ii
ABSTRACT In the past, studies and literature surrounding bipedal skipping gaits have
suggested that it could lead to significant advancements in bipedal locomotion in low-
gravity conditions and motor control development in humans. This thesis presents an
analytical model of the bipedal skipping gait and further analyzes the model’s
characteristics. The model incorporates multiple phases for a single stride with each
phase building upon its predecessor. In addition, the model is intended to generate
realistic results integrating adjustable parameters such as height and weight. Of
particular interest is the comparison of the model’s simulated results to experimental
data. Through analysis of the model, a better understanding of the gait can lead to
breakthroughs in motor development in children’s coordination skills and those of
patients recovering from strokes or other, similar illnesses.
iii
ACKNOWLEDEMENTS
I would first like to thank Dr. Jim Schmiedeler for his continued support throughout this
project. He gave me an opportunity to do research and provided me an exciting and
interesting avenue. I would also like to thank Dr. Eric Westervelt along with the
members of the Locomotion and Biomechanics Lab at the Ohio State University. They
have also given assistance and input into the project.
iv
VITA
August 22, 1984 …………………………… Born – Rochester, Michigan 2002 ………………………………………. High School, Eisenhower High School 2002 – present …………………………….. Undergraduate Researcher, The Ohio State University
FIELDS OF STUDY Major Field: Mechanical Engineering
v
TABLE OF CONTENTS ABSTRACT ……………………………………………………………………………...ii FIELDS OF STUDY ……………………………………………………………….……iv Chapter 1: Introduction ………………………………………………………….……..1 1.1. Background ………………………………………………………………………..1 1.2. Previous Work …………………………………………………………………….1 1.3. Project Motivation ………………………………………………………………...2 1.4. Approach ………………………………………………………………………….3 1.5. Outline of Thesis ………………………………………………………………….5 Chapter 2: Skipping Model …………………………………………………….………7 2.1. Description of Pendulum and Spring Legs ………………………………………..7 2.2. Description of the Point Mass ……………………………………………………..7 2.3. Description of the Five Phases …………………………………………………….8 2.4. Assumptions ………………………………………………………………………11 Chapter 3: Developing the Equations of Motion ………………………...…………..12 3.1. Development of Constants …………………………………………………….…12 3.2. Development of the Equations of Motion ………………………………………..12 3.2.1. First Phase ……………………………………………………………..…12 3.2.2. Second Phase …………………………………………………………..…14 3.2.3. Third and Fourth Phase ………………………………………………...…16 3.2.4. Fifth Phase ……………………………………………………………..…18 3.3. MATLAB Programming …………………………………………………………18 Chapter 4: Experimental Data Collection and Analysis ………………..………...…21 4.1. Motivation ……………………………………………………………………...…21 4.2. Description of Experimental Setup ……………………………………………….21 4.3. Analysis of the Footage ………………………………………………………..…23 4.4. Comparison of Bilateral and Unilateral Skipping Data …………………………..25 Chapter 5: Explanation of Results …………………………...……………………….27 5.1. Inputs into the System ……………………………..……………………………..27 5.2. Steady-State Solution …………………………………………………………….27 5.3. Comparison of Experimental and Simulated Results …………………………….28 5.4. Problems Encountered ……………………………………………………………30 Chapter 6: Project Conclusions ………………..……………………………………...32 6.1. Closing Remarks ………………………………………………………………….32 6.2. Future Work ………………………………………………………………...…….32
vi
LIST OF TABLES
Table 4.1. Bilateral Skipping for a 20 Foot Range ………………………………..…….24
Table 4.2. Bilateral Spring Leg …………………...……………………….…….….…...24
Table 4.3. Bilateral Pendulum Leg ..................................................................................24
Table 4.4. Experimental Time Data for Bilateral Skipping .............................................25
Table 5.1. Verification of Steady-State Solution …………………….………………….28
Table 5.2. Experimental and Simulated Results for Time ………………………………29
vii
LIST OF FIGURES
Figure 1.1. Skipping Gait Cycle ………………………………………………………….4 Figure 2.1. Skipping Model: Flight Phase ………………………………………………..8 Figure 2.2. Pendulum Leg Touchdown ………………………………………….……..…9 Figure 2.3. Spring Leg Touchdown ……………………………………………….……...9 Figure 2.4. Compression of the Spring Leg ……………………………………………..10 Figure 3.1. Flight Phase …………………………………………………………………13 Figure 3.2. Touchdown Phase …………………………………………………………...15 Figure 3.3. Steady-State Nature of the Spring Leg ……………………………………...18 Figure 4.1. Experimental Setup for First and Second Shots …………………………….22 Figure 4.2. Bilateral Skipping over a 20 Foot Span ………………………….………….23 Figure 4.3. Analysis of Spring Leg Angles ………………………….………….……….25
Figure 4.4. Pendulum Leg Touchdown for Bilateral and Unilateral Skipping ………….25
1
CHAPTER 1 INTRODUCTION
1.1. Background
The goal of this project is to develop an analytical model of bipedal skipping gaits
consistent with experimental data. A skipping gait involves keeping one foot forward, as
the other trailing foot hits the ground first and propels the body into its next stride. It’s a
high-speed gait where one leg takes a step similar to walking, while the other leg takes a
step similar to running. Skipping gaits can be either unilateral or bilateral. The most
common form is bilateral where the leading leg alterna tes for each stride. Unilateral
skipping is when the leading leg remains the same for each stride. Typically, skipping is
learned at a young age, but following walking and running. This is because skipping is a
non-spontaneous gait that requires concentration or even practice. Many have suggested
that it’s an excellent form of adult exercise, athletic training, and even enjoyable [9].
1.2. Previous Work
Typically, the topic of skipping has not been heavily researched, which can be
attributed in part to the fact that people abandon these gaits as a means of locomotion at
an early age [1]. This project will expand upon the existing work that has focused on
skipping, drawing from other research focused on similar gaits as well. Minetti [2] took
data from human subjects and created a simulation model. His research also focused on
bilateral skipping and was concerned with specific outputs of skipping such as stride
frequency. In contrast, the model developed in this project is analytical, including the
equations of motion for the skipping gait. The current model developed in this research
is used to analyze steady-state skipping. In the future, it will compare the energy
2
consumed at different speeds with the energy required for walking and running at the
same speeds and create the outputs to compare the different environments of Earth and
other surfaces. Those outputs can vary from position and velocity of the body to work
and force components of the motion. Also unique to this project, this model allows
certain parameters, such as weight and height, to be integrated into the analysis to acquire
realistic outputs for each individual.
