The Pennsylvania State University. The Graduate School. Department of Mechanical and Nuclear Engineering

Transcription

1 The Pennsylvania State University The Graduate School Department of Mechanical and Nuclear Engineering DEVELOPMENT OF A STATISTICAL MODEL OF REACH EXERTION MAGNITUDE PERCEPTION FOR USE IN DESIGNING TRUCK CABS A Thesis in Mechanical Engineering by Brittany A. Klein Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 212

2 The thesis of Brittany A. Klein was reviewed and approved* by the following: Matthew B. Parkinson Associate Professor of Mechanical Engineering and Engineering Design Thesis Advisor H. Joseph Sommer III Professor of Mechanical Engineering Karen A. Thole Professor of Mechanical Engineering Department Head of Mechanical and Nuclear Engineering * Signatures are on file in the Graduate School

3 ABSTRACT The goal of this work was to develop a statistical model to describe a user s perception of the difficulty of performing various reaching tasks in a truck cab as a function of the reach location. Towards this end, a piecewise model that would yield a surface of points corresponding to the desired difficulty rating for a specified user as a function of his or her anthropometry was developed. This model consisted of a linear combination of the first four spherical harmonics, with the coefficients of each term in the model changing according to the desired difficulty rating. The model was developed using data gathered from an experimental study performed by the University of Michigan Transportation Research Institute (UMTRI) in which volunteers performed reaches to a variety of points and rated each point according to their perception of the reach task s difficulty. Nonlinear regression techniques and individual subject anthropometry were used to fit to fit the model to the data. The model successfully replicated the experimental results, with residual sum of squares values varying from.624 for the lowest difficulty rating to.992 for the highest. Concerns over unexpected ordering of the iso-rating surfaces generated by the model led to a revision of the model that classified reach locations as easy, acceptable, or difficult to provide a more generalized model that was useful for predicting difficulty ratings of individuals not present in the experiment. An application of the anthropometry of individuals recorded in the U.S. Army Anthropometry Survey (ANSUR) to the model indicated that the model can be used to predict reach surfaces for populations of individuals other than the experimental group. Future iterations of the model are expected to benefit from including additional data such as the force exerted during the reach tasks in the model calculations. In addition, changing the scale of difficulty ratings used in experiments may improve accuracy if a less subjective scale, possibly with fewer divisions, could be implemented successfully. iii

4 TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES v vi ACKNOWLEDGEMENTS vii CHAPTER I. INTRODUCTION II. BACKGROUND AND LITERATURE REVIEW Truck cab design Reach Envelopes Early uses of the reach envelope Improving utility with models Beyond measuring reach Spherical harmonics III. METHODOLOGY University of Michigan Reach Study Development of model Linear regression model Spherical harmonics difficulty model Spherical harmonics radius model Piecewise spherical harmonics radius model IV. DATA AND RESULTS Assessment of model fit Revision of model parameters V. DISCUSSION AND CONCLUSIONS Model results Limitations and future work A. PARTICIPANT DIFFICULTY RATINGS B. REACH ENVELOPES FOR STUDY VOLUNTEERS BIBLIOGRAPHY iv

5 LIST OF FIGURES 3.1 A diagram showing the handle locations in the experimental setup. The H-point appears in red (UMTRI, 29) The first four spherical harmonics. Green indicates positive values and orange indicates negative values. (Jarosz, 28) A diagram of the ten reach surfaces of volunteer denoted Pilot_MML1. The x denotes the H-point The two-surface surface model of volunteer denoted Pilot_MML1. The x denotes the H-point The distribution of difficulty ranks in the experiment Comparison of point ratings from experimental data with model surface plots Stature/Sitting Height Distribution for Experimental Population Stature/Sitting Height Distribution for ANSUR Population Minimum Stature Male Results Median Stature Male Results Maximum Stature Male Results Minimum Stature Female Results Median Stature Female Results Maximum Stature Female Results v

6 LIST OF TABLES 3.1 Target Participant Groups (UMTRI, 29) Anthropometric Measures of Participants Locations for each of the handles with respect to the H-point Spherical coordinates (mm and rad) of the eleven reach tasks in relation to the H-point Goodness of Fit Results for Varying Normalization Factors Model Coefficients Participant Statures and Sitting Heights Used to Test Fit Residual sum of square values observed for each difficulty model Alternate goodness of fit values observed for each difficulty model Average Difference Between Calculated and Experimental Radius Values for Individual Reach Difficulties (mm) Reach Task Counts for Each Difficulty Rating Final Model Coefficients Difficulty Ratings of Each Handle for ANSUR Population Estimated Difficulty Ratings of Each Handle for Experimental Population. 34 vi

