yiZ and MP.x

. This equation is also based on five parameters, but three are directly measurable (Az, F^, r )^. Therefore, the risk of error accumulation and propagation is less.

by using kinetic and kinematic methods, and the exponential curve fit of the kinetic method. The results show that one does not have sufficient reliability at the beginning and end of the propulsion phase for the calculated cp, 48 as similarly reported by Cooper et al. [52]. This is likely because of the lack of constant stability during the initial period of the propulsion phase (roughly the first 20%) when the hand impacts the handrim. During the later part of the propulsion phase (roughly the last 15%) the grip on the handrim becomes soft, and the propulsive moment begins to decrease. Therefore, it is reasonable to attribute these instabilities to the making and breaking of the hand contact with the handrim. During much of the propulsion phase, there is a reasonable relationship between the two calculated (p by using the kinetic and kinematic methods. To improve the reliability of the results,

by using the kinetic method and the exponential curve fit of the kinetic method, respectively (Figures 2.12-13). The figures show an absolute error of about ±3° for

using the kinetic method. Now, using q> and Equation 2.6f one can calculate M/,z. Figure 2.14 illustrates the behavior of the global propulsive moment and the hand moment in the z direction during the propulsion phase. It can be seen that they act in opposite directions, meaning that Mhz reduces the propulsive moment. This situation is unavoidable and necessary for the natural stability of the propulsion. 50 40 30 __, is a reasonable assumption. 20 40 60 Propulsion phase (% 80 100 Mhx Mhy Mhz •Curve fit for Mhx •Curve fit for Mhy •Curve fit for Mhz Figure 2.15 Components of the user's hand moment. Microsoft Excel , MATLAB and LabVIEW software were used to calculate all forces, moments, and __

__3 = d__

__/BMgy=l/D ( 2 . 2 0 b ) D5 = d__

__7 = d__

__)2£/2 Ao =(rh cos(p)2Ul +(F^ cosr^)2^2 +(Fgyr/l sin„) 2t/ 2 (2.24c) (2.24d) UMtt=[D9+Dl0] 1/2 (2.24f) 64 As the results of the tests presented low values for hand moments (Figure 2.15), their uncertainties were not calculated. In the next section, the instrumented wheel system is verified by using an experimental technique, and system specifications are determined by applying statistical methods. 2.6 System Verification To obtain the degree of reliability of the results obtained from the designed and fabricated instrumented wheel, an experimental technique was used to determine the system specifications by performing static and dynamic tests. Four different setups were used for these tests, and both qualitative and quantitative analyses were conducted. Pearson correlation and coefficient of variation techniques were used to determine linearity and repeatability, respectively, as key system specifications. Also, the error for quantitative analysis was estimated. Three different angular velocities were used in the dynamic tests. The static and dynamic tests were performed at different levels of loading on the handrim at four different loading positions. 2.6.1 Experimental Setup The following four different loading setups were used: a first vertical loading for static tests, a second vertical loading for static tests, a horizontal loading for static tests, and a dynamic loading. 65 2.6.1.1 First Vertical Loading Setup The first vertical loading setup for static tests was used to apply the selected vertical loads at four loading points (one at a time) when they were placed in turn at the loading position of point 1 (Figures 2.20 and 2.21) on the handrim. Points 1 and 2 were at the intersections of a horizontal line passing through the handrim and its center. A loading disk was connected via a wire cable to the handrim with a clamp. Six different weights (22.27, 44.48, 66.76, 89.04, 111.50 and 133.30 N) were used in this setup. The level of resolution for the weights depends on the resolution of the sensor—in our case, 0.01 N. The range of weights covers the typical loads applied on the handrim during the propulsion. The first local coordinate system, which is attached to the load transducer, turned in unison with points 2, 3 or 4, when they turned into the position of point 1. Rotation of the wheels during the static loading was prevented by locking the shaft of one of the rollers. Figure 2.20 First vertical loading setup for static tests. 66 2.6.1.2 Second Vertical Loading Setup The second vertical loading setup for static tests applied the selected vertical loads at the four points when they were placed at the position of point 3 (Figure 2.21) on the handrim. Points 3 and 4 were at the intersections of the handrim and a vertical line passing through its center. A load-holding disk was hung from a horizontal bar 2 m in length and 1.5 cm in diameter. One side of the bar was hung from the handrim using a rope so that only a normal load was transmitted. The other side of the bar rested perpendicularly on a bar 2 cm in length and 0.5 cm in diameter, which itself rested on a smooth horizontal surface. This combination provided a rolling effect and eliminated horizontal frictional loads on the long bar due to deformation and shortening of its span after loading. The set of six weights used in this setup was the same as the set used in the initial vertical loading setup. Figure 2.21 Second vertical loading setup for static tests. 67 2.6.1.3 Horizontal Loading Setup The horizontal loading setup was used to apply horizontal static test loads at four different points located 90° apart from each other on the handrim's outer circumference. In fact, these were the same four points used in the previous tests. Four loading points were used to cover the entire circumference of the handrim. These points were at the intersections of the x and y axes of the first local coordinate system and the handrim, and were used to apply pure axial loads. A loading disk was connected to the loading point on the handrim through a pulley using a 2 mm wire cable (Figure 2.22). Six different weights (4.50, 9.02, 13.49, 16.41, 19.31 and 22.23 N) were used in this setup for each point. During manual wheelchair propulsion, the subject applied a lower load in the direction of the axle of the wheel compared with the loads in the plane of the wheel. Therefore, we used a new set of the loads in the horizontal static tests. Figure 2.22 The horizontal loading setup for static tests. 68 2.6.1.4 Dynamic Loading Setup The dynamic loading setup applied centrifugal test forces at four loading points, which were also the same as the loading points in the static setups. Three different weights (4.50, 8.95 and 13.39 N) were used as loads. The loads were attached individually to the handrim's lateral surface with a very powerful magnet (Figure 2.23). The AC motor was used to mechanically turn the wheel at three different speeds. Figure 2.23 The dynamic loading setup. Increasing the number of weights, loading points and tests can yield more data for different parts of the system, but it also increases the calculation time. Therefore, the 69 number of weights, loading points and tests were chosen such that proper statistical analysis could be performed within a reasonable time and with sufficient accuracy. The test loads were not meant to reproduce the level of the loads applied by the wheelchair user. 2.6.2 Verification Tests Protocol After the design and fabrication of the instrumented wheel and determining the transformation equations for the applied forces and moments [56], the system had to be verified. For this purpose, both qualitative and quantitative analyses were performed for the output of the experiments. Two of the most important system specifications for qualitative analysis—linearity and repeatability-— were determined using Pearson correlation and descriptive analysis, respectively. The error for the quantitative analysis was also estimated. Given the dynamic nature of the real situations, both static and dynamic conditions had to be considered to verify the system. 2.6.2.1 Static Verification To verify the system under static conditions, the wheelchair was placed on and securely strapped to the roller-rig. Three different test setups, described in Section 2.6.1, were used to apply loads in three different directions (x, y and z) of the first local coordinate system (Figure 2.2). For vertical loading in the static tests, six different weights (22.27, 44.48, 66.76, 89.04, 111.50 and 133.30 N) were suspended independently from points 1,2,3 and 4 on the handrim circumference using two vertical loading setups (Figures 2.20 and 2.21). The loading positions were 90° apart in the -x', -y', x', and y' 70 d i r e c t i o n s o f t he f i r s t l o c a l c o o r d i n a t e s y s t e m ( F i g u r e 2 .2 ) . T h e b a s e l i n e o f t he l o a d h o l d i n g d i s k ' s o w n w e i g h t w a s m e a s u r e d b y p e r f o r m i n g a n o - l o a d test , a n d the r e s u l t s s u b t r a c t e d f r o m the m e a s u r e d l o a d s a c c o r d i n g l y . M e a s u r e m e n t s w e r e r e p e a t e d th ree T o d e t e r m i n e the s p e c i f i c a t i o n s f o r q u a l i t a t i v e a n a l y s i s , t he P e a r s o n c o r r e l a t i o n c o e f f i c i e n t ( r ) w a s u s e d , w h i c h i s d e f i n e d as w h e r e sx a n d sy a re the s t a n d a r d d e v i a t i o n s o f the i n d e p e n d e n t a n d d e p e n d e n t v a r i a b l e s a n d the v a l u e bo i s d e t e r m i n e d as w h e r e x , i s t he c a s e v a l u e f o r the i n d e p e n d e n t v a r i a b l e , x i s the m e a n o f t he i n d e p e n d e n t v a r i a b l e , yt i s t he c a s e v a l u e f o r the d e p e n d e n t v a r i a b l e , y i s the m e a n o f t he d e p e n d e n t v a r i a b l e , N i s t he n u m b e r o f c a s e s a n d sx i s t he v a r i a n c e o f the i n d e p e n d e n t v a r i a b l e [ 6 3 ] . I n t h i s s t u d y , d e p e n d e n t v a r i a b l e s a re the m e a s u r e d f o r c e s a n d m o m e n t s a n d i n d e p e n d e n t v a r i a b l e s a re the a p p l i e d l o a d s at d i f f e r e n t l o a d i n g p o i n t s . T h e P e a r s o n c o r r e l a t i o n c o e f f i c i e n t m e t h o d w a s u s e d t o o b t a i n t he l i n e a r i t y o f t h e s y s t e m . T h e c o e f f i c i e n t o f v a r i a t i o n w a s u s e d f o r a l l d i f f e r e n t tes ts t o d e t e r m i n e s y s t e m r e p e a t a b i l i t y , a n d t o c o m p a r e the v a r i a b i l i t y o f d i f f e r e n t p a r a m e t e r s w i t h d i f f e r e n t u n i t s . t i m e s at f o u r d i f f e r e n t l o a d i n g p o i n t s w i t h r e s p e c t to t he f i r s t l o c a l c o o r d i n a t e s y s t e m . r = b0x(sx/sy) ( 2 . 2 5 ) ( 2 . 