To date, there has been significant research done to characterize running and also
comparisons between running and skipping. Whitall and Caldwell [6] analyzed both
human running and galloping, in contrast to the more common study of animal galloping.
Their research has helped validate certain variables and constants used within this
project’s model. Other researchers, in particular Farley [7], developed spring-mass
systems to model the hopping gait in animals. It uses a point mass to represent the body.
A similar spring-mass system is incorporated into this project’s model, as see in Fig. 1.1.
1.3. Project Motivation
A formal analysis of the skipping gait could lead to significant advancements in
two areas: biped locomotion in low-gravity conditions and motor control development in
humans.
There is evidence that skipping is a preferred bipedal gait in reduced gravity
environments [3]. In other words, if a human were on the surface of a different planet
(i.e. Mars), their most efficient form of rapid locomotion would in fact be skipping, or a
very similar gait. In these different conditions, there is a reduction in potential energy,
which makes walking difficult, or much less efficient, because it involves the exchange
of potential and kinetic energy. In lower gravity conditions, there is less potential energy
3
to be used for walking. With less potential energy to transfer into kinetic energy, the
walking step becomes much more difficult, if not impossible. There is also a lack of
normal forces, which decreases the amount of friction, causing the foot to slip when
running. When running, the leg makes a sharper angle to the ground, thus relying more
heavily on the friction force. Friction force is dependent on normal forces due to gravity,
and that value is greatly reduced on surfaces like Mars (1/3 of that on Earth) or the Moon
(1/6 of that on Earth). Minetti [2] cites post- flight debriefing transcripts from NASA
concerning astronauts’ experiences on the Moon to support these claims. This makes the
project worthwhile to the space industry and would come at a time when the president
has outlined the goal of sending astronauts back to the Moon. The analytical model
developed in this work will lead to a better understanding of skipping gaits such that
training practices and machines can be developed for astronauts to enhance their ability
to locomote quickly and efficiently in reduced gravity.
Walking and running are considered spontaneous gaits. They are natural
movements that don’t necessarily require significant thought or skill for the average
person. In contrast, skipping is a non-spontaneous gait. It requires a person to
concentrate and maybe even practice. Many athletes, typically in track and field, use
skipping in their daily workouts to train and prepare for competition [5]. In a similar
manner, this also means that it can be used to develop children’s motor skills at a young
age [4]. With the proposed analysis, a better understanding of the gait could lead to
helping children develop improved coordination skills. In addition, rehabilitation of
patients recovering from strokes or other illnesses could benefit from use of this gait.
1.4. Approach
4
The first step of this project was to review the relevant literature. This helped
verify important variables and constants. It also helped validate assumptions made in the
development of the model. The next step was to begin creating the model. The concept
is to model one leg as a spring (similar to running) and one as a rigid pendulum (similar
to walking), as shown in Fig. 1.1. The model is then broken down into five phases,
starting with “touchdown” as the pendulum leg makes contact with the ground. It is then
followed by touchdown of the spring leg. After this phase, the spring leg compresses and
decompresses to simulate the human leg propelling the body off the ground. The last
phase involves the body returning to its original position at the top of flight. This entire
cycle is featured in Fig. 1.1 to better illustrate the model. The equations of motion were
constructed for each phase, and then their results were incorporated into the following
phase. For solving more complicated phases, MATLAB's built in software SIMULINK
was applied. This was specifically helpful in determining initial conditions for different
phases of the gait, such as velocity in the horizontal and vertical directions after specific
phases.
Figure 1.1. Skipping Gait Cycle.
5
After the model and its equations for steady-state motion were fully developed, it
was entered into a MATLAB script file. This script includes realistic inputs, such as
horizontal velocity, lengths of legs, and angles between the ground and body. As the
script runs, it produces outputs such as time intervals and velocities, which develop a
basis for whether the model is physically realistic. The output of time, in particular, will
be used to verify that the outputs are consistent. For example, if the model concludes that
it takes 2 minutes for one skipping stride, then the model and its equations of motion
need to be re-examined to correct any inconsistencies. Other inputs, such as body mass,
will also be put through a sensitivity analysis to verify that adjusting those parameters
still produce realistic outputs when changed.
The next phase involved acquiring video footage of a human subject performing
the gait. The camera was positioned in order to attain a side view as a subject moved
across the floor, helping to determine parameters like horizontal velocity and angles
between the legs and ground. The footage was also analyzed to find different time values
that were used as a comparison tool to validate the model.
Finally, further analysis will need to be conducted on the model to find values for
energy, work, and force. The more information about the skipping gait’s dynamics, the
more likely training exercises or machines, such as treadmills, can be developed that
emphasize the benefits touched on earlier.
1.5. Outline of Thesis
This thesis begins with an introduction into the background and motivation
behind the project. This is followed by a brief summary of the approach to developing a
working, analytical model. Next, the model is discussed in detail focusing on the
6
different phases and the respective equations of motion. This includes an explanation of
the variables, constants, assumptions, and the MATLAB script located in Appendix C.
To ensure the model’s accuracy, the video footage experiment and its results will also be
discussed in a separate chapter. The overall results will then be highlighted, along with
the conclusions that can be drawn from the project. Finally, future work will be briefly
presented.
7
CHAPTER 2 SKIPPING MODEL
2.1. Description of Pendulum and Spring Legs
The model developed in this project is used to represent the human skipping gait.
It is displayed and labeled in Fig. 2.1. As touched on earlier, it incorporates a pendulum
leg, modeled as being rigid, that represents the walking step. The second leg is modeled
as a spring that represents the running step. Each leg is assumed to be massless and is
missing typical characteristics of the human leg which include: knee joints, ankles, feet,
or a distinct separation between the thigh and shank.