7 ACKNOWLEDGEMENTS The author would like to thank her parents, Gary and Priscilla Klein, and Brett Holbert for their tireless love and support during the creation of this thesis. vii

8 CHAPTER I INTRODUCTION The design of truck cabs provides a unique challenge to tractor-trailer manufacturers. The compounding factors of high employee turnover in the trucking industry and long working hours lead to the necessity of a product that can accommodate a large variety of potential users for extended periods of time. Design research has been conducted to determine the best methods for ensuring accommodation through seat adjustments and proper control placement using reach envelopes. A reach envelope is the boundary that encloses all points within the motion capability of a given user. By using reach envelopes to determine which controls in a truck cab are within the reach of different portions of the population, the level of accommodation (the proportion of users that can successfully operate the truck) can be calculated. Currently the majority of reach envelopes measure maximum reach capabilities that measure only if a point is attainable or not attainable without consideration if certain locations may be easier for the user to reach than others. The practice of considering ease of use as well as accommodation is relatively new and will yield promising results as designers will be enabled to consider more than the simple criteria of what is acceptable for a large population to how to make the design comfortable to use. The first chapter of this thesis will describe the current state of truck cab design and the concept of reach envelopes. The extension from modeling reach to modeling exertion will be discussed. The second chapter will discuss the methodology used to collect the data and develop the model. Chapter 3 will present the results and Chapter 4 will present conclusions, limitations, and potential for future work. 1

9 CHAPTER II BACKGROUND AND LITERATURE REVIEW The trucking industry is responsible for nearly 7 percent of all freight shipped in the United States. An individual truck cab can be expected to travel over 1, miles in a single year delivering freight (Berwick and Frank, 1997). Nearly every consumer product is transported on a truck at some point during its fabrication (Harrison and Pierce, 29), making their operation critical to the health of the economy. 2.1 Truck cab design A truck cab, when properly maintained, will be in service for decades, driving hundreds of thousands of miles as it tows freight across the country. The large fleet companies that maintain these trucks, however, often see a turnover rate of over one hundred percent in a single year (Harrison and Pierce, 29). Unlike a consumer car, which can be selected for the comfort and ease of use of one or two drivers, a truck cab must perform effectively for several dozen drivers during its lifetime. These truck drivers have widely varying anthropometry; for example, currently 5.9% of drivers are women, an increasing population that is completely different from the classic stereotype of large male drivers (Harrison and Pierce, 29). The Federal Motor Carrier Safety Administration regulations prevent truckers from driving more than 11 cumulative hours in a 24-hour period. Therefore, the challenge in designing a truck cab is to determine a combination of seating and control locations that will accommodate dozens of different people using it continuously for at least 11 hours a day over a span of decades. One key aspect to accommodating a variety of users is ensuring that the seat can be adjusted to a comfortable position for each driver. 2

10 Adding adjustability to the seat is expensive and the amount that can be included is often limited by the small size of the truck cabs. In light of size and cost constraints, the design challenge is to find the most suitable range for the seat adjustability to cover. One study used models to determine the locations for the driver s seat and steering wheel that would accommodate the largest range of drivers in a cab of a specific size and with a maximum allowable range of adjustability (Parkinson et al., 27). A variety of techniques, from constructing full-size prototypes for volunteers to test to constructing virtual populations of potential users and determining their preferred settings can be utilized to judge the accommodation level of a design. One such technique is the use of reach envelopes. 2.2 Reach Envelopes The reach envelope is a tool that has been used in the design field for decades (King, 1948). In its simplest form, a reach envelope describes the farthest distance an individual is able to reach. For the purposes of design, any point on its surface of the envelope or contained within it is considered to be accessible by the user. Over time this simple concept has evolved into a sophisticated design tool that not only addresses accessibility but comfort. The prevailing design attitudes progressed from the goal of ensuring that ninety-five percent of the population can reach surfaces in a design properly, to finding the surfaces ninety-five percent of the population can reach and designing new surfaces accordingly, to moving beyond the basic reach requirement and addressing issues such as ergonomics and comfort Early uses of the reach envelope The concept of using human dimensions to set the boundaries of a workspace was discussed in papers as early as 1948 when King et al. measured the extreme fingertip reach of U.S. men restrained in a pilot s seat by a seatbelt (King, 1948). By the 197 s, however, the focus shifted towards measuring dimensions within specific workspaces, rather than building workspaces off of data from experiments designed to measure maximum reach. Margaret Bullock wrote in 1974, Static anthropometric measurements 3