2 6 ) 71 The coefficient of variation expresses the standard deviation as a percentage of the mean. This allows one to compare the variability of different parameters. The coefficient of variation is given by Coefficient of variation = {^tandard deviati°^mear^100 (2.27) where mean is the mean of the variable of interest. To determine the specifications of the instrumented wheel from the quantitative analysis, the actual values were compared with the measured values. SPSS® 11.0 and Microsoft Excel® software were used to analyze the data and calculate the system specifications. All r values were calculated by using the results of the first series of tests. Table 2.5 shows r due to static verification. The "Position" column gives the different load application points, and the "Channel" row gives different measurements. The values of r show high linearity (above 0.9) at different loading points and for different measuring channels in the static situation. Table 2.5 Pearson correlation coefficient r (static verification). ~^~"~~~----~^ _____Channel Position Fx Py Fz Mx My Mz 1 1.000 1.000 0.999 1.000 1.000 1.000 2 1.000 0.994 0.998 0.993 1.000 0.985 3 1.000 1.000 0.997 1.000 1.000 1.000 4 1.000 1.000 0.999 1.000 1.000 1.000 72 Table 2.6 shows the mean of the percentages of the coefficient of variation for different measured loads at the four loading points. The "Load" column gives the different loading forces used during the tests. The loads differ for channel Fz because they did not reach high values during propulsion. These values indicate low coefficients of variation (less than 2%), and were calculated using the measured values of the three different tests. The entries in Tables 2.6 show high repeatability of the instrumented wheel. Tables 2.5 and 2.6 present the results for the qualitative analysis and collectively show reliable values for system specification. The average of the results from three series of the repeated tests has been used to calculate the mean errors. Table 2.6 Mean coefficient of variation of measured loads (%; static verification). ^•^^Channel L o a d ( N j ^ \ Fx Fy My Mz "~^~~^Channel L o a d ( N ) ^ \ Fz 22.273 0.110 0.166 1.547 1.784 0.133 4.50 1.736 44.482 0.045 0.147 1.545 0.398 0.166 9.02 1.293 66.755 0.059 0.174 1.401 0.289 0.082 13.49 0.863 89.043 0.070 0.053 1.314 0.355 0.067 16.41 0.747 111.504 0.102 0.117 1.192 0.381 0.563 19.31 0.895 133.299 0.096 0.106 0.234 0.485 0.117 22.23 1.325 Table 2.7 presents the results of the quantitative analysis and lists the mean errors of the measured forces and moments as percentages of the loads. The values indicate low mean error (mostly less than 5%) for different loads on all channels. Some errors were 73 expected because of the effect of other sources of errors, such as human or experimental errors. The low levels of the errors indicate that the parameters measured by the instrumented wheel are reliable. Table 2.7 Mean errors as percentage of loads (static verification). ^ ~ \ C h a n n e l L o a d ( N T \ ^ Fx Fy Mx My Mz " ^ \ ^ C h a n n e l Load ( N T \ ^ Fz 22.273 0.857 0.070 1.608 2.969 0.344 4.504 3.572 44.482 0.583 0.291 8.422 4.556 0.113 9.015 1.374 66.755 0.640 0.144 2.586 1.088 0.179 13.489 2.037 89.043 0.576 0.170 1.922 1.420 0.081 16.406 2.680 111.504 0.726 0.074 3.187 0.881 0.259 19.308 3.718 133.299 0.666 0.097 3.383 0.614 0.128 22.225 7.401 The results of qualitative and quantitative analyses for the mstrumented wheel in the static situation show a reliable range of the values for all system specifications. 2.6.2.2 Dynamic Verification Dynamic verification was more challenging than static verification. The local coordinate system of the transducer spun with the wheel and the loadings were weights, so the loads (in the global coordinate system) could not be measured directly. An encoder was used to determine the position of the load attached to the wheel with respect to the global coordinate system. The wheelchair was placed on the roller-rig, and the A C motor 74 rotated the driving roller. Three different angular velocities (3.0, 3.8, and 4.8 [rad/s]) were used for the dynamic tests to cover the wheeling speeds of the user. Three different weights (4.50, 8.95 and 13.39 N) and one powerful magnet were used for loading at points 1 to 4 (Figures 2.20, 2.21 and 2.22) on the handrim lateral surface. The loading positions were the same as for the static verification tests. The measured forces and moments of three successive cycles were used to verify the system repeatability. The baseline of the attachment's own weight was set to zero by using the method described in the static verification tests. The actual values were compared with the measured values to obtain the specifications for quantitative analysis. The actual values were determined using the inverse dynamics method. The angular motion of the loaded wheel was considered in the vertical plane, where the centripetal force, Fs was determined as Here, mw is the mass of the weight that was attached to the wheel, o, is the moment arm (handrim radius), 0 is the wheel angular velocity and g is the acceleration of gravity. There was no force component in the z direction because the object had a planar motion (x-y) and the wheel camber angle was zero with respect to the global coordinate system. The x and y planar components are as follows: Fs =rnwrhe2+mwg (2.28) (2.29a) F —m ru02 cos0 — m g sy w h wo (2.29b) 75 where 6 is the angular position of the wheel (or load for these tests), and Fs.Xwy are the components of the centripetal force. Equations 2.25, 2.26 and 2.29 were used to determine the specifications for qualitative analysis in the dynamic tests with three different angular velocities. As the nature of the manual wheelchair propulsion is dynamic, qualitative and quantitative analyses were performed for the instrumented wheel under dynamic situations. These analyses were carried out for three different angular velocities (3, 3.8 and 4.8 rad/s). Table 2.8 shows the Pearson correlation coefficient for the tests conducted. These values mostly show high correlation (r above 0.9) between different angular velocities and loadings implying very good linearity. Channel Fz was not considered for dynamic verification because there was no appreciable load on this channel, due to the nature of dynamic loading. The values in Tables 2.9 show a low mean coefficient of variation for different measured loads (less than 4%) at four loading points, and high repeatability of the instrumented wheel. Tables 2.8 and 2.9 show the results of the qualitative analysis. They indicate reliable values for system specification in the dynamic verification tests. The mean errors produced by the instrumented wheel as a percentage of loads are presented in Table 2.10 (quantitative analysis). The low mean error values (mostly less than 6%) indicate that, the parameters measured by the instrumented wheel are equally reliable for the dynamic situations. 76 Table 2.8 Pearson correlation coefficient r (dynamic verification). ^•^^Channel Position"""---^ Fx Fy Mx My Mz 0= 3 rad/s 1 1.000 1.000 0.999 1.000 1.000 2 1.000 1.000 1.000 1.000 0.992 3 1.000 1.000 1.000 0.997 0.998 4 1.000 1.000 0.993 1.000 0.987 (9=3.8 rad/s 1 1.000 1.000 0.999 1.000 1.000 2 0.999 1.000 0.999 0.996 0.982 3 1.000 1.000 0.994 1.000 0.995 4 1.000 1.000 0.996 1.000 0.989 (9=4.8 rad/s 1 1.000 1.000 0.993 0.985 1.000 2 0.998 1.000 0.993 0.989 0.991 3 1.000 1.000 0.998 1.000 0.989 4 0.997 1.000 0.999 0.999 0.991 Table 2.9 Mean coefficient of variation of measured loads (%; dynamic verification). ^^-^Qiannel L o a d ( T ^ T ) \ ^ Fx Fy Mx My Mz 0=3 rad/s 4.50 2.170 1.467 1.547 0.967 0.842 8.95 1.871 1.752 1.236 1.226 0.144 13.39 1.230 0.954 0.638 0.761 0.115 (9=3.8 rad/s 4.50 2.511 3.196 2.892 1.475 0.445 8.95 2.144 1.498 2.781 1.915 0.300 13.39 1.054 0.968 0.896 0.937 0.078 .9=4.8 rad/s 4.50 3.986 3.404 2.191 3.051 0.844 8.95 2.529 1.604 2.247 3.695 0.456 13.39 1.691 0.856 3.380 1.721 1.347 77 Table 2.10 Mean errors as percentage of loads (dynamic verification). ^^^--^Channel L o a d ( N 5 ^ \ Fy Mx My Mz 0=3 rad/s 4.50 5.675 7.724 3.276 4.514 4.395 8.95 6.532 6.820 1.924 5.227 6.005 13.39 6.723 7.038 2.944 6.878 5.638 (9=3.8 rad/s 4.50 4.115 4.833 3.762 4.116 3.134 8.95 5.581 6.636 4.798 2.867 5.530 13.39 7.430 6.909 5.579 3.259 5.640 •9=4.8 rad/s 4.50 5.837 6.200 3.079 6.189 4.643 8.95 5.675 7.211 5.794 4.329 4.300 13.39 5.590 8.079 2.281 3.467 4.639 Given the actual performance for the instrumented wheel and its measurements, Figures 2.24 and 2.25 show the measured and predicted values for Fx, Fy, Mx, My and Mz with respect to the global coordinate system. As mentioned previously, Fz was not considered in the dynamic measurements because there was no significant load on this channel due to the nature of loadings for dynamic tests. These figures show that the patterns of the measured and predicted curves of the data for forces and moments are highly compatible with typical results measured by other researchers [49]. 78 .-| 0 I 1 1 1 ! I 1 ! I I L 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) Figure 2.24 Measured and predicted global sample force components. 2.5 _2 51 i i i i i i i i i i_ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) Figure 2.25 Measured and predicted global sample moment components. 2.7 Conclusions In this chapter, a general uncertainty analysis was performed to determine the uncertainty equations for the local and global forces and moments, the local hand forces and moments, and the hand-contact angular position in MWP. The uncertainty values for the local and global forces and moments and the hand-contact forces were then calculated. The results provided an estimation of the errors and uncertainty in the output of the instrumented wheel. The uncertainties were found to vary from 1.40 to 3.40 N for the local forces and from 0.20 to 0.70 N.m for the local moments. The maximum and minimum of the uncertainties for global values were about the same as the uncertainties for the local values, but the patterns of variation were different. Uncertainties determined by Cooper et al. [52] for the forces and moments are in the range of 1.1-2.5 N and 0.03-0.19 N.m in the plane of the handrim, and 0.93 N and 2.24 N.m in the wheel axle direction, respectively. Our results show uncertainty for the forces and moments in the range of 1.40-1.70 N and 0.58-0.68 N.m in the plane of the handrim, and about 3.40 N and 0.25 N.m in the wheel axle direction, respectively. For our system, however, the uncertainty values for the important load components, namely the planar forces and the axial moment, are low. The absolute error for hand-contact position was determined as ± 3° or ± 1 cm for most of the propulsion phase. Cooper et al. reported uncertainties between 1.8° and 16° for the hand-contact position using their Smart w h e e l [52]. A complete experimental technique was designed and performed under different static and dynamic conditions to determine the specification of the instrumented wheel. The verification techniques, which were highlighted and demonstrated step-by-step, can be implemented in similar wheelchair instrumentation setups. The results of the static and 80 dynamic tests were used for both qualitative and quantitative analyses to determine the system specifications. The static tests showed high linearity (r above 0.9), very low standard deviation (mostly close to zero) and a low mean coefficient of variation for measured loads (less than 2%). These results indicate high repeatability and low mean error (mostly less than 5%) due to the different loading for all load channels. Two cells of Table 2.6 show mean errors above 5%, with one at the maximum horizontal load. Usually, the horizontally applied loads are not so high during manual wheelchair propulsion, and the main idea is that the users try to apply planar loads. Therefore, the system is not proportionally responsive to higher horizontal loading due to its structure, but it had mean errors less than 5% for all other lower horizontal loadings. The other mean error above 5% corresponded to Mx. All of the mean errors for this column were close to 2% or more. Generally, higher values of the mean error (%) were obtained for Mx compared with the mean errors (%) for the other channels. Dynamic tests were performed at three angular velocities and at four loading positions for all measuring channels. The results also showed high linearity (r above 0.9). The low mean coefficient of variation for measured loads (less than 4%) confirmed high repeatability (reliability) of the instrumented wheel. The results showed that most of the mean errors were around 5%. The resultant specifications showed high linearity, repeatability and a low percentage for errors. The results presented in this chapter collectively show that it is possible to reliably obtain the essential information required for manual wheelchair propulsion analysis, including the applied forces and moments, using the designed and fabricated instrumented wheel. The tests with one able-bodied subject reproduced patterns and 81 overall behavior comparable to the available data, ensuring that the system can be used for the designed experiments. It is worth emphasizing that a system developed in-house allows flexibility in enhancing the experimental scope. The instrumented and verified wheel can now be used to determine the kinetic aspects of wheelchair propulsion. Varying the seat position with respect to the wheel axle affects all the forces and moments, as well as the mechanics, of propulsion. Determining and prescribing optimum positions is expected to reduce pain and help prevent injury of manual wheelchair users, and may improve the gross mechanical efficiency of propulsion. In the next chapter, the proposed indices for efficiency and injury assessment of MWP are described. These indices are used as criteria to determine the optimum wheelchair variables. 