2.2. Description of the Point Mass
It is assumed the entire mass of the body is concentrated at the hips, where the
two legs join together, represented as a point mass. This was done not only to simplify
the model, but also to track the point mass throughout the analysis. In other words, the
position and time of the point mass is particularly important in the analysis. Because of
this, the body has two degrees of freedom. The ability of each leg to rotate produces a
total of four degrees of freedom in the model. From the orientation of the model in Fig.
2.1, the body will move from right to left, or, in the negative x-direction. The design of
the model allows for both the weight and height to be adjusted. In other words, the leg
length and the body’s mass are not fixed values. This allows the model’s outputs to be
tailored to the individual and to be compared to experimental data.
8
Figure 2.1. Skipping Model: Flight Phase
2.3. Description of the Five Phases
After the model was created, it was used to outline five distinct phases of a single
skipping stride. These phases are the foundation for the equations of motion that are used
to analyze characteristics of a skipping stride.
The first phase of the model is flight, shown in Fig. 2.1. This phase begins as the
body is at its highest vertical position and ends when the pendulum leg makes contact
with the ground. During this phase, the body is moving at a constant horizontal speed as
it descends downward.
As shown in Fig. 2.2, the pendulum leg is the first to contact the ground. It is
assumed that the leg proceeds downward at an angle perpendicular to the surface. This
assumption simplifies the problem by eliminating another potential variable in the model.
It has also been confirmed through observation of the gait that this perpendicular angle is
a reasonable value. At this point of contact, the model enters into its second phase,
touchdown. The touchdown phase can be defined by the duration of time between each
leg touching down. As one can see in Figs. 2.2 and 2.3, the body pivots about the
pendulum leg upon touchdown, which is also referred to as an inverted pendulum. As the
9
spring leg touches down, it is assumed there is an instantaneous transfer of support from
the pendulum to spring leg. During this same point in time, the pendulum leg loses
contact with the ground, and the body now pivots about the spring leg. Between
touchdown of the pendulum and spring legs, there is an assumption that energy is
injected into the system. It is assumed that when the spring leg contacts the ground the
body is still moving at the same horizontal velocity as it had in the first phase. By
specifying this velocity, there is a creation of energy since the magnitude of velocity
should be less. The mathematical expression below illustrates the amount of energy.
[2.1]
The third phase, compression of the spring leg, begins as the pendulum leg loses
contact with the surface. The end of the phase occurs when the spring leg has fully
compressed following its touchdown. This compression of the leg is used to represent
the human leg contracting or bending. At maximum compression, it is assumed the
spring leg should be perpendicular to the ground, marking the end of the third phase.
Figure 2.2. Pendulum Leg Touchdown Figure 2.3. Spring Leg Touchdown
( )
( )
2 212
kinetic flight real
flight
E m x x
mx assumed constant
s
m=mass kg
∆ = −
=
& &
&
10
The fourth phase is the decompression of the spring leg. The phase begins at
maximum compression of the leg and ends when it loses contact with the ground. This
mimics the physical “push” a human exerts as he or she propels him/herself back into
flight. During both the compression and decompression phases, the body pivots about
the spring leg. In the case of bilateral skipping, the pendulum leg swings through and
transitions into the lead leg during these two phases.
Phase five is the final stage of a skipping stride. It is defined as the duration of
time between when the spring leg lifts off the ground to when the body has reached its
top of flight. The body can be thought of as a projectile, similar to the first phase.
These five phases combine to create a simplified version of the human skipping
gait. From these phases, the equations of motion are developed and used to validate the
model’s outputs. In addition, by presenting these phases graphically, one can visualize
the walking and running segments as distinct features of the skipping gait.
Figure 2.4. Compression of the Spring Leg
11
2.4. Assumptions
The development of the skipping model has unveiled several assumptions used to
simplify the problem. The first assumption is defining the legs as massless and centering
the entire mass of the body at the hips. This point mass becomes the focal point of the
model, tracking its position and velocity components with respect to time. As touched on
earlier, there are missing characteristics of a typical human leg which include: knee
joints, ankles, feet, or a distinct separation between the thigh and shank. This was done
to limit the number of degrees of freedom, making the equations of motion easier to
solve.
Another assumption is the orientation of the pendulum leg upon contact. It is
assumed that the leg makes a perpendicular angle with the surface as it touches down.
This is also later confirmed in the experimental data presented in Chapter 4.
All the assumptions mentioned are helpful for developing an analytical model for
skipping by reducing the analytical work and eliminating potential variables. It is
important to note that they were carefully chosen to ensure key dynamics of the gait were
still obtainable.
12
CHAPTER 3 DEVELOPING THE EQUATIONS OF MOTION
3.1. Development of Constants
The constants selected in this model play an important role in the development of
the equations of motion. They serve as the inputs that allow the model to be
mathematically simplified. These values include: horizontal velocity, spring leg
touchdown angle, leg length, and mass. The model was designed to have as few selected
constants as possible in order to make its results more accurate and less dependent on
these assumptions.
While each of these values helped simplify the solution, they are also typically
consistent from stride to stride. Horizontal velocity and spring touchdown angle have a
low standard deviation in the experimental data discussed in Chapter 4. Mass and leg
length are necessary constants in order to tailor the model to an individual.
3.2. Development of Equations of Motion
3.2.1. First Phase
As explained in the previous chapter, the first phase of the model is the flight phase.
During this time, the body is assumed to be moving at a constant horizontal velocity
of flightx& . This value is defined as negative when motion is in the forward direction.
At the beginning of the phase, the vertical displacement of the body is defined as topz ,
with its vertical velocity being zero.
13
Figure 3.1. Flight Phase
These values reflect the assumption that the body is at maximum displacement and is
beginning its downward approach towards the surface. The lengths of the legs are
listed as Lp for the pendulum and Ls for the spring.
The equations of motion for the first phase originate from the idea that the point
mass is acting as a projectile. As the horizontal velocity remains constant, the
magnitude of the vertical velocity continues to increase until the pendulum touches
down.