11 provide necessary information about the range of body dimensions, but the fact that those measurements are taken in conventional, standardized positions means that they cannot be applied to dynamic situations such as those which exist during flight (Bullock, 1974). After conducting her own literature review on the topic, she expressed her belief that the best way to solve design problems would be to take measurements in conditions as near as possible to those prevailing in the actual work situation and to use this information to select values that would be satisfactory for a large percentage of potential operators. Bullock s work and other reach envelope experiments of the time, such as the SAE Controls Reach Study, shared many similar characteristics in their attempts to closely mimic real-life conditions with experimental setups. The SAE study included an equal number of male and female participants performing reach tasks in three test fixtures representing a sport car, passenger car, and family truck in order to obtain reach surfaces for the most commonly used vehicle configurations (Hammond and Roe, 1972). Bullock s study with the University of Queensland was designed to take functional arm reach measurements of pilots in the cockpits of light aircraft. The subjects were selected according to the height and gender distribution of Australian pilots and they performed reach tasks in a representative seat and harness configuration with adjustability mimicking that found in most light aircraft (Bullock, 1974). Both studies had subjects reach for points at a variety of angles and heights from their starting position and utilized this experimental data to develop reach envelopes that were applicable for later designs of the type of test environment modeled (e.g. heavy truck reach envelope data to determine an appropriate location for radio controls on the dashboard of an 18-wheeler) Improving utility with models One of the major limitations of reach envelope research conducted in the 197s is that the data were only useful for the design of artifacts similar to the ones used when collecting the experimental data. The dimensions of the work space (such as back angle and steering wheel location) greatly affect the user s capability to perform reach tasks, and the SAE study quantified this work space as a series of nine package variables" (Hammond and Roe, 1972). A survey on the package variables most common vehicles in use at the time 4

12 was conducted to determine the best configurations for the test environments, but out of practicality the survey was limited to "most vehicle packages currently produced or likely to be used in the foreseeable future." This can pose difficulties when trying to utilize the SAE data to develop drastically new vehicle designs. An engineer designing a vehicle type other than one used in the SAE test would find that the reach envelope data has limited utility for a design with different package variables. It would be difficult to determine if poor accommodation was a result of design flaws that needed to be addressed or simply because the designer was attempting to fit data from the test to a vehicle design with completely different geometry. The designer would need to interpolate the existing information to his or her design or conduct an entirely new experiment. It s therefore desirable that a reach envelope be created that is independent of vehicle geometry, and is based solely on the anthropometry of the potential user. When the reach envelope is associated with the user instead of the product, it becomes applicable to any environment with which that user could interact. Boundary manikins, representative figures of specified anthropometry, are often used to determine reach envelopes without relying on experimental data. These manikins are used to stand in for a potential user and are posed in positions that the user would be expected to assume while interacting with a model artifact, a process known as posturing. For example, by posturing a variety of manikins of different statures inside a model truck cab, the designer can intuitively determine what points within the cab are accessible simply by moving the manikin s arm to see if it can reach the location while in a sitting position. Manikin use has evolved from physical articulated human models placed inside product mock-ups to a digital process. Software packages such as Siemens Tecnomatix Jack create digital manikins of specified anthropometry which can then be postured inside of imported CAD models of the product, saving the designer the time and cost of making multiple manikins and new product models whenever the design changes. One limitation of using manikins, however, is that the reach envelopes generated are directly a function of how the designer postures the manikin. While the design will benefit from determining the reach envelopes of manikins with varying anthropometry, the user s posture and position is also critical to determining reach (Reed et al., 23). 5

13 With only the designer to posture the manikins, variations in posture such as those between people of similar anthropometry will not appear. Furthermore, the algorithms used to generate reach envelopes based on manikins include joint ranges of motion using a few sets of data configured into the model directly, as most human modeling software packages do not account for the distribution of joint ranges of motion across individuals and the association between joint range of motion and anthropometry. This leads to most reach envelopes being calculated with a variety of anthropometry but limited joint angle possibilities (Reed et al., 23). It is expected therefore that the most useful reach envelopes would be those that are both accurate in predicting the reach of actual human beings, with variability present between similarly-sized individuals, and applicable to a wide variety of situations. Thus, the newest incarnation of reach envelopes are evolving into generalized models of reach. One such method of modeling was detailed by Reed, et. al. (23). Volunteers performed a series of reach tasks to targets of varying location and distance and rated the reach difficulty from 1 to 1, with impossible targets coded as 11. The data could be fit to a linear function with an exponential term which provided the difficulty rating for that particular person as a function of the reach point location (Reed et al., 23). This function was then used to plot reach surfaces of specified difficulty. From this model, many reach surfaces could be obtained by using the model to simulate a large number of drivers with varying anthropometry and including the random residual variance determined from the linear regression (Reed et al., 23). This model, generated by the data gathered from a few participants, can then be applied to the anthropometry of the target consumer base and generate reach surfaces for potential users not measured in the study without the need for large-scale testing. Parkinson and Reed (27) proposed an alternative method of creating models that bypasses the need for experimental data altogether. Instead, a simulation based on biomechanics models was utilized. A database of potential users generated with anthropometry measures that match the distribution of the target user base (a virtual population ) was created using previously-available anthropometric data (in this case, the U.S. Army Anthropometry Survey) and kinematic simulation techniques such as realistic 6