82 CHAPTER 3 Efficiency and Injury Assessment 3.1 Introduction Previous studies have reported a low value of about 10% for the efficiency of Manual Wheelchair Propulsion (MWP) [22]. The Gross Mechanical Efficiency (GME) for human movement is defined as the ratio of the work accomplished to the amount of the corresponding Metabolic Energy Expenditure (MEE) [23]. However, few studies have focused on the mechanical factors only and have not considered the physiological aspects of the MWP in calculating the efficiency [64]. Although the reasons for this low efficiency have not been sufficiently addressed in many studies [8, 65,66], Veeger, et al. [8] reported that it can be partially attributed to nonoptimal tuning of the wheelchair to the functional abilities of the user. Braking torques at the start and end of the propulsion phase [21], and suboptimal direction of the propulsion force [8] are reported as the other possible causes for the low efficiency of 83 MWP. Whereas, de Groot et al. [9] reported that even the applied tangential force acting in the optimal direction tends to decrease G M E because of the conflict that was explained in Section 1.5.1. Injuries due to MWP are usually consistent with pain. It has been determined that the pain is a limiting factor in the daily activities for MWUs [67]. Roach et al. [68] developed the Shoulder Pain and Disability Index (SPADI) to quantify shoulder pain and difficulties during the functional activities in an ambulatory population. Curtis et al. [67] developed the Wheelchair User's Shoulder Pain Index (WUSPI) to measure the severity of the shoulder pain associated with functional activity in the individuals who use wheelchairs. SPADI and WUSPI are in fact 38-item and 15-item questionnaires, respectively. A score is given for the response to each item. The higher the total score, the greater pain interference of activities. These indices do not measure pain intensity, but pain interference. Wheelchair Propulsion Strength Rate (WPSR) is a reported injury index and is the ratio of the joint moment generated during propulsion and during the maximum isometric strength test in different directions [17]. It has been hypothesized that larger values of this ratio indicates a high risk of injury. In this chapter, a new index for efficiency assessment during MWP is proposed. This index uses the heart rate of the subject as a factor to estimate the variation of the M E E . This is followed by developing two new injury indices to estimate the level of the probable injury due to MWP. 84 3.2 Efficiency Assessment Wheelchair-user system is a combination of a mechanical device and a human body. A major requirement for reliability of the efficiency assessment for MWP is to consider both physiological and mechanical aspects of the motion. Measuring the oxygen uptake is one way of estimating M E E , but some of the subjects may feel uncomfortable to have the device on their face during the tests, which may affect their natural performance. Finding an alternative method could be helpful for some of the studies that have above consideration for the subjects or do not have access to the respective equipment. However, one should consider that the calculated results using the heart rate can be used for within-subject analysis. hi Section 1.5.4, we reviewed some studies, which reported that the physiological cost of the body can be predicted by measuring heart rate. Figure 3.1 shows the variation of heart rate versus time, during the start, performance and finishing a steady-state exercise [39]. Area 2 represents the total number of the subject's heartbeats above the resting level during the steady-state phase of the test. The area 2 is determined by subtracting the numbers of heartbeats during the exercise from the resting level, multiplied by the seconds of the test period. In this study, the area 2 will be determined for the tests. While the areas 1 and 2 represent the total heartbeats during the exercise above the resting level, the areas 1, 2 and 4 show the total number of the heartbeats during the exercise, including the resting level. Area 3 represents the extra heartbeats during the recovery phase. Areas 3 and 5 together represent the total heartbeats during the recovery. The resting heartbeats during the exercise and recovery are defined by areas 4 and 5, respectively. 85 Gross Mechanical Efficiency (GME) considers the biological aspects of the manual wheelchair propulsion [9], and is defined as Gross Mechanical Efficiency = Useful Energy Out / Metabolic Energy Expenditure or: GME = Mz • A0/MEE (3.1) where Mz is the average propulsive moment applied on the hub of the wheel, and AO is the angular displacement of the wheel; both during the test period. Heart rate Steady-state heart rate Resting heart rate Figure 3.1 Variation of heart rate versus time, from start to completion of a steady-state exercise and back to rest (inspired by [39]). 86 Methodology— The linear relationship between the heartbeats and the oxygen uptake profile for steady state and non-steady state situations allows us to use an alternative and convenient method for measuring M E E . We prefer the heart rate measurement over oxygen uptake measurement because it is more comfortable, cheaper and the equipment is lighter. Continuous heart rate monitoring is now possible with the development of portable heart rate monitors. We have proposed Equivalent Biomedical Index (EBI) for efficiency assessment, which reflects the behavior of G M E and is given by Equivalent Biomedical Index = Useful Energy Out / Total Heartbeats or: EBI = M Z A0 /THB (3.2) where THB is the area 2 in Figure 3.1. A comparison of the Equations 3.1 and 3.2 shows that their numerators are the same and their denominator have linear relationship with one another. Therefore, EBI can reflect the behavior of GME. 3.3 Injury Assessment The referenced studies in Chapter 1 indicated that MWP is associated with injuries. To predict probable injuries during MWP, two injury indices were proposed. These indices consist of the important factors that may lead to injury. 87 As reported by other researchers, higher applied forces and moments on the handrim may increase the risk of RSI or over-use injury [17,69]. Therefore, the applied forces and moments can be considered as key factors that can cause the injury. It has been reported that the subject's weight is related to pushrim forces and the median nerve function [23]. Also, previous studies have shown that the Body Mass Index (BMI) is significantly related to the shoulder injury [70,71]. BMI is based on the anthropometric data of the subject and can be considered as another factor in injury assessment. As it is assumed that repetitive motion puts a person at the risk of RSI, therefore the pushing frequency of the M W U has direct influence on the wheelchair user's joint injury. Boninger et al. [35] suggested that decreasing the frequency of propulsion may help to prevent median nerve injury and thus CTS. Methodology—Wheelchair User Joint Injury Index (WUJH) is our first proposed injury index and reflects a value that is representative of the level of possible M W U joint injury. A general idea for linking WUJII to the above factors can be expressed as WUJn = F m M m B M I - / p (3.3) Variables Fm and Mm are the maximum total force and moment applied on the hub of the wheel during the propulsion phase, respectively, and fp is the pushing frequency. BMI is defined as 88 BMI = mjh2s (3-4) where ms and hs are the mass and the height of the subject, respectively. Substituting Equation 3.4 in Equation 3.3, WUJU is related to individual parameters as follows: WVm = Fm-Mm-fp-ms/h> (3.5) To be able to compare this index between different subjects, the index is normalized with respect to the subject arm length (anthropometric parameter), and the total weight of the subject and the wheelchair, which affect the applied moment and force on the upper limb's joints: F Mm — —-fp^s W W • L w u j n = — — - — T x IOO A, (3.6a) or F -Mm • fn • m;. WUJII= m 2m2J p sxl00 ( 3.6b) where La is the arm length, and Wt is the weight of the user and wheelchair combined. The index was multiplied by 100 to avoid presenting the values as a percentage. A pre-test showed small numerical value of the index without using a hundred as a coefficient. 89 WUJII can be used for estimating the injury at the shoulder, elbow or wrist joints by using the joint loads instead of the applied loads in Equation 3.6b. The modified form is presented as WUJII, = m-ms-fp-Fmi-Mn ^•h)-Lai (3.7) / = 1,2,3 Variable / represents the corresponding joint as follows: 1-shoulder, 2-elbow, and 3-wrist. Fmi and Mmi are the maximum total force and moment applied on the joint / during the propulsion, and Lai is the length of the upper limb segments connected to the distal part of the joint /, such that La\ is the total length of the arm and hand, La2 is the total length of the forearm and the hand, and Las is the length of the hand. As BMI is not a perfect index for all cases (compare BMI between two persons, who have the same height and weight but one has more fat, and the other has stronger muscles), WUJIT is proposed by using calculated %BF as equations 3.8a and 3.8b for general form and the joints injury analysis, respectively. , 100 • %BF • f • F • M WUJII'= ^ — - (3.8a) W, • L„ WUJII ; = i = 1,2,3 100-%BF-/ / ,F W J . -M„ W,2-Lai (3.8b) Percentage of Body Fat is the ratio of the fat to the total body mass, and therefore is unitless. A variety of techniques have been developed to measure this parameter such 90 as using calipers (skinfold measurements - anthropometry), bioelectrical impedance analysis, hydrodensitometry weighting, near-infrared interactance, magnetic resonance imaging, computed tomography, total body electrical conductivity, and Dual Energy X -ray Absorptiometry (DEXA) [72-80]. The hand-held caliper that exerts a standard pressure was used in the previous studies for SCI subjects, and the skinfold thickness was measured at following body locations: Triceps, Biceps, Subscapula and Supraspinal [72-74]. In this study, the same method and measuring sites were used. Linear regression equation for the estimation of body density (kg/m ) has been reported by Dumin et al. [75] as follows: Density = A0-B0x log Stotal (3.9) where Ao and Bo are the constants and their values differ for different genders and ages, and Stotai is the sum of the skinfold measurements at four sites. In the same report, %BF has been determined as %BF ( s k i n f o l d ) = ( 5 ^ - 4 - 5 0 ) - 1 0 0 (3-10> where %BF( S k i nf 0i d) is the value of %BF that is measured by using the skinfold method. Maggioni et al. [73] reported that measured %BF for people with spinal cord injury using the skinfold method is under-estimated. They introduced Equation 3.11 that shows the relationship between %BF calculated by the skinfold and D E X A methods. 91 D E X A is known as the best method to determine % B F . In this study, the values of % B F were modified by using the following equation as % B F , (DEXA) = 1 . 4 5 % B F , (Skinfold) + 2.58 (3.11) where %BF(DEXA) is the value of % B F that is measured by using the D E X A method. Wheelchair velocity has also been reported as a factor that can affect the efficiency of MWP [4]. Veeger et al. [8] performed manual wheelchair exercise tests on a stationary ergometer for nine able-bodied subjects and determined that G M E increases with lower tangential velocities of the handrim, whereas another study reported that propulsion speed slightly lower than the freely chosen speed is energy efficient [38]. In this study, to verify the effect of velocity on the indices, several tests were performed, which, are described in the next chapter. It was found that the injury indices increase by raising the Average Linear Wheelchair Velocity (would be referred as velocity) during the propulsion phase. Considering the dependency of the injury indices on the velocity, the relations of the injury indices were modified to include the velocity as one of their parameters. This allows the injury indices to be stand alone measures for the level of possible joint injury due to MWP. The modified injury indices for general evaluation are then stated as WUJII = lOO-Fm-Mm-fp-Vrms (3-12) 92 wu jn , lOO-%BF-fp-VrFm-Mn 2 (3-13) where V, is the symbol that represents the velocity. To determine the level of the possible injury at the joints, the modified injury indices are given by WUJIL = \00-ms-fp-VrFmrMmi W2-h)-Lai (3.14) t "s "ai i = 1,2,3 , 100 • %BF • fp-V, • F j • M WUJII = Jp I m W,2.Lai (3.15) »" = U,3 The units of WUJII and WUJII' are kg/(m.s2) and m/s2, respectively. The risk of the injury may increases with higher values of WUJII or WUJII'. The effects of changing the seat position on the proposed indices were investigated in this research. The minimum value for each injury index corresponds to the optimum seat position. We think that WUJII' can be a more realistic index and will discuss it more in Chapter 5. 