21( )
2topz t z gt= − [3.1]
( )z t gt= −& [3.2] ( )z t g= −&& [3.3]
Upon touchdown, the z-coordinate of the point mass should be equal to the length of
the pendulum leg. As suggested in Figs. 3.1 and 3.2, at this point in time the
pendulum leg is assumed to be perpendicular to the ground. This assumption was
helpful in simplifying the proceeding equations of motion because it eliminated a
14
variable. As the pendulum leg touches down, there is an instantaneous change in
vertical velocity, as it becomes zero.
pt : time of pendulum touchdown
( )p pz t L= [3.4]
Finally, the value of time pt is solved for using Eqs. 3.5 and 3.6 below. This value of
time is then used to generate inputs for the following phases and their respective
equations.
212top p pz gt L− = [3.5]
2( )top pp
z Lt
g−
= [3.6]
3.2.2. Second Phase
The second phase of the model has been described as the touchdown phase,
defined by the duration of time it takes for the spring leg to contact the surface
following the pendulum leg. At time pt , the pendulum leg has contacted the ground,
and the point mass pivots about its foot. The pendulum leg is now behaving as an
inverted pendulum.
When the spring leg touches down, it makes an angle of sdθ with the ground. This
angle is assumed to be a known value in order to simplify the equations of motion.
pdθ is defined as the angle that the pendulum leg makes with the positive x-axis. It
initially has a value of 90 degrees, but slowly increases as the leg pivots about the
foot. Angular velocity and angular displacement are found through integration of the
15
angular acceleration. As shown below, by taking the moment of the body about the
pendulum foot, the angular acceleration was found.
o o pM I θ=∑ && [3.7] 2
o pI mL= [3.8]
coso p pM mgL θ= −∑ [3.9]
cos cosp p pp
o p
mgL gI L
θ θθ
− −= =&& [3.10]
cos p flight
pp p
g xt
L Lθ
θ−
= −&& [3.11]
2cos2 2
p flightp
p p
g xt t
L Lθ π
θ−
= − +&
[3.12]
dt : time of spring leg touchdown
As demonstrated above, the constants used in the integration were horizontal
velocity and the perpendicular angle of the pendulum leg upon touchdown, again
emphasizing their importance. From the law of sines, the angle pdθ was found in
terms of s dθ and the length of the two legs.
Figure 3.2. Touchdown Phase
16
Substituting dt and pdθ into the expression for pθ , the time of spring leg touchdown,
dt , was calculated.
sin( ) sinsin pd pdsd
p s sL L Lπ θ θθ −
= = [3.13]
1 sinsin ( )s sd
pdp
LL
θθ π −= − [3.14]
2 1cos sinsin 0
2 2pd flight s sd
p p p
g x Lt t
L L L
θ θπ − + + − =
& [3.15]
To find the roots of Eq. 3.15, the quadratic formula was used. As expected, this
method calculated two values for time, dt . The root chosen for dt corresponded
with experimental data and was positive.
2 42d
b b act
a− − −
= [3.16]
At time dt , it is assumed that there is an instantaneous transfer of support from the
pendulum leg to the spring leg, as the pendulum leg begins to lift off the ground.
Similar to the previous phase, the outputs of the second phase serve as inputs to the
third phase.
3.2.3. Third and Fourth Phase
The third phase of the model, compression, begins with the spring leg’s
touchdown and ends when it is fully compressed. This compression occurs as the
point mass pivots about the spring foot. The fourth phase, decompression, is defined
as the duration of time between maximum compression and when the spring leg
loses contact with the surface. Decompression is representative of the human leg
pushing the body back into flight.
17
In the model, phases three and four have been combined together because their
equations could not be solved explicitly. It was noticed that as the spring leg touches
down it has position and velocity components. As the spring leg moves through its
compression and decompression, it should lift off the ground with identical values of
leg length and horizontal velocity, flightx& . Fig. 3.3 displays the symmetric motion of
the spring leg with respect to the starting and ending position. From Fig. 3.3 and the
known constants, the following equations of acceleration were developed to simulate
this symmetric motion.
2
2 2 2(1 )sLd x k
x xdt m x z
= = −+
&& [3.17]
2
2 2 2( 1)sLd z k
z z gdt m x z
= = − −+
&& [3.18]
The free length of the spring leg is defined as Ls, as noted earlier in the chapter. The
variable k is the spring stiffness of the leg, which is determined during the analysis.
The transition between the two phases can be seen in Fig. 3.3 as the spring leg
2
2
2
2
2 2
2 2 2 2
( )sin
( )cos
sin cos
s
s
d x kL l
dt m
d z kL l g
dt m
l x zx z
x z x z
θ
θ
θ θ
= −
= − −
= +−
= =+ +
18
Figure 3.3. Steady-State Nature of the Spring Leg becomes fully compressed and is perpendicular to the ground. At this point in time,
the leg length and horizontal velocity decrease in magnitude.
3.2.4. Fifth Phase
The final phase is defined by the duration of time between liftoff of the spring
leg and the body reaching its top of flight. Similar to the first phase, the body acts as
a projectile with the same constant, horizontal velocity when the model has produced
a steady-state solution for the gait. Along with position components, liftoff angles,
and spring stiffness, the initial vertical velocity, initialz& , is an output given by the third
and fourth phases. When this velocity becomes zero, the body has reached its top of
flight, and its vertical displacement is equal to topz . If this topz is consistent with the
value from the first phase, then there is a steady-state solution for a skipping gait
cycle.
et : time at which the body reaches the top of flight
( )e topz t z=
( )e flightx t x=& &
initiale
zt
g=
& [3.19]
21( )
2top initial e ez z t gt z= − +& [3.20]
19
3.3. MATLAB Programming
The equations of motion formulated in the previous sections were then entered
into a MATLAB script file, which can be referenced in the Appendix C. This file is
designed to output several characteristics of the skipping gait model, but in particular, the
time for each respective phase. In addition to time, different velocities and positions of
the point mass are recorded. These values are helpful in validating the model. They can
be compared back to experimental data of actual skipping.
The MATLAB script begins with the identification and placement of a number of
inputs. Mass and leg length are two constants that allow the script to be tailored to a
unique individual. Horizontal velocity and spring touchdown angle are assumed
constants that are used to formulate the equations of motion. Finally, a spring stiffness
and maximum vertical position are given initial values, but later are redefined as the
script iterates to find a steady-state solution. Following the script’s inputs, the model
begins working through each respective phase, in the same order they were presented in
the previous chapters. Each phase’s outputs serve as inputs for the next phase. This
becomes increasingly important for the compression and decompression section.
