ABSTRACT. In the past, studies and literature surrounding bipedal skipping gaits have

ABSTRACT In the past, studies and literature surrounding bipedal skipping gaits have suggested that it could lead to significant advancements in bipedal locomotion in lowgravity 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. ii

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. iii

VITA August 22, 1984 Born Rochester, Michigan 2002. High School, Eisenhower High School 2002 present.. Undergraduate Researcher, The Ohio State University Major Field: Mechanical Engineering FIELDS OF STUDY iv

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 v

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 vi

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 vii

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 alternates 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 1

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 2

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 3

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. 4

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 5

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. 6

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. 7

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 8

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. 1 2 2 Ekinetic = m( x flight x real ) 2 m x flight = assumed constant s m=mass kg ( ) [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 9

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 10

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. 11

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 ẋ flight. 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 z top, with its vertical velocity being zero. 12

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 L p for the pendulum and L s 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. 1 2 zt () = ztop gt 2 [3.1] zt () = gt [3.2] zt () = 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 13

variable. As the pendulum leg touches down, there is an instantaneous change in vertical velocity, as it becomes zero. t p: time of pendulum touchdown zt ( ) = L [3.4] p p Finally, the value of time t p 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. 1 2 ztop gtp = Lp 2 [3.5] 2( ztop Lp) t p = 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 t p, 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 θsd with the ground. This angle is assumed to be a known value in order to simplify the equations of motion. θ pd 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 14

angular acceleration. As shown below, by taking the moment of the body about the pendulum foot, the angular acceleration was found. Mo = Ioθ p [3.7] 2 Io = ml p [3.8] M o = mgl p cos θ p [3.9] mglp cosθp gcosθ p θ p = = [3.10] I L o p gcosθ p x flight θ p = t [3.11] L L θ p gcosθ x π 2 2 p 2 flight p = t t+ Lp Lp t d : time of spring leg touchdown p [3.12] 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 θ pd was found in terms ofθ sd and the length of the two legs. Figure 3.2. Touchdown Phase 15

Substituting td and θ pd into the expression forθ p, the time of spring leg touchdown, t d, was calculated. θ sin( π θ ) sinθ sd = = [3.13] L L L sin pd pd θ pd p s s 1 Lssinθsd = π sin ( ) [3.14] L p gcosθ pd x 2 flight π 1 Lssinθ sd t + t+ sin = 0 2L p L p 2 L p [3.15] To find the roots of Eq. 3.15, the quadratic formula was used. As expected, this method calculated two values for time, t d. The root chosen for t d corresponded with experimental data and was positive. t d 2 b b 4ac = [3.16] 2a At time t d, 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. 16

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, ẋ flight. 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 d x k = 2 L s l dt ( )sinθ m 2 d z k = ( )cos 2 L s l θ g dt m l = x + z 2 2 x z sin θ = cosθ = x + z x + z 2 2 2 2 2 dx k L s 2 dt m 2 2 x + z x = = (1 ) x [3.17] 2 dz k L s 2 dt m 2 2 x + z z = = ( 1) z g [3.18] The free length of the spring leg is defined as L s, 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 17

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, ż initial, 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 z top. If this z top is consistent with the value from the first phase, then there is a steady-state solution for a skipping gait cycle. t e : time at which the body reaches the top of flight zt ( ) = z e xt ( ) = x e top flight zinitial te = g [3.19] 1 2 ztop = ( z initialte gte ) + z 2 [3.20] 18

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 Newton. Meter 19

This increment was originally varied, but 1.0 was chosen based on the script s outputs and hardware limitations. With increments below 1.0 Newton, the computer was unable to Meter simulate the script. Increments above 1.0 Newton Meter were not as accurate in their results. Within the MATLAB script, there is a final conditional statement used to find the correct ztop value. Because there was no desired z top value based on experimental data, the researcher allowed the script to calculate a value. As the entire script runs, it continues to iterate as z top is incremented by 0.001 meters. The 0.001 meter increment ensures that when comparing the first and final phase, the z top value is consistent to the thousands place. Finally, the script produces a final spring stiffness and z top 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. 20

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 trials 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. 21

1 19.5 59.5 2 41.25 59.5 9 20 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. 22