3.4 Chapter Summary Most of the previous research on the efficiency assessment has been conducted for sport wheelchairs, and some did not consider the physiological aspects of MWP. All 93 of the previous studies about MWP that we found have used oxygen uptake to determine MEE for efficiency assessment. Two reported indices in the previous studies measure the interference of the pain, and they are based on questionnaires. Previous reported injury index was based only on the joint moments during MWP and the maximum isometric strength of the muscles. This injury index does not consider the other factors like the applied force, the frequency of the propulsion, and the weight of the subject that have effects on the injury during MWP. In this chapter, a new index was presented for efficiency assessment of MWP. The use of the subject's heart rate in the proposed index represents the biological factors. EBI was considered as a good alternative to estimate the efficiency of MWP because it uses the heartbeats of the user to estimate the variation of the MEE. Measuring the heartbeats by using the newly developed heart rate monitors is cheaper, more comfortable and the required equipment is lighter. To measure the level of the possible injuries, two indices were proposed. The general forms of the indices determine the level of the probable injury totally at the upper limb of the subject. The indices were modified to be used for injury assessment at each joint of the upper body of the user. Also, there is a choice of using BMI or %BF to calculate the injury indices. In the next chapter, the test procedures are explained and the calculated values of the proposed indices are given. The sensitivities of the indices are verified, and the optimum seat positions for two users are determined by presenting the values of the indices in 3D graphs. 94 CHAPTER 4 Optimum Seat Position 4.1 Introduction The wheelchair-user interface, based on the design and settings of the wheelchair and the physical and habitual characteristics of the user, will affect the pattern of the applied loads, cardio-respiratory factors, kinetic and kinematic parameters, pushing angle, pushing frequency, and joint loads during MWP [13,17,19,35,36,81]. Inappropriate settings can lead to RSI [82]. Furthermore, the results from at least one study have confirmed the possibility of reducing, or even eliminating, back pain and discomfort related to wheelchair seating by individually adjusting the settings of the subjects' wheelchair (p<0.001) [83]. Actually they did not explain their adjustment procedure. Also, another study presented a significant relationship between the wheelchair seat's tilt angle and the biomechanical efficiency [84]. 95 Determining the optimal seating positions for MWUs is a major challenge for researchers. Masee et al. [85] found a low position to be optimal for smoother upper limb motion, less electromyogram (EMG) activity and lower pushing frequency, while another study reported greater upper limb motion in a low position [86], and Kotajarvi et al. [11] did not find lower pushing frequency at the low seat height positions. These contradictions imply that more work is required in this area. In this chapter, a new method for determining the optimum seat position for a user is described, followed by explaining the experimental setup and a subject model. Also, the optimum seat position is determined using an analytical method, and the reliability of the indices are investigated by sensitivity analysis. Four research questions concerning the optimum seat position and the relationship between linear wheelchair velocity and the proposed indices are addressed. 4.2 Modeling and Analysis Approach The data acquired from this study covers the kinetic, kinematic, and part of the anthropometric and physiological information for the subject group of the manual wheelchair users. The above parameters were measured and/or monitored using several devices and instruments that we provided in the Human Measurement Studio at the Institute for Computing, Information and Cognitive System (ICICS), the University of British Columbia (UBC). All measurements were taken at the same time to acquire synchronized data. A medical student assisted in the measurement of the heart rate, blood pressure and the %BF of the subjects. 96 The following method was devised and applied to determine the optimum seat position for each user by analyzing the calculated values of the indices at different seat positions. The subsequent sections of this chapter describe in details the study population, study design, test protocol, setup and modeling. 4.2.1 A n Overview In this research, new injury and efficiency indices were proposed as criteria to determine the optimum wheelchair variables for each manual wheelchair user. To accomplish these, the parameters that are measurable and have significant effect on the factors in the proposed indices were investigated. Possible wheelchair parameters are the horizontal and vertical positions of the seat with respect to the wheel axle, the backrest position or angle, the seat angle, the footrest position and the camber angle of the wheels. Choosing the proper parameters depends on the situations and characteristics of different wheelchairs. If the specifications of the wheelchair are flexible enough for setting and measuring these variables, and the subjects can handle the requirements of the tests, one can use the new method with different combinations of the parameters listed above to determine the optimum position. In this study, the vertical and horizontal positions of the seat with respect to the wheel axle were considered as the adjustable variables of the seat position, which have significant effects on the factors in the proposed indices. The lower and upper bound values of the seat position can be different for various manual wheelchairs. 97 4.2.2 Study Population This study was approved by the Clinical Research Ethics Board at U B C (Appendix). Subjects were recruited through a database provided by Dr. Bonita Sawatzky and the Spinal Cord Injury Research Registry at the GF Strong Rehabilitation Center (Vancouver, BC). All of the subjects were of legal age (> 18 years) and they signed a consent form. Subjects were provided with a $50 honorarium upon completion of the protocol to primarily cover transportation related expenses. All of the subjects met the inclusion and exclusion criteria for the study. The inclusion criteria for the subjects in this study were as follows: • Males with a spinal cord injury for longer than one year. • Age between 19 and 59 years. • Height ideally between 160 and 190 cm. • Dominant right hand side. • Fit into a 16" wide wheelchair or use the same size wheelchair. • Can independently use a manual wheelchair for 50% of the day. Potential subjects were excluded from this study if they: • had been previously diagnosed with any kind of heart or lung disease. • had lesion level higher than the sixth thoracic vertebrae (T6; see Section 1.5.4). • had significant shoulder pain during wheeling. • have had surgery within three months prior to the tests. • were unable to transfer themselves independently from their wheelchair to the test wheelchair. 98 In Chapters 1 and 3, the linear relationship between the heart rate and M E E was explained. Also, it was discussed that among SCI subjects, this linear relationship only works for the individuals with lesions at T6 or lower. Eight adult male MWUs (n= 8) participated in this study. Demographic data are given in Table 4.1. Table 4.1 Demographic data for the manual wheelchair user subjects. Subject code Gender Age (years) Diagnosis Level of lesion 1 M 52 Paraplegia Tll-12 2 M 27 Paraplegia T10 3 M 20 Spina Bifida Lumbar 4 M 48 Paraplegia T10 5 M 59 Paraplegia Tll-12 6 M 49 Paraplegia T6-7 7 M 24 Paraplegia T6 8 M 34 Paraplegia T10 M = Male; Tn = The nth thoracic vertebrae. 4.2.3 Study Design In this research, the relationship between the proposed indices and the seat position for eight MWUs were analyzed and their sensitivity was evaluated. To do this, experiments were designed and implemented in two categories: 1- Fixed seat position; 2- Constant wheelchair velocity. 99 I n t he f i x e d seat p o s i t i o n tes ts , the e x p e r i m e n t s w e r e p e r f o r m e d at th ree d i f f e r e n t v e l o c i t i e s f o r a l l s u b j e c t s i n s e a r c h o f a m e a n i n g f u l r e l a t i o n s h i p b e t w e e n t h e i n d i c e s a n d the v e l o c i t y f o r M W U s . I n t he c o n s t a n t v e l o c i t y tes ts , the e x p e r i m e n t s w e r e p e r f o r m e d at t w o v e r t i c a l a n d th ree h o r i z o n t a l p o s i t i o n s o f t he seat ( s i x tests) f o r a l l s u b j e c t s , to c o n d u c t s e n s i t i v i t y a n a l y s i s a n d to v e r i f y t he p o s s i b i l i t y o f e s t a b l i s h i n g s o m e g e n e r i c r u l e s to e s t i m a t e the o p t i m u m seat p o s i t i o n s f o r a l l u s e r s . P e r f o r m i n g at l eas t n i n e tests e n a b l e s o n e to a c q u i r e the r e q u i r e d d a t a f o r a r e s p o n s e s u r f a c e , w h i c h re l a tes the i n d i c e s t o t he seat p o s i t i o n . T o d e t e r m i n e the r e s p o n s e s u r f a c e , t w o s u b j e c t s p a r t i c i p a t e d i n n i n e tes ts . T h e m e t h o d o l o g y i s e l a b o r a t e d i n S e c t i o n 4 . 3 . 4.2.4 Test Protocol F i r s t , t he s t u d y p r o c e d u r e w a s e x p l a i n e d i n d e t a i l e d a n d a n e a s y t o f o l l o w f o r m a t to t he s u b j e c t s , a n d t h e y w e r e a s k e d to s i g n t he e t h i c a l c o n s e n t f o r m . S o m e i n d i v i d u a l s w i t h s p i n a l c o r d i n j u r y h a v e a c o n d i t i o n k n o w n as " A u t o n o m i c D y s r e f l e x i a ( A D ) " , w h i c h a f f e c t s t h e i r h e a r t ra te i f t he b l a d d e r i s f u l l [ 8 7 , 8 8 ] . A l t h o u g h , w e c h o s e s u b j e c t s w i t h l e s i o n l e v e l s b e l o w T 5 , w h i c h i s the cu t o f f l e v e l f o r a p o s s i b i l i t y o f A D , w e s t i l l a s k e d t h e s u b j e c t s t o v o i d t h e i r b l a d d e r s b e f o r e t he tests . T h i s w a s c o n s i d e r e d as a m a t t e r o f c o n v e n i e n c e f o r t e s t i n g , as w e l l as to p r e v e n t i t s p r o b a b l e e f f ec t s o n the b l o o d p r e s s u r e , a n d to a v o i d d i s t r a c t i o n d u r i n g the tes ts . T h e s u b j e c t s ' a n t h r o p o m e t r i c d a t a ( h e i g h t , a r m l e n g t h , j o i n t c i r c u m f e r e n c e , a n d h a n d w i d t h ) w e r e m e a s u r e d . T h e y w e r e a s k e d t o s t r e t c h t h e i r a r m s s t r a i gh t to t he s i d e s s u c h that the d i s t a n c e f r o m the t i p o f t h e i r l e f t m i d d l e f i n g e r t o t he t i p o f right m i d d l e f i n g e r c a n b e m e a s u r e d . T h i s m e a s u r e m e n t i s a g o o d 100 representative of the height of a subject, which is normally hard to measure in seated individuals. Previous studies have reported a high correlation (0.73-0.89) between the arm-span and height for different genders and ethnicities [89,90]. The subjects' %BFs were determined using a caliper for skinfolds test at Triceps, Biceps, Subscapula and Supraspinale. The measurements were taken three times and the average values were used. Then using a custom made scale the weights of the wheelchair users were obtained. Ten semi-spherical passive markers were attached to the upper limb and trunk landmarks, and two markers were attached to the instrumented wheel (Figure 4.1). The markers reflected the infrared waves emitted from the six surrounding cameras of an advanced Motion Analysis System (VICON). The joint positions and the motion of the upper limb of the subjects were determined by tracking the markers through the VICON system and a digital camcorder. All measurements were non-invasive. The subjects wore sleeveless shirts (tank top) during the tests. They transferred themselves onto the stationary instrumented test wheelchair. As part of subject calibration for the motion analysis system, they were asked to take a "T" pose (stretch arms horizontally) for a few seconds; and turn their upper limb segments starting from their hand, forearm, and upper arm around their joints for about a minute in front of the cameras. The base line of the heart rate and the blood pressure of the subjects were measured just prior to the tests (Figure 4.1). The wheelchair seat could be positioned at three vertical positions (Y l , Y2 and Y3) and three horizontal positions (XI, X2 and X3) (Figure 4.2). There were two different sets of the tests. In the first set, three velocities of 0.9, 1.1 and 1.3 m/s were used, at a fixed position (X2 and Y2) for eight subjects. The 101 selected velocities are typical wheeling velocities for MWUs. In the second set, the tests were performed at a fixed velocity of 0.9 m/s. In this category, the tests were conducted at six different combinations resulting from two incremental vertical seat positions (Yl and Y2) and three incremental horizontal seat positions (XI, X2 and X3) for all eight subjects. Two subjects had three more tests at three different combinations of the seat position (XI, X2, and X3; all at Y3). A speedometer measured the speed of the wheelchair. The order of the tests was selected randomly for each subject. The magnitude of X I , X2, and X3 were 11, 14, and 17cm, and Y l , Y2, and Y3 were 15, 18, and 20cm with respect to wheel axle, respectively. X values were negative. Figure 4.2 illustrate these positions with respect to the wheel axle. Figure 4.1 A subject on the instrumented wheelchair and roller-rig during blood pressure measurement. The marker on the left hip is not shown. 