14 joint movement and location of the center of balance were employed to pose the models. By determining the overall cost of each posture (in this case modeled by the joint angle of the torso), the difficulty of the associated reach can be estimated (Parkinson and Reed, 27). While not as accurate as having exact reach data from real subjects in the situation desired, this approach can be useful to quickly quantify how different designs compare to one another, or as a means to model potential users of rare anthropometry who may not have been present in experimental studies Beyond measuring reach As the collective ability to model the most basic of reaching tasks improves, focus has been shifting to develop reach envelopes that give more information than simply which points are easiest to reach. Several studies have been performed which are similar to those employed by King and Bullock, but with a stronger emphasis on the discomfort of the test subject as opposed to maximum reach capabilities. The targets were normalized in relation to the stature of the subject so all targets were reachable, leaving discomfort as the main variable under study. A detailed scale of 1-5 was used to rate discomfort, but may have been too fine as the results were not very repeatable (Chevalot and Wang, 24). It was determined that the most comfortable points vertically were located halfway between the knee and shoulder (and increased in difficulty quadratically as the height to reach grew lower and higher), and that difficulty of reach increased with distance (Chevalot and Wang, 24). Later, surface fitting was performed with orthogonal polynomials to obtain the reach surfaces at different levels of difficulty, so that the discomfort at a point could be expressed as a function of the point s latitude and longitude (Wang et al., 27). These studies and others note a nonlinear relationship between reach distance and the degree of discomfort experienced. It is also noted that easier reaches tend to occur when the torso can remain stationary, whereas when a subject must lean into the reach, it is rated more difficult (Parkinson and Reed, 26). There have been some studies investigating the particular contributions of the torso s position to reach difficulty. The virtual population modeling method described in the previous section utilized the torso position as a predictor of the cost, and by extension, the difficulty, of a given reach task. 7

15 It was discovered that the pelvis is a major component of reaches, as it often rolls to allow someone to reach slightly forward. As expected, torso motion occurs when the target is further away than the length of the participant s arm (Reed et al., 24). By creating reach envelopes with models that contain the kinematic chain of pelvis, thorax, clavicle, shoulder, elbow, and wrist, more realistic results could be obtained, possibly lessening the need for experimental data. Balance is also a factor in perceived difficulty. A point that is within a subject s reach envelope but is at a location that throws him or her off balance will be rated as difficult. Changes in position to perform reach tasks alter the location of a person s center of pressure, and in general people are unwilling to perform reaches farther than those that result in displacing their COP more than approximately 7% in any direction from it s initial position (Holbein-Jenny et al., 27). Therefore, particularly distant reaches may be unattainable not because of a lack of ability to reach the point, but because such a task would cause the participant to lose his or her balance. Therefore, if the center of pressure is not being measured experimentally (such as when one is working with a virtual population) and balance is desired as a factor in determining reach difficulty, the center or pressure must be calculated and accounted for in the model (Parkinson and Reed, 26). This center of pressure value can then be determined for various reaches in opposition to the expected center of pressure limit where the user would barely be balanced to determine if the user is balanced, and thus accommodated for that point. Generally the center of pressure excursion capacity can be modeled as a function of stature, age, hip breadth, and the reach azimuth (Parkinson et al., 26). The model described in this thesis takes an alternative approach from that of (Chevalot and Wang, 24), in that it employs spherical harmonics instead of surface polynomials, and that the data used came from an experimental setup in which subjects were able to reach the points, so that the resulting model describes only perceived difficulty, and not the envelope itself. 8

