Figure A. 2 Magnetic Source Diagram 
A.3 Sensors 
The sensor is the smaller device whose position and orientation is measured 
relative to the source. The sensor is dimensionally shown in Figure A.3, including the 
position of the electrical center. The sensor package provides two mounting holes for #4 
nylon screws in the event that sensor mounting is required.   
 
 150 
 
Figure A. 3 Sensor Diagram 
 
Table A. 2 Technical Specifications of Optotrak 3020 Unit 
A.4 Optotrack 3020 
Dimensions 1110 mm ?315 mm ? 215 mm 
Weight 40 kg 
Bracket Weight 5 kg 
Power Requirements 100/120 VAC, 60 Hz or 
220/240 VAC, 50 Hz 
Maximum Marker Rate 3500 Hz 
Frame Rate (raw) 750 Hz  
( 3D) 450 Hz  
(3D with optional hardware) 750 Hz 
(6D with optional hardware) 145 Hz 
RMS Accuracy at 2.25 m distance 0.1 mm for x, y coordinates 
0.15 mm for z coordinate 
3-D Resolution at 2.25 m distance 0.01 mm 
 151 
Sensor Resolution 1:200,000 
Supported Platforms  
Pentium Class PC MS Windows 95/98/NT/2000 
SGI (requires SCSI) Irix 5.x or higher (o32-bit),  
Irix 6.x (n32-bit, n64-bit) 
HP (requires SCSI) HP-UX 9.x or higher 
SUN (requires SCSI) Solaris 2.x or higher 
Linux support also available for Linux 2.x or higher 
 
Table A. 3 Technical Specifications of System Control Unit 
 
A.5 System Control Unit (SCU) 
Dimensions 297mm ?236mm ?91 mm 
Weight 3.4 kg 
Power Requirements 100-240 VAC 50/60 Hz, 1.25 A 
Connections for Host computer PCI card, SCSI Ethernet 
Number of Strober Ports 3 
 
 
 
 
 
 
 152 
A.6 Optotrak Data Acquisition Unit II 
The Optotrak Data Acquisition Unit II (ODAU II) is an optional component to the 
Optotrak system. The ODAU II allows you to collect analog and digital data that is 
synchronized with the Optotrak position data.  Pre-conditioned signals from devices such 
as EMG sensors, force plates, accelerometers, and strain gauges may be viewed in real-
time and stored to data files for later analysis. You have feedback control to the 
subject/object of study through 8 digital I/O channels or 2 analog output channels. 
 
 
Figure A. 4 Optotrak Data Acquisition Unit II 
 
 
 
 
 
 
 
 
 
 
 153 
B. MATLAB Programs 
 
B.1 MATLAB Program for Filtering Data 
clear all; 
clc; 
A=xlsread('Timeseries')/1000; 
framerate=100; 
Q=((length(A)/framerate)); 
W=(Q/length(A)); 
t=(0:W:Q-W)'; 
[m,w]=butter(1,10/50); 
disp=filtfilt(m,w,A); 
f = fittype('smoothingspline'); 
fit1 = fit(t,disp,f); 
[vel,accel] = differentiate(fit1,t); 
figure(1) 
hold on;         
plot(t,disp,'r'); 
plot(t,vel,'g'); 
plot(t,accel,'b'); 
legend('Displacement(deg)','Velocity(deg/sec)','Acceleration(deg/sec2)') 
xlabel('Time(sec)') 
ylabel('Disp,Vel,Acc') 
figure(2) 
hold on 
plot(disp,vel) 
xlabel('Displacement') 
ylabel('Velocity') 
figure(3) 
hold on 
plot3(disp,vel,accel) 
xlabel('Displacement') 
ylabel('Velocity') 
zlabel('Acceleration') 
 
 
 
 
 
 
 154 
B.2 MATLAB Program to Calculate Range of Motion and Velocity 
 
clear all 
clc 
A=xlsread('smooth',-1); 
plot(A) 
[x,y]=ginput(45); 
for i=1:length(x)-1 
    T(i)=x(i+1)-x(i); 
    R(i)=((y(i+1))-(y(i))); 
    len(i)=sqrt(1+(R(i)/T(i))^2); 
    clength(i)=((len(i)*y(i+1))-(len(i)*y(i))); 
end 
curvelength=((clength)'); 
ROM=(curvelength(1:2:end)); 
for k=1:length(T) 
Time(k)=T(k)/120; 
end 
for e=1:length(x)-1 
    Velo(e)=curvelength(e)/Time(e); 
end 
Velocity=Velo'; 
for y=2:length(x)-1 
    vel(y)=(Velo(y)-Velo(y-1)); 
     
end 
vel(1)=Velo(1); 
for u=1:length(x)-1 
    Accel(u)=vel(u)/Time(u); 
end 
Acceleration=Accel'; 
 
 
 
 
 
 
 
 155 
B.3 MATLAB Program to Calculate Approximate Entropy (ApEn) 
clear all; 
clc; 
a=0; 
S=xlsread('Timeseries'); 
r=2; 
g=5; 
N=length(S); 
for m=g:g+1 
a=a+1; 
pen=N-m+1; 
[row, cols] = size(S); 
p = zeros(cols-(m-1), m); 
for i = 1:cols -(m-1); 
    p(i, :) = S(i:i+(m-1)); 
end 
P = num2cell(p, 2); 
Y=cell((length(P))*(length(P)),1); 
 n=0; 
 for j=1:size(P); 
  for k=1:size(P); 
    for  s=n; 
      n=s+1; 
       Y{n}=P{j}-P{k}; 
    end 
  end 
 end 
 e=0; 
 u=0; 
 while u<((length(P)*length(P))-(length(P)-1)); 
v=u+1; 
count=0; 
e=e+1; 
 for u=v:v+(length(P)-1); 
if (abs(Y{u})<r); 
    B{u}=Y{u}; 
      if  prod(B{u})<1; 
    count=count+1; 
      end 
end 
  
 end 
 C(e)=count/pen; 
 end 
 Cal(a)=mean(C); 
 156 
 clearvars -except Cal a N S r 
 end  
  
ApEn=log(Cal(1)/Cal(2)) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 157 
C. Documentation 
 
C1. Auburn IRB 
 
 
 
 
 158 
 
 
C2. Biomechanics Test Log Form 
 
 159 
 
 
 
 
 
 
 
 160 
C3. Accelerometer Calibration Sheet