102 To change the seat position, it was necessary for the subjects to transfer themselves in and out of the wheelchair a number of times. Each test took 3 minutes and the data was collected during the final minute of the test. The subjects rested between the tests, and prior to each test their heart rate and blood pressure were measured to ensure they had returned to the baseline levels. Figure 4.2 Possible seat and backrest positions. The seat position is set at XI and Y l in this figure. Dimensions are not to scale. 103 The forces and moments that the subjects apply during propulsion are needed for the analysis, and were measured with the instrumented wheel. During the tests, the heart rate, and the kinetic and kinematic data of the subjects were recorded, simultaneously. 4.2.5 Anthropometric Data The anthropometric dimensions of the upper limb were obtained using a tape measure. A platform was designed for this study to determine the weight of the subjects as well as the weight of the instrumented wheel, separately (see Section 4.2.6.6). BMI and %BF were calculated using the Equations 3.4 and 3.11, respectively. 4.2.6 Experimental Setup In this study, it was necessary to measure a number of physical and biological parameters; therefore several devices were used to acquire such data. The wheelchair, the instrumented wheel, the roller-rig and the two computers for kinetic data acquisition were described in Chapter 2. This section outlines the rest of the equipment used. 4.2.6.1 Motion Analysis System ®VICON Motion Analysis System was used to acquire the kinematic data. VICON is equipped with infrared cameras, which are more accurate compared with the conventional video cameras (Figure 4.3). To conduct a 3D kinematic study of the upper body during MWP, at least four cameras are necessary for a good all around coverage of the subject for data acquisition, with less possibility of any marker being missed. Figure 4.4 illustrates a schematic view 104 of the multi-camera setup. We used six cameras in our tests to have more confidence in covering all landmarks and to ensure redundancy in data acquisition. Figure 4.3 VICON infrared camera. Figure 4.4 Positions of six infrared cameras, a subject, the wheelchair and the roller-rig for stationary MWP. 105 4.2.6.2 Heart Rate Monitor A heart rate monitor (HR-Polar S610™) was used to measure the heart rate of the subjects. This was one of the data required to determine EBI for the subjects (Figure 4.5). (a) (b) Figure 4.5 HR-polar heart rate monitor: (a) Heart rate sensor and transmitter; (b) recorder. 4.2.6.3 Blood Pressure Monitor An automatic blood pressure monitor (© 2005 A&D Medical) was used to determine the blood pressure of the subjects at rest and prior to each test (Figure 4.6). Figure 4.6 Blood pressure monitor. 106 4.2.6.4 Fat Caliper Skinfold tests were performed using a Slim Guide fat caliper (SLrMGUIDE®) to determine %BF for each subject (Figure 4.7). Figure 4.7 Fat caliper. 4.2.6.5 Speedometer A speedometer (Filzer dB4L) was used to measure the linear propulsion speed of the wheelchair during the tests (Figure 4.8). The subjects were able to see the speed on the digital display of the speedometer. This online feedback helped them to adjust their propulsion and maintain the desired constant speed during the tests. 4.2.6.6 Weighting Scale A special scale was designed and fabricated to determine the combined weight of the wheelchair and the sitting subject (Figure 4.9). We then subtracted the weight of the wheelchair from the total weight to obtain the net weight of the subject. 107 (a) (b) (c) Figure 4.8 Speedometer: (a) Cycling computer; (b) Holding magnet; (c) Wiring kit and sensor. Figure 4.9 The setup for measuring the wheelchair user weight. 108 4.2.6.7 Global View of the Experimental Setup Figure 4.10 is a schematic sketch of the physical experimental setup and its electronic connections. The subject wears the heart rate monitor not shown in this diagram. Common mouse Figure 4.10 Global schematic rear view of the kinetic and kinematic data acquisition system and its connections. 109 4.2.7 Modeling Although, there may be some differences between the left and right side of the subjects, but since we had a right-side instrumented wheel, we studied subjects with dominant right hand side. The right upper limb was emulated as a linkage system with three links (upper arm, forearm, and hand) and three joints (shoulder, elbow, and wrist) (Figure 4.11). (a) (b) Figure 4.11 The Model of the upper limbs: (a) Sagittal view of the half body; (b) the linkage model. Numbers 1-3 represent upper arm, forearm, and hand, respectively. Twelve semi-spherical passive camera markers were used to determine the positions of 10 anatomical landmarks (cervical 7, acromion, medial and lateral epicondyle, radial and olnar styloid, second and fifth metacarp, left and right greater trochanter), and 2 points on the wheel (wheel axle and wheel angular position) (Figure 110 4.12). The neck and hip information were used to construct the "prismatic-box model" in the VICON software system, and the wheel markers were used to determine the angular position of the wheel. Al l of the markers were attached on the skin of the subject. The prismatic-box model was designed using the subjects' anthropometric dimensions. The VICON system resizes the designed model using the acquired data through a subject calibration test as explained in Section 4.2.4. Cervical 7 Medial epicondyle Lateral epicondyle Radial styloid Olnar styloid Second metacarp Fifth metacarp Figure 4.12 Landmark positions on the upper limbs and trunk of the subject, and the wheel. Figure 4.13 shows the designed 3D model of the upper body of the subject, which was used for kinematic data acquisition. VICON IQ2 software was used to construct this model and re-play animations of the tests. During the re-play mode, it was possible to turn the viewing camera around the prismatic-box model to see the details of the wheelchair propulsion. I l l Acromion Two markers on both Greater trochanters (The left marker, which is not shown, is on the opposite side of the right one). Wheel angular position Wheel center Figure 4.13 A 3D model of the upper body and wheel, developed by using the VICON system. 4.3 Analytical Methodology To determine the optimum seat position for each user, focusing on the efficiency aspect, it is necessary to obtain an equation, which relates seat position (Xand Y) to EBI. It has been reported that heart rate and propulsive moment are related to seat position [19]. Considering Equation 3.2, it is assumed that THB and Mz are related to X and Y, whereas AO is a constant, because the experiments are conducted for a pre-determined period of time and velocity. THB and Mz are related to X and Y as follows: 112 THB = gl(X,Y) Mz=g2(X,Y) (4.1a) (4.1b) where gt stands for function /. EBI is the related to THB and Mz as EBI = g i (THB, Mz) (4.2) EBI is now obtained using Equations 4.1 and 4.2: EBI = g 4 (X,7) (4.3) To determine the optimum seat position for each user considering the injury aspect, it is also necessary to obtain an equation that relates X and Y to WUJII or WUJIF. In Equations 3.12-3.15, ms, W,, hs, La, V,, and %BF are constant. Also, Fm, Mm, and fp are related to Xand Y. Fm, Mm and fp are related to A!"and Fas follow: Fm=g5(X,Y) (4.4a) Mm=g6(X,Y) (4.4b) fp=g7(X,Y) (4.4c) Equation 4.5 shows the relationship amongst WUJII, Fm, Mm and fp. 113 WUJII = gs(Fm,Mm,fp) (4.5) Using Equations 4.4 and 4.5, WUJU is re-stated as WUJII = g 9 ( X , T ) (4.6) In a similar fashion, WUJII' is determined as Wjm' = g10(X,Y) (4.7) Human responses are not exactly the same in repeated tests. Therefore, determining the Std. Dev. (tr) can provide a measure of the variability of the results. We defined z as a general function of Xand Y, which represents EBI, WUJII, or WUJII'. The test were repeated five times at each position with the same velocity to determine the average value for each index (z ) and its standard deviations (az). z and <7Z are functions of X and Y (Equations 4.8), and we called them the response model. One needs these equations to determine the optimum position for a wheelchair user. z = g(X,Y) (4.8a) oz=u(X,Y) (4.8b) 114 4.3.1 Design of Experiments Clearly, a large number of physical experiments requires long time and costs more. Also, because of the physiological and/or anatomical limitations of the subjects, it is not always possible to perform many experiments. Specially, the number of the experiments has to be reduced to a practical one. In statistical analysis, the problem of choosing a suitable sample of design variables is referred to as Experimental Design or Design of Experiments (DOE) [91,92]. When a required parameter is related to two variables the resulting function is called the response surface. To have a more reliable response surface, one has to increase the number of tests and have the variables reasonably distributed over the possible range. In this study, the subject fatigue, total test time, and the possible range for X and Y were the factors that constrained the number of the experiments. Therefore, a DOE method was used to build the response model that related the biomedical indices to the seat position. DOE methods reduce the number of the experiments required. Using a DOE method, one can generate a set of representative input parameters that uniformly cover the entire design surface. The response model is used as a surrogate model to substitute the actual response. 4.3.1.1 Grid-base Design In this study, three levels for the vertical and horizontal positions of the seat were considered. As the number of the experiments was limited, four sub-areas were defined with four data points (JP,) on the corners of each sub-area. The grid for this experimental design is shown in Figure 4.14. The values of the variables have been normalized. 115 This model presents nine data points for the experiments. Two more tests with different velocities at position X2 and Y2 were needed that increased the total number of the tests to eleven. It was not possible for all subjects to follow all eleven tests continuously, due to considerations given to possible fatigue. Therefore, the above model was used for two subjects, who could complete the eleven tests, and another design was used with two vertical seat positions (Yl and Y2) and three horizontal seat positions (XI, X2 and X3) for the rest. The other design required eight tests consisting of six experiments at six different positions and two additional tests at the position X2 and Y2 for different velocities. LOO 1 Normalized Y P4 0.00 Pi 0.00 T;t. p2 Ps P 3 Pe P9 1.00 Normalized X Figure 4.14 Grid-base design for two variables with four sub-areas and nine data points. Dimensions are not to scale. 