16 2.3 Spherical harmonics Multiple models have been employed to describe the shape of reach envelopes. Linear models with additional terms (Reed et al., 23), polynomial surface regression (Chevalot and Wang, 24) and spline regression (Boydstun et al., 198) have all been used to create reach envelope models from experimental data. Another approach to modeling reach envelopes involves the use of spherical harmonic regression. Spherical harmonics are the set of harmonic functions obtained by solving the angular portion of Laplace s equations in spherical coordinates. They can be described as a three dimensional solution to the Fourier series while the latter represents the fundamental modes of vibration in a one-dimensional element, spherical harmonics describe the vibration of a sphere. While a direct solution of Laplace s equation yields a complex solution, the data collected for this model is strictly real-valued; for the purposes of this work, Euler s formula was used to yield a purely real-valued collection of spherical harmonics. For a model of order l and degree m, the real spherical harmonics basis functions are defined as: where K m l Y m l (θ, φ) = is defined as 2K m l cos mφp m l (cos θ) : m > K l P l 2K m l (cos θ) : m = sin mφp m l (cos θ) : m < (2.1) K m l = (2l + 1) (l m )! 4π (l + m )! (2.2) and Pl m are the associated Legendre polynomials (Jarosz, 28). Traditionally, spherical harmonic regression using minimax design has been used to model natural processes like the steady-state heat flow over the surface of the earth (Kupper, 1972). More recently, spherical harmonic regression has been used in computer graphics to model the effect of light reflecting off of objects and as a means for developing reach envelopes. Unlike more traditional methods such as polynomial or 9

17 spline regression, spherical harmonics are periodic in θ and are completely independent of θ at the poles, eliminating resolution problems (Boydstun et al., 198). With the exception of estimates at the poles, spline regression and spherical harmonics regression give comparable reach envelope estimates, but the designer will often find spherical harmonics easier to use as they involve calculations and do not require complicated programing structures to implement (Boydstun et al., 198). Unlike spline regression, spherical harmonics do require the prior development of an underlying knot structure. In addition, their basis functions can be computed directly instead of recursively as required by splines (Boydstun et al., 198). A lack of recursion allows for easier development of the model on the part of the designer. As the shapes created by linear combinations of spherical harmonic basis functions are close to the natural shapes of reach envelopes obtained through data collection, the model creation process is simple and easy to tailor to different individuals. 1

18 CHAPTER III METHODOLOGY 3.1 University of Michigan Reach Study The University of Michigan Transportation Research Institute (UMTRI) conducted a study in which 26 participants sat in an test rig consisting of the pedals, seat, and steering wheel of a truck cab and performed reaching tasks. The study was approved by the University of Michigan s Health Sciences and Behavioral Sciences Institutional Review board. The volunteer population consisted of 15 males, 13 of whom had been professional truck drivers within the last two years, and 11 females, 2 of whom had been professional truck drivers (UMTRI, 29). A wide variety of anthropometry was desired and to this end specific participant groups were targeted to ensure a varied set of participants. Individuals with body mass indices (BMI) both greater and less than 3 were desired, as well as individuals in the 2 th and 8 th percentiles of stature for their gender. A summary of these target groups is seen in Table 3.1. Table 3.1: Target Participant Groups (UMTRI, 29) Gender BMI Stature (mm) Count Male Female <3 >=3 <3 >=3 < > < > < > < >

19 Figure 3.1: A diagram showing the handle locations in the experimental setup. The H-point appears in red (UMTRI, 29). This targeted sampling allows the model produced from this data to be relevant for potential users of a variety of height and weights. Had the study been conducted with the most readily available volunteers without targeting specific groups, its highly probable that there would be a disproportionate amount of the most populous group in the vicinity of the institute (i.e. physically fit males of average stature) present in the data and the model would not be as effective for predicting reaches for less common groups. A variety of anthropometric measures were also taken from each participant, as summarized in Table 3.2. The experimental setup included a movable handle instrumented with a load cell. The handle was placed in a vertical orientation for ten of the reach tasks with the eleventh handle configuration horizontal. A diagram of the experimental platform with the handle locations labeled appears in Figure 3.1. The handle locations in millimeters with respect to the truck seat H-point are detailed in Table 3.3. Each participant grasped and pulled the handles at the eleven locations, varying the force exerted from % to 1% of their maximum capabilities. Following the tasks, participants rated their perceptions of the difficulty they experienced reaching for 12

20 each handle on a scale from 1 to 1, 1 being the highest. The model was fitted to the collected anthropometric and force perception data. 3.2 Development of model Several different approaches to developing a difficulty model were employed before the model reached its current iteration Linear regression model The first iteration was a linear regression to verify that the data were sufficiently nonlinear that alternative modeling practices were necessary. Table 3.2: Anthropometric Measures of Participants Subject Anthropometry Date of birth Upper Arm Length Weight with shoes Forearm Length Stature with shoes Buttock-knee length Weight without shoes Chest depth at armpit height Stature without shoes Chest depth at umbilicus height (sitting) Erect sitting height Chest depth at umbilicus height (standing) Acromial height Pelvis depth Knee height Arm span Biacromial breadth Wrist-wall length Bideltoid breadth Wrist-grip axis length Hip breadth (seated) Bi-ASIS breadth Table 3.3: Locations for each of the handles with respect to the H-point Condition X Y Z Horizontal