For the compression and decompression phases, a SIMULINK diagram was
created because the equations could not be solved explicitly. This SIMULINK diagram,
developed from Eqs. 3.17 and 3.18, can be found in Appendix C. As the diagram runs,
two conditional statements were developed to perform iterations until specific conditions
are met. These conditions specified that the beginning of phase three and end of phase
four needed to have identical spring leg length and also identical horizontal velocity. At
this point in the script, the spring stiffness was increased by increments of 1.0 NewtonMeter
.
20
This increment was originally varied, but 1.0 was chosen based on the script’s outputs
and hardware limitations. With increments below 1.0 NewtonMeter
, the computer was unable to
simulate the script. Increments above 1.0 NewtonMeter
were not as accurate in their results.
Within the MATLAB script, there is a final conditional statement used to find the
correct topz value. Because there was no desired topz value based on experimental data,
the researcher allowed the script to calculate a value. As the entire script runs, it
continues to iterate as topz is incremented by 0.001 meters. The 0.001 meter increment
ensures that when comparing the first and final phase, the topz value is consistent to the
thousands place. Finally, the script produces a final spring stiffness and topz value that
correspond with the steady-state nature of the skipping gait. From the outputs of the third
and fourth phases, the final time is calculated for the fifth phase.
21
CHAPTER 4 EXPERIMENTAL DATA COLLECTION AND ANALYSIS
4.1. Motivation
In order to validate the results of the model and MATLAB script, there was a
need for experimental data from a skipping gait. By collecting video footage, the
researcher was able to obtain realistic values for common characteristics of the gait that
include: horizontal velocity, touchdown angles, and phase times. In addition, horizontal
velocity and spring touchdown angle were used as inputs to the model. The time for each
phase approximated by the footage proved to be an excellent tool for comparison. Stride
period was eventually estimated from the video footage.
4.2. Description of Experimental Setup
The experiment was designed to obtain two distinct shots with a Sony digital
video camera. Each shot would have several different tria ls explained later in the
chapter. The camera was attached to a tri-pod and positioned in a specified location.
Following the placement of the camera, two identification makers were placed diagonally
19.5 feet from the camera position for the first setup. The first shot setup had a
horizontal span between the two markers of nine feet. The experimental setup can be
seen in Fig. 4.1. The camera, for both shots, was 3.33 feet above the ground. This setup
is designed to capture the body as a human subject performs the gait across the nine-foot
span. The span allows for at least two strides to be recorded.
22
Figure 4.1. Experimental Setup for First and Second Shots. Using the first shot setup, three different gaits were performed that included bilateral
skipping, unilateral skipping, and exaggerated bilateral skipping. Each separate form of
the gait had four trials, making sure to include the body moving back and forth.
The second shot moved the markers 41.25 feet from the camera, allowing them
to be 20 feet apart from each other. The camera height remained 3.33 feet above the
surface, still able to capture the entire body moving along the 20 foot range. The
experimental setup for this shot can be seen in Fig. 4.1. The reasoning behind the larger
range was to receive a better estimate of horizontal velocity. Identical to the first shot
experiments, the subject performed the same three forms with four trials each, just simply
over a longer distance. Fig. 4.2 depicts a trial run where the span was 20 feet.
There was also a third shot used during this experiment concerning the frontal and
rear planes. The subject performed the gait either towards or away from the camera
position. Originally, it was assumed this footage would capture the vertical displacement
of the body and potentially the vertical velocity.
19.5”…59.5’
9”
1 41.25”…59.5’ 2
20”
23
Figure 4.2. Bilateral Skipping over a 20 Foot Span.
Although recorded, it was decided this footage was not helpful in determining these
points of interest. The software being used was unable to calculate displacement from
the video footage. Without displacement, the vertical velocity could not be found.
4.3. Analysis of the Footage
After the experiment was complete, the video footage was converted into a video
file. This file was then imported into Microsoft’s Windows Movie Maker. Using this
software, the footage could be analyzed to obtain the quantities of interest. Horizontal
velocity was first calculated by dividing the measured distance between markers (9 to 20
feet) over the time.
ker1 ker2mar mar
span spanv
t t t= =
− ∆ [4.1]
This value was calculated in metric units (ms
). Table 4.1 displays an example of
experimental data for horizontal velocity, including the mean and standard deviation.
24
Table 4.1. Bilateral Skipping for a 20 Foot Range.
Bilateral time (s) distance (ft) distance (m) speed (m/s) Trial 1 1.73 20 6.1 3.524Trial 2 1.80 20 6.1 3.387Trial 3 1.80 20 6.1 3.387
Trial 4 1.80 20 6.1 3.387
Mean 1.78 20 6.1 3.421
Standard Dev. 0.035 0 0 0.069
Next, the angles of touchdown for both the pendulum and spring legs were
evaluated. This required the researcher to take digital snapshots of the video footage
using Windows Movie Maker. Each snapshot, or now jpeg, was then analyzed with
Adobe Illustrator. A line was drawn from the hips (point mass) to the point of contact
with the surface, as shown in Fig. 4.3. This was because the model does not account for
both feet and knees, but the footage obviously includes these characteristics. After the
line was in place, the software allowed the researcher to record the angle each leg made
with the surface.
From all the collected data that is located in the Appendix B, a mean and standard
deviation were calculated for each parameter. Standard deviation for horizontal velocity
and spring touchdown angle provided the researcher a small range to adjust these two
assumed constants. An example of recorded data is presented in Tables 4.2 and 4.3
which includes the statistical analysis.
Table 4.2. Bilateral Spring Leg. Table 4.3. Bilateral Pendulum Leg.
Bilateral Angle (deg.) Bilateral Angle (deg.) TD* 1 69 TD* 1 88.6TD 2 63 TD 2 84.6TD 3 61 TD 3 87.4
TD 4 67 TD 4 89.5
Average 65 Average 87.5
STD 3.65 STD 2.13*Note. TD represents touchdown.
25
Figure 4.3. Analysis of Spring Leg Angles.
Lastly, by being able to analyze the footage frame by frame, the time for each phase was
also recorded for bilateral and exaggerated bilateral skipping using setup one. Since the
model’s phases can be defined by a duration of time, this experimental data is extremely
important in verifying the model’s simulated data. Table 4.4 displays the recorded time
data for bilateral skipping over the nine-foot span.