4.3.2 Response Equation Using the results of the experiments, the responses were calculated at the designed data points. The response equations can be determined using one of the following three approximation methods: 116 • Artificial Neural Network (ANN) • Local interpolation of the discrete database [93] • Bivariate Quadratic Function (BQF) 4.3.2.1 Artificial Neural Network ANN method is very versatile approach and there are many applications of it in areas such as signal processing, controls, pattern recognition, medicine, business, speech recognition and production. ANN is an information-processing system, and is a generalization of mathematical models of human cognition or neural biology [94]. In this research, ANN was not used because there was insufficient data to train and check the network. 4.3.2.2 Local Interpolation In this method the response at a query point (Po) is calculated as follows. First the closest pair of data points of the database (Pi, Pi) are identified and the distances d\2 between them, and d\ and d2 from the points P\ and P 2 to the query point P 0 are determined. The weights h\ and h2 are calculated as hx =d2ldn h2=djdn ( 4 - 9 ) Normalized weights are shown by w\ and w2 and given as 117 w2 = h2l{hx +h2) (4.10) The response is obtained as R(P0) = wlR(Pl) + w2R(P2) (4.11) where R(Po) is the response at the query point P0, and R(P\) and R(P2) are the known responses at data points Pi and Pj, respectively [93]. In this method, if the database is sufficiently dense and if the query point Po is located such that the distance d\ or di are less than dn, the calculated response is a good approximation of the real case. Again, since the database was not sufficiently dense, this method was not used. 4.3.2.3 Bivariate Quadratic Function (BQF) In this study, each of the response equations was estimated by using a BQF [95] as follows: where b\, bz, bj, b$, bs and be are unknown constants and were determined by having the values of X, Y, and z for n' data points or tests and m' unknown, and using the m'-equation-n'-variable method, z represents the response surface and gives the value of the corresponding index at different seat positions. BQF is a practical method that can be z=g{X,Y) = bxX2 +b2Y2 +b3X + b4Y + b5XY + b6 (4.12) 118 used for the cases with a small number of the data points. However, increasing the number of the tests will increase the reliability of the results. Although, this method can be used with equal number of the equations and the variables, in case the data points are at the border of the sampling region there is no solution and more data points are needed. This conflict is because of the singularity that may occur in the solution. 4.3.3 The Big Picture The big picture of the entire test process is given as a flowchart in Figure 4.15. It shows the steps, which are followed to determine the optimum seat position of a manual wheelchair for a user. This method determines the procedure, which can be used to prescribe a more suitable manual wheelchair considering the injury priorities, conditions and concerns, of the subject. 119 Recruit the subject Measure anthropometric data of the subject Set the seat position (Xand Y) Train the subject 3 £ Record the data of at least five consecutive propulsions at this seat position Let the subject rest, and check his blood pressure and heart rate to ensure he is back to the baseline level Return to the third block and set a new seat position, continue the process until all planned positions have been used Post-process the raw kinematic data Measure Fm Mm, THB, and %BF Calculate z and oz at each data point Obtain the response equations using BQF Determine the values of Xand Tat optimum z Figure 4.15 Flowchart for the entire test process to determine the optimum positions of a wheelchair for a M W U . 120 4.4 Results and Discussion In the sections that follow, different categories of the results are presented. First, the heart rate, blood pressure and the anthropometric data of the subjects are given. Then, four research questions are explained and answered by addressing the following issues: • Relationship between the biomedical indices and the propulsion velocity. • Sensitivity of the biomedical indices to the seat position. • Generic rules for estimating the optimum seat position for all users. • Optimum seat position for a particular user. 4.4.1 Heart Rate, Blood Pressure and Anthropometric Data The measured and calculated resting level of heart rate and blood pressure, and anthropometric data for the subjects are presented in Tables 4.2a and 4.2b, respectively. These data were used in the kinetic and kinematic analysis. Also, the subject's limb segment lengths were required to design the prismatic-box model (see Sections 4.2.5 and 4.2.7). Table 4.2a. Heart rate and blood pressure for the subjects. Subject Heart rate Systolic blood pressure Diastolic blood pressure (Beats/min) (mmHg) (mmHg) 1 72 129 76 2 94 141 85 3 45 130 82 4 71 122 78 5 100 173 91 6 87 94 66 7 80 112 64 8 76 118 62 121 Table 4.2b Anthropometric data for the subjects. Subject Height Upper arm Forearm Hand Hand Shoulder joint Elbow joint Wrist joint Mass BMI % BF code length length length width circumference circumference circumference (m) (m) (m) (m) (m) (m) (m) (m) (kg) (kg/m2) ~T~ 1.85 0.32 0.29 Ojfl 0.10 0.45 0.26 0.16 ~88 25.71 28.9 2 1.62 0.28 0.25 0.08 0.09 0.38 0.26 0.15 58 22.10 27.1 3 1.65 0.27 0.26 0.08 0.09 0.40 0.25 0.17 53 19.47 21.5 4 1.93 0.29 0.30 0.10 0.09 0.42 0.28 0.17 100 26.85 24.0 5 1.80 0.30 0.26 0.10 0.09 0.44 0.29 0.19 94 29.01 32.2 6 1.80 0.28 0.28 0.09 0.09 0.42 0.29 0.18 80 24.69 29.5 7 1.77 0.29 0.27 0.09 0.08 0.36 0.25 0.16 58 18.51 8.8 8 1.91 0.31 0.30 0.10 0.10 0.43 0.30 0.19 87 23.84 22.8 4.4.2 Fixed Seat Position In this category, the fixed positions at X2 and Y2, and the velocities of 0.9, 1.1 and 1.3 m/s for eight subjects were used to determine the relationship between the biomedical indices and the speed. The mean and Std. Dev. of five consecutive pushing phases were analyzed for each test. In this study, because of the small sample size of the subjects we performed the statistical analysis for estimating the Type I or Alfa error. The results from statistical analysis using repeated-measures Analysis of Variance (ANOVA) showed that velocity alter the injury indices significantly (p<0.01). Figures 4.16 and 4.17 illustrate the variation of the mean values of WUJII and WUJII' (WUJII and WUJII') with respect to the velocity using Equations 3.6b and 3.8a, respectively. The figures show that these two indices increase by increasing the velocity for all subjects. Tables 4.3 and 4.4 present the mean and Std. Dev. of WUJII and WUJII', which confirm the above finding. So, the relations of the injury indices were modified to include the velocity as one of their parameters. The results from repeated-measures A N O V A did not show significant relationship between the velocity and the mean value of EBI (EBI). Figure 4.18 presents the variation of EBI with respect to the velocity. The figure shows that EBI increases for five subjects by increasing the velocity up to about 1.1 m/s. Three subjects have their maximum EBI at the middle speed. As this seems case dependent, we cannot determine a specific rule for variation of EBI with respect to the velocity. This result confirm the findings of Mukherjee et al. [38] that efficient propulsion velocity is case dependent and is not related to lower or higher speeds. The mean and Std. Dev. of EBI are shown in Table 4.5. 123 5 8 7 6 5 4 3 2 1 0 0.85 -0.95 1.05 1.15 Velocity (m/s) 1.25 — Subject 1 — « — Subject 2 A Subject 3 — X - Subject 4 — Subject 5 - : - Subject 6 — r - — Subject 7 - Subject 8 1.35 Figure 4.16 Variation of WUJII versus velocity. Table 4.3 Mean and Std. Dev. of WUJU for the subjects for three velocities. Subject code Velocit (m/s) 1.553 1.809 2.453 2.970 2.416 1.608 0.869 2.369 0.9 ±0.195 ±0.396 ±0.576 ±0.220 ±0.249 ±0.255 ±0.108 ±0.171 1.1 2.343 2.552 3.399 4.126 4.632 3.622 1.356 3.797 ±0.232 ±0.335 ±0.513 ±0.425 ±0.660 ±0.618 ±0.131 ±0.429 1.3 3.272 3.274 3.543 4.813 5.087 5.441 2.87 6.800 ±0.658 ±0.158 ±0.563 ±0.801 ±0.974 ±0.594 ±0.553 ±1.047 124 7 6 5 & 4 3 3 2 1 0 — Subject 1 — • — Subject 2 A Subject 3 —K - Subject 4 — — Subject 5 - o - Subject 6 *— Subject 7 - Subject 8 0.85 0.95 1.05 1.15 Velocity (m/s) 1.25 1.35 Figure 4.17 Variation of WUJU' versus velocity. Table 4.4 Mean and Std. Dev. of WUJII' for the subjects for three velocities. Subject code Velocit i m / s l 1.748 2.233 2.741 2.660 2.684 1.923 0.429 2.265 0.9 ±0.219 ±0.488 ±0.643 ±0.197 ±0.277 ±0.304 ±0.050 ±0.164 1.1 2.637 3.151 3.798 3.696 5.145 4.331 0.645 3.629 ±0.261 ±0.414 ±0.574 ±0.381 ±0.732 ±0.740 ±0.062 ±0.410 1.3 3.682 4.042 3.959 4.311 5.650 6.506 1.326 6.500 ±0.740 ±0.195 ±0.630 ±0.717 ±1.082 ±0.710 ±0.191 ±1.001 125 0.85 0.95 1.05 1.15 Velocity (m/s) 1.25 • Subject 1 — • - Subject 2 • Subject 3 —X - Subject 4 )l( Subject 5 - o -Subject6 —4—* Subject 7 — - Subject 8 1.35 Figure 4.18 Variation of EBI versus velocity. Table 4.5 Mean and Std. Dev. of EBI for the subjects for three velocities Subject code Velocit (m/s) 8 19.152 15.428 20.308 28.194 15.378 18.443 8.016 38.019 0.9 ±1.933 ±1.647 ±3.429 ±1.333 ±1.962 ±0.896 ±0.455 ±2.217 1.1 22.641 14.327 19.731 28.520 24.374 18.157 11.868 40.522 ±1.320 ±2.298 ±2.177 ±2.840 ±2.019 ±4.328 ±0.780 ±0.643 1.3 24.328 16.318 15.967 25.539 20.582 19.176 15.670 39.785 ±3.688 ±1.251 ±3.921 ±1.933 ±2.284 ±1.287 ±0.758 ±1.407 126 4.4.3 Constant Wheelchair Velocity In the second category of the experiments, the tests were performed at three X and two Y settings, and the velocity of 0.9 m/s for all subjects. Five consecutive pushing phases were analyzed for each test. 4.4.3.1 Seat Height Y l Using repeated-measures A N O V A the results showed that the horizontal position of the seat was significantly related to the indices at low seat position Y l (p<0.05). Figures 4.19, 4.20 and 4.21 show the variations of WUJII, WUJII', and EBI against the ratio of X to the arm length (X-ratio) at Y l , respectively. X-ratio was used to normalize the horizontal seat position amongst the subjects. It is seen that WUJII and WUJII' increase by moving the seat forward, except for the subjects 1, 4 and 7 in Figure 4.19 and subjects 1, 4, 6 and 7 in Figure 4.20. Figure 4.21 shows that, except subjects 1 and 4, the other subjects had their minimum values of EBI at most backward seat position, and six subjects had their maximum value of EBI at most forward seat position. Tables 4.6, 4.7 and 4.8 present the mean and Std. Dev. of WUJB, WUJII' and EBI for the subjects at three X-ratios at seat height Y l , respectively. The Std. Dev. of WUJII and WUJE' vary between 0.053-0.783, and 0.025-0.967, respectively, but are predominantly under 0.400. The Std. Dev. of EBI varies between 0.75-3.06 and is mostly below 2.00. The results indicated that the average values of the injury indices and EBI at low seat height Y l , can vary between 5-27.5% and between 3.1-21.1%, respectively. 127 -X o Subject 1 Subject 2 Subject 3 Subject 4 • Subject 5 Subject 6 Subject 7 Subject 8 Seat height Y l - --30 -28 -26 -24 -22 -20 -18 -16 X-ratio* 10~2 r 7 : 6 : 5 s : 4 i 3 1=3 ; 2 \ i : 0 •14 Figure 4.19 Variation of WUJU against X-ratio at Y l . Minimum values encircled. Seat height Y l o Subject 1 1 — • — Subject 2 A Subject 3 I — X - Subject 4 I * Subject 5 - Subject 6 I „ Subject 7 I — - Subject 8 -30 -28 -26 -24 -22 -20 -18 -16 -14 X-ratio* 10"2 Figure 4.20 Variation of WUJU' against X-ratio at Y l . Minimum values encircled. 128 — Subject 1 — • — Subject 2 A Subject 3 — X - Subject 4 - - Subject 6 • Subject 7 — - Subject 8 Seat height Y l 7 ^ --30 -28 -26 -24 -22 -X-ratio* 10' 20 2 • 18 -16 60 50 - t 40 ff u 30 £Q 20 10 0 ffl •14 Figure 4.21 EBI with respect to the X-ratio at Y l . Maximum values encircled. Table 4.6 Mean and Std. Dev. of WUJII for the subjects at three X-ratios and seat height Y l . Subject Seat^^code X-position 8 1.370 3.414 2.068 2.529 5.708 2.315 0.551 2.734 ±0.099 ±0.738 ±0.361 ±0.612 ±0.568 ±0.154 ±0.053 ±0.245 XI X2 X3 0.934 2.849 1.415 2.317 4.905 1.941 1.085 1.790 ±0.093 ±0.783 ±0.355 ±0.167 ±0.576 ±0.500 ±0.152 ±0.201 1.134 1.566 1.472 2.766 3.034 1.981 0.929 1.589 ±0.114 ±0.308 ±0.074 ±0.383 ±0.261 ±0.346 ±0.110 ±0.114 129 Table 4.7 Mean and Std. Dev. of WUJII' for the subjects at three X-ratios and seat height Y l . Subject Seat^\code 1 2 3 4 5 6 7 8 X - p o s i t i o n \ 1.542 4.214 2.311 2.265 . 