21 A linear model was fitted to the reported difficulties of each of the individual reach tasks recorded for each study participant using the spherical coordinates of each point and the participant s stature. This model yielded a poor fit, with a residual sum of squares of only Several other combinations of participant anthropometry were introduced into the model, without greatly improving the fit. A spherical harmonics-based model was then selected from a review of the literature reporting success with its use in similar applications Spherical harmonics difficulty model The second approach was to use a linear combination of spherical harmonics to predict the difficulty of each reach point. A model for each study participant was developed by using the least-squares method to solve for a set of coefficients β that would yield the recorded difficulty values for each reach location. These individual models were to be combined to create a generalized model by performing regressions on the coefficients. This was not possible, however, due to the form of the model and the variability in the data. Spherical harmonics are functions of two variables, θ and φ, and the difficulty model was expected to be a function of at least four variables, three to represent the handle locations and at least one to represent participant anthropometry. By fitting the data to a linear combination of spherical harmonics, the radius and anthropometry of the participant were accounted for in the model coefficients. While attempts were made to normalize the coefficients with respect to participant anthropometry and to add in terms containing the point radius, all attempts at forming a single model failed. Due to the large amount of variability between participants in both difficulty ratings for the same handle locations and in anthropometry differences, the coefficients for each term in the model were too different and no correlation between them could be found that would allow a generalized model to be made Spherical harmonics radius model The next iteration differed from the previous two iterations in that it took the form of a discrete model instead of a continuous function. As the spherical harmonics describe 14

22 three-dimensional shapes in space, it was determined that it was more appropriate to use spherical harmonics to solve for the radius of a point of a given difficulty. The goal then shifted from developing a model of the form D = f (r, θ, φ, A) (3.1) that returned the difficulty rating D that a given person of anthropometry A would assign to a point at location (r, θ, φ) to the form r = f (θ, φ, A, D) (3.2) This form returns the radius r of a point at location (θ, φ) for which a person of anthropometry A would assign a difficulty value D. The first attempt at forming the model was similar to the work done in the previous iteration in that a version of this piecewise model would be developed for each participant and then the coefficients of model would be used to develop a generalized model that used functions of anthropometry as its coefficients. Maximum Likelihood estimation was used with a logistic regression to calculate the coefficients for each difficulty, but this was ultimately unsuccessful. Logistic regression was determined to be unsuitable as it is used to determine outputs on a scale from to 1. While such a model could potentially be used to describe the odds of a point being rated at a given difficulty, it would not be able to generate points at that difficulty, as desired. As each individual performed only eleven reach tasks, no participants had a dataset in which they used every rating from one to ten. Generally most ranked several reaches as one or two, several as an average value of five or six, then the remaining at a high value. There were not enough data to develop individual participant models for all ten difficulty ratings. Therefore the decision was made to pool all of the reach tasks together and use them to develop the final iteration of the model. 15

23 Figure 3.2: The first four spherical harmonics. Green indicates positive values and orange indicates negative values. (Jarosz, 28) 3.3 Piecewise spherical harmonics radius model The model developed as a result of this process calculates the radial distance from the truck seat s H-point to a reach location as a function of the reaching individual s anthropometry, the desired reach difficulty, and the elevation and azimuth coordinates of the location. This model can be used repeatedly over a range of elevation and azimuth values for a single user and difficulty level to yield a surface of points that the individual would rank as reach tasks of that difficulty level. As the surfaces that represent these difficulty levels are curved but not completely spherical,the model was developed from a first-order linear combination of spherical harmonics. A zeroth order (l = ) spherical harmonic describes the the shape of a sphere. From the review of the literature, it was observed that reach envelopes are never perfect spheres. Therefore it was anticipated that that additional terms would be necessary. A zeroth order spherical harmonic is a constant value, but adding the first order harmonics is the equivalent of adding three first order polynomials in cartesian coordinates, allowing the radius of the resulting shape to vary along its surface. The decision was made to begin the model creation process with a first order (l = 1) model, which would include the first four spherical harmonics. Upon generating results, it would be determined if a higher-order model was necessary to obtain the desired accuracy. The first-order model consisted of 16