Table 4.4. Experimental Time Data for Bilateral Skipping.
Phase Run 1 Run 2 Run 3 Run 4 Average Standard Dev. Pendulum TD 0.17 0.15 0.15 0.13 0.150 0.016Spring TD 0.13 0.13 0.13 0.13 0.130 0.000Compress 0.20 0.22 0.20 0.22 0.210 0.012To top of flight 0.16 0.16 0.15 0.16 0.158 0.005Total 0.66 0.66 0.63 0.64 0.648 0.015
4.4. Comparison of Bilateral and Unilateral Skipping Data
As mentioned earlier in the chapter, experimental data was recorded and analyzed
for both bilateral and unilateral skipping.
26
Figure 4.4. Pendulum Leg Touchdown for Bilateral and Unilateral Skipping.
The model’s value for horizontal velocity is based on the results for bilateral skipping. It
is also important to recognize that there was no distinct difference between the two setup
spans in determining this value. The body moved consistently slower for unilateral
skipping by anywhere from 0.1 to 0.5 ms
. Because of this decrease in horizontal
velocity, the time for a skipping stride naturally increases. Velocity and time proved to
be the preferred comparisons tools since they serve as both an input and evaluation
method for the model. The touchdown angle of the pendulum leg, assumed to be
perpendicular, has consistent results for both bilateral and unilateral skipping. An
example of this angle, for both forms, is featured in Fig. 4.4.
27
CHAPTER 5 EXPLANATION OF RESULTS
5.1. Inputs into the System
As mentioned earlier, there are very few inputs, or assumed constants, into the
model. The results presented in this chapter are unique to the human subject whose
experimental data has already been recorded, analyzed, and discussed. The mass of the
body is 68 kilograms, with leg lengths of 0.9 meters. The analysis in Chapter 4 of the
data from Appendix B, determined the body to be moving at 3.3 ms
, making the results
specifically for bilateral skipping. The spring leg touchdown angle was 69 degrees, also
determined from bilateral skipping data.
5.2. Steady-State Solution
As mentioned in Chapter 3, the model simulates the skipping gait and produces
outputs for characteristics of the gait. The steady-state solution for this gait is achieved
when specific conditions are met throughout the model. The first condition is a
consistent value for maximum vertical displacement of the point mass. This value,
labeled as topz , is an initial condition during the first phase and an output of the final
phase. The second condition is the horizontal velocity of the body. This is an assumed
constant that has been validated by analysis of experimental data. At the beginning and
end of a skipping gait cycle, this ve locity must retain its magnitude. The final condition
involves the length of the spring leg. As the spring leg compresses and decompresses
during the third and fourth phases, it must eventually return to its initial value.
28
Table 5.1. Verification of Steady-State Solution.
Parameter First Phase Final Phase Percent Difference horizontal velocity (m/s) 3.300 3.299 0.030%
leg length (m)* 0.900 0.894 0.667%
vertical displacement (m) 1.010 1.009 0.099%
vertical velocity (m/s)* 1.91 1.84 3.665% *Note: this value occurs at the end of the third/fourth phase.
The different conditions mentioned above are displayed in Table 5.1. The table
compares the change in these four parameters from the first to final phases of the
skipping cycle. The small percent differences indicated the steady-state solution has been
achieved. In addition, the table provides a value of about 10 centimeters for the
maximum vertical displacement.
The model retains these key conditions as it simulates the gait. It should also be
noted that changing the model’s input values still allows steady-state nature to be
achieved. As mentioned Chapter 3, the MATLAB script includes several conditional
statements that force the model to iterate until these conditions are met.
5.3. Comparison of Experimental and Simulated Results
Once the steady-state solution has been confirmed, the model’s results can be
compared to experimental data from Chapter 4. The time of each phase, along with the
overall time for a single cycle, has been chosen as the comparison tool. Table 5.2
includes the results for both the experimental and simulated data. The table is divided
into individual phases and includes percent difference calculations.
The first phase, time between maximum vertical displacement and pendulum
touchdown, is almost identical for both sets of data.
29
Table 5.2. Experimental and Simulated Results for Time.
Phase Model Experimental Data Percent Difference first 0.149 0.150 0.60%
second 0.142 0.130 9.15%
third/fourth 0.216 0.210 2.86%
final 0.188 0.158 19.30%
total 0.695 0.648 7.32%
This small percent difference helps indicate the vertical displacement of 10 centimeters is
a realistic value. Next is the time between pendulum and spring leg touchdowns. The
simulated time was slightly higher, but still within 10 percent. The small discrepancy in
this phase may be the consequence of assuming both the pendulum and spring touchdown
angles. This is because both angles are then used in the phase’s equations of motion.
The third and fourth phases combine together for a time of about 0.21 seconds in
the experimental data. This value is consistent with the model’s results. The final phase
is defined by the duration of time between decompression and the return to maximum
vertical displacement. The model’s output is notably higher than experimental data has
shown. This is likely the result of unaccounted for characteristics of the gait. In other
words, the model does not account for any torque that may be generated from the
pendulum leg or arms swinging.
The overall time was also calculated for both the experimental and simulated data.
The difference between the two is under eight percent, concluding the model is producing
realistic results. Even with slightly higher values for phases two and five, the model is
still within 8 percent because the remaining, more accurate, phases account for longer
periods of time in total.
30
The liftoff angle of the spring leg at the end of the fourth phase is also another
comparison tool. The model produced a value of 69.2 degrees that is consistent with the
73 degrees average for experimental data. The percent difference is 5.48%.
The model also produced a value of 14978NewtonMeter for spring stiffness. Overall, it
is consistently within 10,000 to 15,000NewtonMeter depending on the inputs to the system.
Previous research that is focused on running gives a similar estimate for this parameter
[8]. Cheng and McMahon have suggested spring stiffness for humans is typically
between 11,000 and 12,000NewtonMeter for the running gait. In their research, similar values of
mass and leg length are used.