6.339 2.768 0.262 2.614 XI ±0.112 ±0.911 ±0.403 ±0.548 ±0.631 ±0.184 ±0.025. ±0.234 1.051 3.517 1.582 2.075 5.448 2.321 0.516 1.711 X2 ±0.104 ±0.967 ±0.397 ±0.149 ±0.640 ±0.598 ±0.072 ±0.192 1.275 1.933 1.644 2.477 3.369 2.368 0.442 1.519 X3 ±0.128 ±0.380 ±0.083 ±0.343 ±0.290 ±0.413 ±0.053 ±0.109 Table 4.8 Mean and Std. Dev. of EBI for the subjects at three X-ratios and seat height Y l . Subject Seat^\code X - p o s i t i o n \ 1 2 3 4 5 6 7 8 XI 24.526 ±0.752 15.953 ±1.518 9.193 ±0.881 30.519 ±2.251 25.884 ±1.461 22.470 ±2.375 12.610 ±0.796 53.123 ±1.430 X2 17,211 ±1.114 10.851 ±1.830 12.539 ±2.661 26.653 ±1.123 25.323 ±1.403 29.675 ±3.057 10.289 ±0.933 41.715 ±2.219 X3 18.511 ±1.166 9.325 ±1.075 8.475 ±0.708 30.333 ±2.424 20.303 ± 0 . 6 9 0 19.556 ±1.569 9:235 ±0.706 34.602 ±1.314 To have a better understanding of the variation of the average values of the indices, Figures 4.22, 4.23 and 4.24 show the maximum and minimum values of WUJII , WUJII', EBI and their Std. Dev. among the subjects with respect to the X-ratio at Y l . 130 Seat height Yl • Subject 1 (mean) • Subject 1 (mean* SD) • Subject 2 (mean) Subject 2 (mean± SD) -30 -28 -26 — ! -24 6 5 4 3 2 1 0 s s H -22 -20 -18 -16 -14 X-ratio* 10 -2 Figure 4.22 Maximum and minimum values of WUJII and its Std. Dev. against X-ratio at Y l , among the subjects. Seat height Yl • Subject 1 (mean) Subject 1 (mean+ SD) • Subject 2 (mean) Subject 2 (mean± SD) -30 -28 -26 -24 -22 -20 X-ratio* 10" •18 •16 -14 Figure 4.23 Maximum and minimum values of WUJII' and its Std. Dev. against X-ratio at Y l , among the subjects. 131 • Subject 8 (mean) Subject 8 (mean± SD) • Subject 2 (mean) Subject 2 (mean + SD) Seat height Y l -30 -28 -26 -24 X-ratio* 10 22 -20 -18 2 •16 60 50 40 30 20 10 0 S CQ W •14 Figure 4.24 Maximum and minimum values of EBI and its Std. Dev. against X-ratio at Y l , among the subjects. The above results show that by decreasing the magnitude of X-ratio or moving the seat forward at low seat height Y l , both the average value of the injury indices and EBI may increase. Overall, the results show that the indices are sensitive to horizontal seat position at seat height Y l . 4.4.3.2 Seat Height Y2 Using repeated-measures ANOVA the results showed that horizontal position of the seat was significantly related to the indices at high seat position Y2 (p<0.05). Figures 4.25, 4.26 and 4.27 show the variation of WUJII, WUJII' and EBI with respect to X-ratio at seat height Y2, respectively. Five subjects showed their highest values of the 132 injury indices at the most forward seat position or highest X-ratio, whereas, subject 4 showed the minimum value at this position and the other two subjects did not show significant change. Five subjects showed that their EBI decreases by increasing X-ratio or moving the seat backward, whereas two subjects had their maximum EBI at X2. EBI had insignificant change with respect to the X-ratio for subjects 3. Tables 4.9, 4.10 and 4.11 present the mean and Std. Dev. of WUJn, WUJlT and EBI for the subjects at three X-ratios and seat height Y2, respectively. The Std. Dev. of WUJII and WUJH" vary between 0.098-0.629, and are mostly under 0.40. The Std. Dev. of EBI varies between 0.46-3.62, and is mostly under 2.00. The results indicated that the average values of the injury indices vary between 5.6-29.9% and the average value of EBI varies between 3.7-26.0%. O— Subject 1 - Subject 2 A Subject 3 -x - Subject 4 -^—Subject 5 • - Subject 6 - Subject 7 • Subject 8 Seat height Y2 X-ratk>*10 Figure 4.25 Variation of WUJII against X-ratio at Y2. Minimum values encircled. 133 Seat height Y2 «— Subject 1 • - Subject 2 *—• Subject 3 - Subject 4 *—Subject 5 • - Subject 6 4~— Subject 7 — Subject 8 © 0- - r^L-^1 a~ -30 -28 -26 -24 -22 -20 -18 -16 -14 X-ratio* 10" Figure 4.26 Variation of WUJU' against X-ratio at Y2. Minimum values encircled. Seat height Y2 •30 -28 -26 -24 -22 X-ratio* 10 -20 -2 a s •18 -16 -14 Figure 4.27 Variation of EBI against X-ratio at Y2. Maximum values encircled. 134 Table 4.9 Mean and Std. Dev. of WUJII for the subjects at three X-ratios and seat height Y2. Subject Seat^\code X-positioif^^ 1 2 3 4 5 6 7 8 XI 1.495 ±0.102 1.917 ±0.573 2.508 ±0 .139 2.503 ±0.364 4.642 ±0.629 2.610 ±0.144 0.782 ±0.098 2.736 ±0.322 X2 1.182 ±0.134 1.836 ±0.392 1.176 ±0.144 2.450 ±0.273 1.352 ±0.231 1.180 ±0.184 0.693 ±0.180 2.188 ±0.194 X3 0.957 ±0.179 2.012 ±0.262 1.468 ±0.273 2.762 ±0.337 2.136 ±0.283 1.338 ±0.254 0.794 ±0.112 1.748 ±0.223 Table 4.10 Mean and Std. Dev. of WUJII' for the subjects at three X-ratios and seat height Y2. .^v . Subject S e a i \ c o d e X - p o s i t i o n \ 1 2 3 4 5 6 7 8 X I 1.682 ±0.115 2.366 ±0.707 2.802 ±0.156 2.242 ±0.326 5.156 ±0.699 3.121 ±0.172 0.372 ±0.046 2.615 ±0.308 X2 1.330 ±6.151 2.266 ±0.484 1.918 ±0.161 2.194 ±0.245 1.502 ±0.257 1.411 ±0.220 0.330 ±0.086 2.092 ±0.186 X3 1.077 ±0.202 2.484 ±0.324 1.640 ±0.305 2.474 ±0.302 ; 2.372 ± 0 . 3 1 4 1.600 ±0.304 0.378 ±0.053 1.671 ±0.213 Figures 4.28, 4.29 and 4.30 illustrate the maximum and minimum values of WUJII, WUJII' and EBI and their Std. Devs. among the subjects with respect to the X -ratio at Y2, respectively. 135 Table 4.11 Mean and Std. Dev. of EBI for the subjects at three X-ratios and seat height Y2. Subject S e a t \ « > d e X - p o s i t i o n ^ \ 1 2 3 4 5 6 7 8 1 26.317 ±2.311 11.931 ±2.104 15.548 ±1.019 24.331 ±1.778 26.337 ±2.200 23.813 ±1.427 15.376 ±0.570 40.157 ±0.907 2 19.175 ±1.038 13.902 ±3.621 15.737 ±1.040 28.015 ±1.347 12.634 ±1.699 22.433 ±0.976 10.293 ±1.117 38.019 ±2.217 3 18.894 ±1.048 13.243 ±1.468 14.359 ±2.950 18.463 ±0.977 16.231 ±0.534 20.358 ±0.531 8.016 ±0.455 31.319 ±1.985 Seat height Y2 • Subject 1 (mean) Subject 1 (mean± SD) • Subject 2 (mean) Subject 2 (mean ± SD) 3.5 3.0 2.5 2.0 £ 1.5 1.0 0.5 0.0 Figure 4.28 Maximum and minimum values of WUJII and its Std. Dev. against X-ratio at Y2, among the subjects. 136 Seat height Y2 • Subject 1 (mean) Subject 1 (mean± SD) • Subject 2 (mean) Subject 2 (mean+ SD) 1 1 1 1 1 — 30 -28 -26 -24 -22 -20 X-ratio* 10"2 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 = p •18 -16 • 14 Figure 4.29 Maximum and minimum values of WUJU' and its Std. Dev. against X-ratio at Y2, among the subjects. Seat height Y2 subject 8 (mean) Subject 8 (mean± SD) • • Subject 2 (mean) Subject 2 (mean± SD) 60 50 40 ¥ £3 30 20 10 0 03 UJ J-30 -28 -26 -24 X-ratio* 10 22 -20 -18 -16 2 •14 Figure 4.30 Maximum and minimum values of EBI and its Std. Dev. against X-ratio at Y2, among the subjects. 137 The above results show that the average values of the injury indices may increase at the most forward seat position at seat height Y2. Also, it can be possible that EBI increase by moving the seat forward. The results show that indices are sensitive to horizontal seat position at seat height Y2. 4.4.3.3 Horizontal Seat Position X I Two tests were performed for all subjects at horizontal seat position XI with two possible vertical seat positions Y l and Y2. To normalize the seat height for the subjects, the ratio of Y to the arm length was defined as Y-ratio. Figures 4.31 and 4.32 show that WUJII and WUJII' decrease by increasing the Y-ratio (or seat height) for four subjects at position XI , whereas they increase for subjects 2 and 8. The average values of the injury indices do not show considerable variation for subjects 3 and 4. Figure 4.33 shows that four subjects have their maximum values of EBI at Y l . Three subjects have their maximum values of EBI at Y2. Subject 1 did not show considerable variation by changing the seat height. Performing repeated-measures A N O V A the results did not show significant relationship between that vertical position of the seat and the indices at horizontal seat position XI . 138 Seat position XI 20 22 24 26 Y-ratio* 10" 28 30 — — Subject 1 —"* - Subject 2 * Subject 3 —O - Subject 4 H Subject 5 - + -Subject 6 —o™— Subject 7 • Subject 8 Figure 4.31 Variation of WUJU against Y-ratio at XI . Seat position XI 20 22 24 26 Y-ratio* 10 28 30 — Subject 1 — Subject 2 Subject 3 — Subject 4 — Subject 5 - Subject 6 »~ Subject 7 - Subject 8 -2 139 Figure 4.32 Variation of WUJII' against Y-ratio at X I . Seat position XI 40 30 i D Q m 10 20 +• — Subject 1 — Subject 2 Subject 3 — Subject 4 Subject 5 - • - Subject 6 ~ H — ~ Subject 7 Subject 8 22 24 Y-ratio* 10 26 28 2 30 Figure 4.33 Variation of EBI against Y-position at X I . 4.4.3.4 Horizontal Seat Position X2 At position X2, two tests were performed for all subjects with seat heights Y l and Y2. Figures 4.34 and 4.35 show WUJII and WUJII' versus Y-ratio. For half of the subjects no specific relationship is observed. Figure 4.36 illustrates that EBI increases for four subjects and decreases for three by increasing Y-ratio. Subject 7 does not show considerable change for EBI against Y -ratio. Overall, using repeated-measures A N O V A the results did not show significant relationships between the average values of the indices and the seat height at position X2. 140 Seat position X2 6 5 4 6 0 „ ^ 3 3 2 1 0 20 22 24 26 Y-ratio* 10" 28 30 — Subject 1 — X - Subject 2 EJ Subject 3 — O - Subject 4 — + . — Subject 5 - - • Subject 6 — Subject 7 - Subject 8 Figure 4.34 Variation of WUJII against Y-ratio at X2. 6 5 ^ 4 Es 3 1 0 20 Seat position X2 22 24 26 Y-ratio* 10" 28 30 Subject 1 — • - Subject 2 — * — Subject 3 —X - Subject 4 Subject 5 - • - Subject 6 Subject 7 — - Subject 8 Figure 4.35 Variation of WUJU' against Y-ratio at X2. 141 Seat position X2 • Subject 1 — • — Subject 2 Subject 3 — X - Subject 4 )|( Subject 5 - • - Subject 6 -"4 •— Subject 7 Subject 8 20 22 24 26 28 30 Y-ratio* 10" Figure 4.36 Variation of EBI against Y-ratio at X2. 4.4.3.5 Horizontal Seat Position X3 Figures 4.37 and 4.38 show that both WUJII and WUJII' increase by increasing Y-ratio for six subjects. An increasing trend for the average values of injury indices against Y-ratio at X3 can be seen. Figure 4.39 shows that EBI increases by increasing Y-ratio for four subjects. EBI decreases for three other subjects. Subject 5 did not show considerable variation for EBI against Y-ratio. Using repeated-measures ANOVA the results did not show significant relationship between the indices and the seat height at horizontal seat position X3. 142 S e a t p o s i t i o n X3 OO ^ ^ 2 1 0 20 —o 22 24 26 Y-ratio* 10 28 30 -2 — Subject 1 - Subject 2 • Subject 3 —o - Subject 4 — Subject 5 am mm - Subject 6 — Subject 7 — - Subject 8 Figure 4.37 Variation of WUJII against Y-ratio at X3. S e a t P o s i t i o n X3 X ~ - — - A — X KJJUPlJ 20 22 24 26 Y-ratio* 10" 28 30 — • — Subject 1 —• - Subject 2 A Subject 3 — X - Subject 4 )K Subject 5 - • - Subject 6 ""I1 Subject 7 Subject 8 Figure 4.38 Variation of WUJJJ' against Y-ratio at X3. 143 Seat at position X 3 20 22 24 26 Y-ratio* 10" 28 30 — Subject 1 — — Subject 2 • Subject 3 - Subject 4 — * • - • - Subject 6 "~™4~™~ Subject 7 - Subject 8 Figure 4.39 Variation of EBI against Y-ratio at X3. 4.4.3.6 Optimum Seat Position To determine the optimum seat position for a user at a propulsion velocity the Bivariate Quadratic Function (BQF) was used. BQF method requires at least nine tests. This method was performed for subjects 7 and 8 that were able to perform nine tests at nine seat positions. The approximate locations of XI to X3 and Y l to Y3 are shown in the X-ratio - Y-ratio plane of the following figures for ease of reference. Figure 4.40 presents a saddle surface and illustrates the lowest WUJII, between Y l and Y2, and close to XI for subject 7. The lowest value of WUJU' is seen in Figure 4.41 almost at the same location as the lowest WUJII. Figure 4.42 shows the response surface for EBI against X and Y-ratios for subject 7. It shows that the maximum EBI appears close to Y2 and X I . 144 Figure 4.40 Variation of WUJII versus seat position, for subject 7. Figure 4.41 Variation of WUJU' versus seat position, for subject 7. Figure 4.42 Variation of EBI versus seat position, for subject 7. Considering the general form of BQF as in Equation 4.12 and by using the MATLAB® software, the coefficients and constants of the response equations for subject 7 were determined (Table 4.12). Table 4.12 The coefficients and constants for the response equations that determine the indices at different seat positions for subject 7. b \ Z>2 63 Z>4 bs b(, EBI 401.785 -365.205 273.914 150.773 -213.573 24.520 WUJH -48.251 200.876 -46.751 -83.883 94.303 6.447 WUJII' -22.929 95.426 -22.234 -39.828 44.870 3.055 ^Coefficients & Constant Index 146 The same procedure was performed for subject 8 and the results are presented in Figures 4.43^4.45. Figure 4.43 shows that the maximum value of WUJII occurs close to Y2 and X I . The maximum value of WUJII' is shown in Figure 4.44 at the same location as the maximum WUJII. Figure 4.45 shows that the maximum value of EBI is at XI and Y l for subject 8. The minimum value appears in the opposite side of the maximum value. 