24 the spherical harmonic terms r(θ, φ) = β 1 4π + β 2 4π sin φ sin θ + β 3 4π cos θ + β 4 cos φ sin θ (3.3) 4π where the coefficients β were to be calculated from the collected UMTRI data. Each of the twenty-six participants ranked the difficulty of reaching the same eleven handle locations. As previous attempts to make a continuous model were unsuccessful, the final iteration of the model was piecewise, with a different equation for each difficulty rating. With twenty-six volunteers performing eleven reaches each, there were 286 individual reach tasks in the data set, each described by the anthropometry of the participant, the difficulty of the reach, and the spherical coordinates of the reach point. These points are tabulated in Table 3.4. The reach task data were sorted into ten bins which corresponded to the ten-point scale used to rate the tasks. All tasks that were rated with a difficulty value of 1 were placed into a bin, the tasks rated 2 were placed in the next bin, and so on until there were ten bins, each containing all reach tasks rated at its difficulty. The radius values for each task were then normalized with respect to participant anthropometry. Several combinations of anthropometric values and ratios of these values were tested to determine the most suitable normalization factor. Table 3.4: Spherical coordinates (mm and rad) of the eleven reach tasks in relation to the H-point r θ φ H-point Handle_Con Handle_Con Handle_Con Handle_Con Handle_Con Handle_Con Handle_Con Handle_Con Handle_Con Handle_Con Handle_ConHz

25 A variety of anthropometric measures were used as the normalization factor and residual sum of squares were employed to observe their effect on the model s accuracy. These measures included stature, sitting height, arm span, and BMI. While it was expected that the arm span would highly influence reach capabilities, circumference measures such as BMI were also used to determine if overall body type (i.e. thin vs. large) had an effect as well. As seen in Table 3.5, measures of length such as stature and arm span provided more accurate models than measures of circumference such as chest depth or BMI (which had no correlation with reach difficulty). The ratio of stature to sitting height of the participant yielded the most accurate results and was thus employed in the final model. For this normalization factor S, the model to be fitted for each difficulty level took the form r(θ, φ) S = β 1 4π + β 2 4π sin φ sin θ + β 3 4π cos θ + β 4 cos φ sin θ (3.4) 4π Table 3.5: Goodness of Fit Results for Varying Normalization Factors D Stature/Sitting Height Stature Sitting Height BMI Arm Span Chest Depth R2 18

26 To compute the final model, the first four spherical harmonics for each handle were calculated from the obtained θ and φ values. A series of regressions were performed to determine the coefficients β which would best fit the model to the normalized radii that corresponded to each difficulty value. As the relationship between the difficulty rating of the points and the point locations proved to be nonlinear, standard least-squares methods were ineffective at yielding coefficients that resulted in an accurate model. The MATLAB function nlinfit(), which employs the Levenberg-Marquart algorithm to estimate model parameters, was employed to numerically estimate the coefficients. The normalized radius values were used as the responses and the first four spherical harmonics as predictors, with an initial guess for the coefficients obtained from the least-squared estimates. The resulting piecewise model returns the radial distance of a point rated difficulty D at spherical coordinates θ and φ as rated by a person with a ratio of stature to sitting height S r(θ, φ, S) = S(β 1 4π + β 2 4π sin φ sin θ + β 3 4π cos θ β 4 cos φ sin θ) (3.5) 4π 19

27 CHAPTER IV DATA AND RESULTS 4.1 Assessment of model fit After performing the nonlinear regression to fit the model to the points corresponding to each difficulty rating, a piecewise equation was used to summarize the curves representing all possible difficulty ratings for a given person r(θ, φ, S) = S(β 1 4π + β 2 4π sin φ sin θ + β 3 4π cos θ β 4 cos φ sin θ) (4.1) 4π where S is the ratio of stature to sitting height of the person in question, and values of β for varying difficulties appear as tabulated. Table 4.1: Model Coefficients Difficulty β 1 β 2 β 3 β

28 In order to utilize the model, the designer would obtain the value of S for the potential user in question (see Table 4.2 for a list of ratios from the study participants) and input the values of β from Table 4.1 for the desired difficulty rating at the point of interest (θ,φ). The coefficient values vary highly in both sign and magnitude between difficulty levels. This illustrates that there are more factors than anthropometry and point location affecting the rating of each location. As attempts in earlier model iterations to find a correlation between these coefficients and develop a continuous model were unsuccessful, it is expected that most of the variation is due to personal preference and other unquantifiable factors. Table 4.2: Participant Statures and Sitting Heights Used to Test Fit Participant Stature Sitting Height Stature/Sitting Height FMH FMH FMH FML FSH FSL FSL FSL FSL FTL FTL MMH MMH MMH MML MSH MSH MSL MSL MTH MTH MTL MTL MTL MTL PilotMML