5.4. Problems Encountered
An ongoing problem throughout the research has been the selection of constants
and their respective value. With limited literature on the gait, there was very little
information on any of the characteristics of skipping, ranging from velocity to spring
stiffness. Using simplifying assumptions and MATLAB code, several constants were
transformed into outputs including: spring stiffness, vertical displacement, and vertical
velocity. As mentioned earlier, MATLAB was able to iterate the simulation until
specified conditions were achieved.
When developing the SIMULINK diagram to combine the third and fourth
phases, a number of issues were encountered. In the beginning, spring stiffness was still
an assumed constant that needed to be continually defined manually. As the researcher
tried to verify whether the diagram was working correctly, the spring stiffness was
31
always being adjusting based on changes in other initial conditions or the diagram’s
results. The diagram also had problems working properly because its configuration
parameters were not always correctly defined. For example, the length of time for the
simulation was sometimes entered too small causing MATLAB to produce errors in the
command window. It was trying to continue simulating past the allotted range of time, in
order to meet the conditions.
Another problem was the limitations of the editing software when trying to
evaluate certain characteristics of the video footage. It was difficult to retrieve the exact
angles between the legs and ground because the model did not incorporate a knee or foot.
Finally, vertical displacement of the body was not able to be accurately obtained.
Software that could calculate actual displacement in video footage was not available. In
addition, the experimental tests were not designed directly for, or in a way, that made
finding this value easy.
32
CHAPTER 6 PROJECT CONCLUSIONS
6.1. Closing Remarks
The first conclusion that can be drawn from the results is with respect to the
simplifying assumptions made throughout the model. These assumptions are validated
by the outputs of the model that directly correspond to experimental data. In other words,
they do not distort any of the steady-state dynamics of the skipping gait. The next
conclusion concerns the spring stiffness constant that was originally unknown. Having
documented this result is incredibly important for future research, especially in the case
where spring stiffness is defined as an assumed constant. In addition to spring stiffness,
the model gives an output for topz of about 10 centimeters. This value was of particular
interest since it could not be found from the experimental analysis. It is also useful as a
starting point and comparison tool in future models.
6.2. Future Work
Future work with this project can take a variety of avenues. The first is to further
analyze other dynamics of the model, such as energy, force, and work. These are
parameters that can be found using many of the outputs the model already provides. For
example, kinetic energy is dependent on mass and velocity making it easy to calculate
during different phases of the gait. In addition, a new force-plated treadmill in the
research lab will provide an opportunity to gather more experimental data. Another
direction is to adjust the original model, incorporating two spring legs. This will
complicate and increase the analytical work, but eventually produce an even more
realistic model. Outside the implementation of two spring legs, the model could also
33
integrate feet and/or knees. This would increase the degrees of freedom leading to much
more complicated analytical analysis. There would also be additional phases or segments
of the gait, such as heel and toe touchdown for each individual foot.
Finally, an experimental test should be developed to verify the maximum vertical
displacement of a body skipping. This would be extremely helpful in further validating
the model’s results, as well as future models. Similar footage with a different human
subject would also be interesting to analyze with simulation data.
34
REFERENCES
[1] Farley, C. “Just skip it.” Nature, VOL 394 (1998) pp. 721-723.
[2] Minetti, A. “The biomechanics of skipping gaits: a third locomotion paradigm?”
Proceedings of the Royal Society B 265: 1227-1235.
[3] Cavagna, G.A., P.A. Willems, and N.C. Heglund. “Walking on Mars.” Nature,
VOL 393 (1998) pp. 636.
[4] Clark, J., and J. Whitall. “Changing Patterns of Locomotion: From Walking to
Skipping.” Columbia, S.C.: University of South Carolina Press (1989).
[5] Eck, B. The Plyometrics System. http://www.runnersworld.com/article/0,5033,s6-
78-81-0-5902,00.html.
[6] Whitall, J and G.E. Caldwell. “Coordination of Symmetrical and Asymmetrical
Human Gat.” Journal of Motor Behavior, VOL 24 (1992) pp. 339-353.
[7] Farley, C.T. and J. Glasheen and T.A. McMahon. “Running Springs: Speed and
Animal Size.” Biology, VOL 185 (1993) pp. 71-86.
[8] McMahon, T.A., Cheng, G.C. (1990) Journal of Biomech. 23(Supp), 65-78.
[9] http://www.iskip.com
35
APPENDIX A DIGITAL SNAPSHOTS OF EXPERIMENTAL FOOTAGE
*Please note: snapshots were adjusted and magnified during analysis* A.1. Touchdown Angles: Spring Leg Bilateral Setup 1
36
Bilateral Setup 2
Unilateral Setup 1
37
Unilateral Setup 2
38
A.2. Touchdown Angles: Pendulum Leg Bilateral Setup 1
Bilateral Setup 2
39
Unilateral Setup 1
40
Unilateral Setup 2
A.3. Liftoff Angles: Spring Leg Bilateral Setup 1
41
Bilateral Setup 2
42
Unilateral Setup 1
Unilateral Setup 2
43
44
APPENDIX B EXPERIMENTAL DATA
B.1. Horizontal Velocity Bilateral time (s) distance (ft) distance (m) speed (m/s) Trial 1 0.94 9.00 2.74 2.92Trial 2 0.87 9.00 2.74 3.15Trial 3 0.80 9.00 2.74 3.43Trial 4 0.87 9.00 2.74 3.15Averages 0.87 9.00 2.74 3.16 Bilateral time (s) distance (ft) distance (m) speed (m/s) Trial 1 1.73 20.00 6.10 3.52Trial 2 1.80 20.00 6.10 3.39Trial 3 1.80 20.00 6.10 3.39Trial 4 1.80 20.00 6.10 3.39Averages 1.78 20.00 6.10 3.42 Unilateral time (s) distance (ft) distance (m) speed (m/s) Trial 1 0.93 9.00 2.74 2.95Trial 2 0.93 9.00 2.74 2.95Trial 3 0.94 9.00 2.74 2.92Trial 4 0.94 9.00 2.74 2.92Averages 0.94 9.00 2.74 2.93 Unilateral time (s) distance (ft) distance (m) speed (m/s) Trial 1 1.93 20.00 6.10 3.16Trial 2 2.07 20.00 6.10 2.94Trial 3 1.93 20.00 6.10 3.16Trial 4 1.93 20.00 6.10 3.16Averages 1.97 20.00 6.10 3.11 Bilateral* time (s) distance (ft) distance (m) speed (m/s) Trial 1 1.20 9.00 2.74 2.29Trial 2 1.20 9.00 2.74 2.29Trial 3 1.20 9.00 2.74 2.29Trial 4 1.13 9.00 2.74 2.43Averages 1.18 9.00 2.74 2.32*Note. Bilateral Exaggerated Skipping.