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. span span v = = t t t marker1 mar ker2 [4.1] This value was calculated in metric units ( m s ). Table 4.1 displays an example of experimental data for horizontal velocity, including the mean and standard deviation. 23

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 3.524 Trial 2 1.80 20 6.1 3.387 Trial 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. Bilateral Angle (deg.) Table 4.3. Bilateral Pendulum Leg. Bilateral Angle (deg.) TD* 1 69 TD* 1 88.6 TD 2 63 TD 2 84.6 TD 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. 24

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 0.016 Spring TD 0.13 0.13 0.13 0.13 0.130 0.000 Compress 0.20 0.22 0.20 0.22 0.210 0.012 To top of flight 0.16 0.16 0.15 0.16 0.158 0.005 Total 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. 25

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 m s. 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. 26

5.1. Inputs into the System CHAPTER 5 EXPLANATION OF RESULTS 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 m s, 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 z top, 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 velocity 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. 27

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. 28

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. 29

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 14978 is consistently within 10,000 to 15,000 Newton Meter for spring stiffness. Overall, it Newton Meter 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 12,000 mass and leg length are used. Newton Meter for the running gait. In their research, similar values of 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 30

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. 31

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 z top 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 32

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. 33

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 34

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 35

Bilateral Setup 2 Unilateral Setup 1 36

Unilateral Setup 2 37

A.2. Touchdown Angles: Pendulum Leg Bilateral Setup 1 Bilateral Setup 2 38

Unilateral Setup 1 39

Unilateral Setup 2 A.3. Liftoff Angles: Spring Leg Bilateral Setup 1 40

Bilateral Setup 2 41

Unilateral Setup 1 Unilateral Setup 2 42

43

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 2.92 Trial 2 0.87 9.00 2.74 3.15 Trial 3 0.80 9.00 2.74 3.43 Trial 4 0.87 9.00 2.74 3.15 Averages 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 3.52 Trial 2 1.80 20.00 6.10 3.39 Trial 3 1.80 20.00 6.10 3.39 Trial 4 1.80 20.00 6.10 3.39 Averages 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 2.95 Trial 2 0.93 9.00 2.74 2.95 Trial 3 0.94 9.00 2.74 2.92 Trial 4 0.94 9.00 2.74 2.92 Averages 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 3.16 Trial 2 2.07 20.00 6.10 2.94 Trial 3 1.93 20.00 6.10 3.16 Trial 4 1.93 20.00 6.10 3.16 Averages 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 2.29 Trial 2 1.20 9.00 2.74 2.29 Trial 3 1.20 9.00 2.74 2.29 Trial 4 1.13 9.00 2.74 2.43 Averages 1.18 9.00 2.74 2.32 *Note. Bilateral Exaggerated Skipping. 44

B.2. Touchdown Angles: Spring Leg Bilateral: 9 ft. Angle (deg.) TD 1 69 TD 2 63 TD 3 61 TD 4 67 Average 65 STD 3.65 Bilateral: 20 ft. Angle (deg.) TD 1 75 TD 2 60 TD 3 63 TD 4 64 Average 65.5 STD 6.56 Unilateral: 9 ft. Angle (deg.) TD 1 57 TD 2 65 TD 3 61 TD 4 65 Average 62 STD 3.83 Unilateral: 20 ft. Angle (deg.) TD 1 64 TD 2 65 TD 3 67 TD 4 59 Average 63.75 STD 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 45

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 89.6 TD 2 87.4 TD 3 88.2 TD 4 87.9 Average 88.275 STD 0.94 Unilateral: 20 ft. Angle (deg.) TD 1 86.8 TD 2 89.3 TD 3 88.3 TD 4 87.8 Average 88.05 STD 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 46

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 0.016 Spring TD 0.13 0.13 0.13 0.13 0.130 0.000 Compress 0.2 0.22 0.2 0.22 0.210 0.012 To top of flight 0.16 0.16 0.15 0.16 0.158 0.005 Total 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 0.026 Spring TD 0.13 0.15 0.14 0.17 0.148 0.017 Compress 0.24 0.21 0.24 0.22 0.228 0.015 To top of flight 0.2 0.19 0.16 0.21 0.190 0.022 Total 0.80 0.80 0.80 0.80 0.800 0.000 47

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 48