29 The model was used to calculate the expected radius values of each reach task conducted in the experiment. By computing the residual sum of squares between the radius estimates generated by the model and the radius values of the points used the experiment, the model s accuracy could be determined. The R 2 values varied from.624 to.992, indicating a positive correlation between the model s values and the actual values. As the R 2 test was not designed for use with nonlinear models, an alternative measure of fit proposed in (Haessal, 1978) was employed to verify the results. Calculating cos φ = (y i y)(y i y ) 2 (y i y) 2 (y i y ) 2 (4.2) where y denotes the experimental data and y denotes data calculated from the model yields a value cos φ which varies from to 1 according to the goodness of fit in the same manner as R 2. This value describes variation in deviations about the sample mean and generates identical results to the R 2 measure for linear models while retaining utility for nonlinear models (Haessal, 1978). The results of this test can be seen in Table 4.4. Table 4.3: Residual sum of square values observed for each difficulty model D R

30 The results are identical to those seen in Table 4.3 to two significant figures, indicating that in this case, for the entire dataset R 2 was an appropriate measure of the model s fit and suggests that the model is generally linear in form. The model s performance improved at higher difficulty levels, suggesting that for this particular reach task, the estimates of higher difficulties will be more accurate than those of lower difficulty ratings. The R 2 value for Difficulty 1, however, is unusually high. Examining the distribution of tasks rated at each difficulty level in Table 4.6 reveals that only 13 of the the 286 total reach tasks were rated with a difficulty of 1. Similarly, only 18 tasks were rated as Difficulty 9 and 16 as Difficulty 8. It is possible, therefore that the high R 2 values seen in the highest difficulties may be due to overfitting the model. While this particular model reproduced the results of the experiment well, it was important to test if it was applicable to other data. While Tables 4.3 and 4.4 show the model s goodness of fit for the entire dataset, it was also desirable to assess the accuracy with which it predicted points for each of the participants individually. Table 4.4: Alternate goodness of fit values observed for each difficulty model D R

31 A complete estimate of the model s accuracy for every difficulty level for each individual was impossible to calculate. Not every participant used every difficulty rating and in many cases only used the ratings once, making a goodness-of-fit test between multiple points of the same rating impossible. Therefore, the general accuracy for individuals was calculated by determining the average difference in millimeters between the model s calculated radius values and the actual radius values of the points ranked at each difficulty for each participant. Table 4.5: Average Difference Between Calculated and Experimental Radius Values for Individual Reach Difficulties (mm) Difficulty Level Participant FMH na na na na FMH2 na na na FMH na na na na na FML na na na na na na na FSH1 na na FSL1 na na na na 29.6 na 11.9 FSL na na na FSL na na na na na na na FSL4 na na na FTL na 91.1 na na na na FTL na na na MMH na 68.9 na na na 1.83 MMH na na 26.4 na na na MMH na na na 41 na na MML na 11 na na MSH na na na na MSH na na 42.2 na MSL na 56.5 na na na na 8.33 na MSL na na na na na na MTH na 6.69 na 56 na na na MTH na na na na na na MTL1 na na na na na MTL na na na na MTL na na na na MTL4 na 125 na na na na PilotMML na na 24

32 The differences between the calculated radius values and the experimental values vary from 1.83 mm to 23 mm. From Table 4.5 it can be seen that differences greater than 1 mm are present only in estimations of difficulty ratings 1-3. The range of differences in difficulties 1-3 vary from 2.91 to 23, while difficulties 8-1 have differences ranging from 1.83 to 66.9, supporting the calculation of superior R 2 values at higher difficulty levels. The cause for this wide range in the accuracy be partially explained by the differing proportions of the various difficulty ratings with respect to the total set of experimental data. The most inaccurate predictions were largely concentrated in the lower difficulty ratings. From Table 4.6 it can be seen that more handle locations were rated as difficulties 1 or 2 than any other rating. Therefore the difficulty rating model for these lower ratings is fitting a higher number of points with a larger variance than the higher difficulty models, leading to inaccuracies. For example, from Appendix A, a rating of 2 is assigned by at least one participant to every location except Handle 2. Ratings of 1 only appear for Handles 1, 3, 7, and 1. The portion of the model describing a reach of difficulty 2 is based on four times as many points covering nearly twice as many possible radius values as the model for difficulty 1. This supports the theory that the higher difficulty models are overfit. The majority of research in the development of reach envelopes has focused on maximum reach capabilities, as opposed to perception of reach difficulty at varying locations. Approximately two-thirds of the total reach tasks conducted were ranked as difficulty 4 or below. There is more consensus amongst study participants in the handle locations that are ranked as the most difficult, but the distinction in what separates an easy reach that one would rank as difficulty 1 as opposed to a 3 or 4 is not as easily quantified. 25