45
B.2. Touchdown Angles: Spring Leg Bilateral: 9 ft. Angle (deg.) TD 1 69TD 2 63TD 3 61TD 4 67Average 65STD 3.65 Bilateral: 20 ft. Angle (deg.) TD 1 75TD 2 60TD 3 63TD 4 64Average 65.5STD 6.56 Unilateral: 9 ft. Angle (deg.)
TD 1 57TD 2 65TD 3 61TD 4 65Average 62STD 3.83 Unilateral: 20 ft. Angle (deg.) TD 1 64TD 2 65TD 3 67TD 4 59Average 63.75STD 3.40 B.3. Touchdown Angles: Pendulum Leg Bilateral: 9 ft. Angle (deg.) TD 1 88.6 TD 2 84.6 TD 3 87.4 TD 4 89.5 Average 87.53 STD 2.13
46
Unilateral: 9 ft. Angle (deg.) TD 1 89.0 TD 2 89.8 TD 3 88.2 TD 4 89.5 Average 89.125 STD 0.70 Bilateral: 20 ft. Angle (deg.) TD 1 89.6TD 2 87.4TD 3 88.2TD 4 87.9Average 88.275STD 0.94 Unilateral: 20 ft. Angle (deg.) TD 1 86.8TD 2 89.3TD 3 88.3TD 4 87.8Average 88.05STD 1.04 B.4. Liftoff Angles: Spring Leg Bilateral 9 Feet Push 1 78 Push 2 70 Push 3 76 Push 4 77 Average 75.25 STD 3.59 Bilateral 20 Feet Push 1 71 Push 2 73 Push 3 68 Push 4 72 Average 71 STD 2.16
47
B.5. Time Bilateral Phase: 9 ft. Run 1 Run 2 Run 3 Run 4 Average Standard Dev. Pendulum TD 0.17 0.15 0.15 0.13 0.150 0.016Spring TD 0.13 0.13 0.13 0.13 0.130 0.000Compress 0.2 0.22 0.2 0.22 0.210 0.012To top of flight 0.16 0.16 0.15 0.16 0.158 0.005Total 0.66 0.66 0.63 0.64 0.648 0.015 Bilateral Phase: 9 ft.* Run 1 Run 2 Run 3 Run 4 Average Standard Dev. Pendulum TD 0.23 0.25 0.26 0.2 0.235 0.026Spring TD 0.13 0.15 0.14 0.17 0.148 0.017Compress 0.24 0.21 0.24 0.22 0.228 0.015To top of flight 0.2 0.19 0.16 0.21 0.190 0.022Total 0.80 0.80 0.80 0.80 0.800 0.000
48
APPENDIX C MATLAB CODE
%Pat Saad Test clear clc %Input Parameters L_p=.9; %m L_s=.9; %m g=9.8; %m/s^2 x_flight_dot=-3.3; %m/s m=68; %kg (150 lb. male) theta_s3_d=1.2043; % theta_s3_d=1.1694; %radians (67 degrees) % theta_s3_d=1.1345; %radians (65 degrees) % theta_s3_d=1.08; %radians (62 degrees) k=100; %N/m z_top=.9; %m z_final=.92; while z_top < z_final %flight phase (1) t_p=sqrt(2*(z_top-L_p)/g); x1_p=x_flight_dot*t_p; x1_p_dot=x_flight_dot; z1_p=L_p; %touchdown phase (2) theta_p2_d=pi-asin(L_s*sin(theta_s3_d/L_p)) a=(g*cos(theta_p2_d))/(2*L_p); b=x_flight_dot/L_p; c=pi/2-asin(L_s*sin(theta_s3_d)); t_d=(-b-sqrt((b^2)-(4*a*c)))/(2*a); theta_p2_d_dot=(-g*cos(theta_p2_d)/L_p)*t_d-x_flight_dot/L_p; %Compression and Decompression; Simulink Model %Inputs lo=L_s; z=L_s*sin(theta_s3_d); x=-L_s*cos(theta_s3_d); z_dot=L_p*theta_p2_d_dot*cos(theta_p2_d); %Iteration Test to find correct k value sim saadtest j=2; while l(j) < L_s
49
j=j+1; end while x_dot(j) > -x_flight_dot j=2; sim saadtest while l(j) < L_s j=j+1; end k=k+1; end k t_l_m=time(j); x_horizontal=x_dot(j); l_length=l(j) z_initial=z_dot_var(j); z_var(j); x_var(j); figure(1) plotyy(time,l,time,x_dot) xlabel('time (s)') ylabel('leg length (m)') grid on %Find z_value figure(2) plot(time,z_dot_var) %Decompression to Top of Flight t_e=-z_initial/-g; z_final=(z_initial.*t_e-(0.5.*g.*(t_e.^2)))+z_var(j-1); z_top=z_top+.001 end z_top %Total Time (seconds) t_total=t_p+t_d+t_l_m+t_e; %Display Table of Velocities disp(' ') disp('Compression/Decompression Simulated Results') tableYP(1,:)=[x_flight_dot x_horizontal z_dot z_initial]; disp(' x_flight x_horiz z_dot z_initial') disp(' (m/s) (m/s) ( m/s) (m/s)')
50
disp(tableYP) disp(' ') disp(' ') %Display Table of Times disp('Times') tableYP_2(1,:)=[t_p t_d t_l_m t_e t_total]; disp(' Pendulum Spring Compress To Top Total') disp(' (sec) (sec) (sec) of Flight (sec)') disp(' (sec)') disp(tableYP_2) disp(' ') disp(' ') %Display Table of Positions disp('Steady State Confirmation - z-direction') tableYP_3(1,:)=[z_top z_final z z_var(j)]; disp(' z_top z_final z z_var(j)') disp(' (m) (m) (m) (m)') disp(tableYP_3)