147 Figure 4.44 Variation of WUJII' versus seat position, for subject 8. Figure 4.45 Variation of EBI versus seat position for subject 8. 148 The coefficients and constants of the related response equations for subject 8 have been determined and presented in Table 4.13: Table 4.13 The coefficients and constants for the response equations that determine the indices at different seat positions for subject 8. b\ bi 63 64 bs be EBI -414.777 189.879 488.559 652.596 2079.642 198.779 WUJII 106.198 -718.477 72.073 323.304 -76.75 -27.430 WUJII' 101.662 -686.740 68.995 300.985 -73.546 -26.203 jCoefficients & Constant Index 4.5 Conclusions In this chapter, a new method for determining the optimum seat position for the M W U was introduced. Description of the study population, the demographic and anthropometric data were given. The test protocol and experimental setup were explained. The kinematic and kinetic parameters and values of the proposed indices were calculated for each subject at different seat positions. The optimum positions for the users were determined by using the values of indices at different settings. The results of the experiments answered four research questions. bi this study, because of the small sample size of the subjects we performed the statistical analysis for estimating me Type I or Alfa error. We can decrease the Type II or Beta error by increasing the sample size of the subjects. 149 The results showed that the average values of the injury indices for all subjects increase considerably by increasing the linear wheelchair velocity (p<0.01). This result verifies the direct effect of velocity on the injury indices. Therefore, one may conclude that higher propulsion velocity will increase the risk of injury. Boninger et al. [4,69] reported that the flexion/extension and rotation angles for the shoulder and elbow joint of MWUs, as well as applied radial force, increase with increasing propulsion speeds confirming our results. However, a significant relationship was not observed between the values of EBI and velocity. Efficient propulsion velocity is therefore case dependent, and relates to the physiological, anatomical and technical characteristics of the subjects. Efficient propulsion velocity for one subject would not necessarily be the same for the others, and should be determined individually. This result corroborates the report of Mukherjee et al. [38] that there is no specific relationship between the propulsion speed and efficiency, and that the energy-efficient propulsion speed is related to the user's freely chosen speed as a characteristic of the subjects. Considering the results for the constant speed and fixed seat height experiments, one may say it is possible that the values of EBI increase by moving the seat forward related to the wheel axle (p<0.05). Whereas, the average values of the injury indices may decrease by moving the seat backward (p<0.05). We know of no other research that investigated the variation of the G M E with respect to the horizontal seat position with constant speed of propulsion. However, the report of Cooper [12, page 271] confirms our results for the injury indices. He explained that if the seat is too far forward, the shoulder will be excessively extended and internally rotated, which may lead to rotator cuff injury. 150 Also, the results of Boninger et al. [35] report that more forward seat position can increase median verve injury that compliment above results. Our results indicated that the average values of the injury indices and EBI can be 5.6-29.9% and 5-27.5%, respectively. Lower seat height showed lower variation for the results. The higher variations mostly are related to the subjects whose index values changed significantly with respect to the X and Y-positions. Therefore, the indices appear to be sensitive to the seat position. The response equations were determined for subjects 7 and 8 by using the BQF method. These equations can be used to determine the optimum seat position. The presented 3D-graphs for these two subjects show the optimum seat positions and indicate that the positions determined by using EBI and the average values of injury indices are not necessarily the same. These graphs illustrate the probable average values and variations of the indices at different positions by using the BQF method. Subject 5 had the highest BMI and %BF, and subject 7 had the lowest, at the time of the experiments. They presented the highest and lowest average values of the injury indices for most of the test situations, respectively. This supports the work of Boninger et al. [23] and other researches [70,71] that BMI affects CTS and shoulder injuries. Also, the above figures showed that subjects 2, 3, and 4, who were younger and had less wheelchair experience, had lower EBIs. In the next chapter, a method is introduced to determine the values of the injury indices at the upper limb joints of a wheelchair user. This method helps to determine the optimum wheelchair variables by considering the regarding concerns and priorities. 151 CHAPTER 5 Injury Assessment for the Upper Limb Joints 5.1 Introduction In Chapter 1, the prevalence of pain in the upper limb joints of MWUs was discussed. Previous studies presented methods for measuring pain or injury were explained in Chapter 3. There are very few studies that determine the pain or injury at the upper limb joints. The previous methods have shortcomings as they use questionnaires rather than direct measurements, or focus only on measuring a specific factor. In this chapter, the values of the proposed injury indices are calculated for different joints of the upper body for subjects 7 and 8. To do this, a 3D rigid-body dynamic model for the upper limb is presented, and a method (Method I) for calculating the upper limb joint forces and moments is described. Cooper [12] introduced the 152 structure of this method in 1995, and Vrongistinos [46] presented it with some differences in 2001. This method is reproduced here with some changes to the load calculations. The joint loads are calculated using the new method and are used as part of the required data for determining the values of the injury indices. 5.2 Method I In Chapter 4, the test setup, the model, and the methods were explained. The upper limb segments were assumed as rigid-bodies to be able to use the dynamics of the rigid body. hi this chapter, the focus is on determining the optimum wheelchair variables for each user considering the risk of the injury for the upper limb joints. Subjects 7 and 8 participated in this part of the experiments. To define the optimum position, nine tests at different combinations of three X-positions and three Y-positions for each user were performed at a constant propulsion speed. WUJII and WUJII' using Equations 3.14 and 3.15 were used as criteria to determine the optimum position. The values of the injury indices were determined at shoulder, elbow and wrist joints for each subject. The kinematic data acquired by VICON motion analysis system, the kinetic data measured by the instrumented wheel, and the subject's anthropometric data were used in the model to determine the joint loads as part of the required data to calculate the injury indices. The optimum positions determined minimize the probable injuries at different joints for each subject. 153 Considering the general form of Bivariate Quadratic Function (BQF) and using the MATLAB® software, the coefficients and constants of the related equations were obtained for the subjects. 5.2.1 The Kinematics of the Upper Limb To determine the orientation of each body segment with respect to a fixed coordinate system in a 3D analysis, a frame is attached to the Center Of Mass (COM) of the segment, and then a description of this frame is given relative to the reference system. Figure 5.1 shows local frame B ({B}) attached to C O M of a segment. A description of {B} relative to global frame A ({A}) gives the orientation of the segment. To describe the orientation of {B}, the unit vectors of its three principal axes with respect to {A} are determined. APBORG is the vector that determines the position of the origin of {B} with respect to {A}. XB, YB, and ZB are the unit vectors giving the principal directions of {B}. When written in terms of {A}, they are shown as AXB, AYB, and AZB. If one stacks these three unit vectors together as columns of a 3><3 matrix, a new matrix AR is obtained, which is referred to as the rotation matrix (Equation 5.1) [96]. ^ Figure 5.1 Local and global frames for a rigid body. Segment 154 where, rn_33 are the components of AXB, AYB, and AZB. Figure 5.2 shows a randomly selected point E in {B} and its relationship to {A}. P E represents the position vector of point E with respect to {A}, and P E is the vector that shows the position of point E with respect to {B}. A P E = BR B P E + A P B O R G (5-2) In the case where the local and global coordinate systems are concentric and one needs the information with respect to the global coordinate system, the following equation is used: APE = ABR BPE (5.3) Figure 5.2 General transformation of a vector. 155 5.2.1.1 Velocity of the Segment The linear velocity of a point with respect to {A} is obtained at any instant by using the position vectors as Ay - ± A p y E ( ; + A Q - y E ( Q K B " A ^ b " 1 S J At { 5 A ) where E is a point in {B}, A represent {A}, and P is its position vector, and AV% is the linear velocity of point E with respect to {A} [96]. Euler angles, Bryant angles, and Euler parameter are different methods, which are commonly used to derive the rotation matrix. In this study, three non-collinear markers on each segment, and the Euler angles were used to define the segments in the local coordinate system [12,46]. 5.2.1.2 Z - Y - X Euler Angles One method of describing the orientation of {B} with respect to {A} is as follows. Start with the frame coinciding with known {A}, first rotate {B} about ZB by an angle a to obtain {A1}, then rotate about the new YB by an angle /? to find {A"}, and finally rotate about the last^B by an angle y [96,97]. Frames A' and A" are the intermediate frames for transforming {A} to {B}. These rotations give us BR as BR = ROT(BX,y) R0T(BY,J3) ROT(BZ,a) (5.5) 156 Because"/? =^i?, we can compute BR as R = ROT(BZ,a) ROT(BY,p) ROT(BX,y) R = ca - soc 0 sec ca 0 0 0 1 c/J 0 sp 0 1 0 -sp 0 cP 1 0 0 " 0 cy -sy 0 5 7 ca.cP ca.sp.sy-sa.cy ca.sp.cy+ sa.sy sa.cp sa.sp.sy+ca.cy sa.sPry-ca.sy -sP cp.sy cp.cy (5.6a) (5.6b) (5.6c) where ca = cos(a) and s a = sin(a), etc. g i? is determined by using Equations 5.1 and 5.6c as ru 12 ra ca.cP ca.sp.sy-sa.cy ca.sp.cy+sa.sy ABR = r22 r23 = sa.cp sa.sp.sy+ca.cy sa.sp.cy-ca.sy (5.7) / 3 1 r32 r33_ -sp cp.sy cfi.cy The results for Z - Y - X Euler angles from a known rotation matrix are as follows: a = Atan2 (r2l,rn) P^Atan2(-r3l,±^rx]+r2]) y=Atan2 (rn,r33) (5.8a) (5.8b) (5.8c) Considering the positive square root in the formula for 6, a single solution was computed. The range for R was -90 <= /? <= 90. 157 5.2.1.3 Determining the Directional Cosines In this study, three markers were placed on three non-collinear landmarks to determine the position and orientation of each segment of the upper body of the M W U . The positions of the markers were tracked during the tests using the VICON Motion Analysis System with a sampling frequency of 100 Hz. Points 1, 2, and 3 represent the three markers on a typical segment, and E\, E2 are the position vectors between points 3 and 1, and between points 3 and 2, respectively. Directional cosines (r^ ) were determined for the rotation matrix, which transforms the coordinates from the local frame to another, with the origin of the local and the orientation of the global coordinate system. {B} is transformed to the center of mass of the segment. {A} and {B} are concentric. Figure 5.3 shows the directional cosines of the x-axis of {B} with respect to the three axes of {A}. Figure 5.3 Directional cosines of rotation matrix for the axes of {B} with respect to {A}. To determine r\j, the unit vectors of the axes of frames {A} and {B} are calculated. The unit vectors of the axes of the global coordinate system {A} are 158 T "0" "0" 1 = 0 , J = 1 , K = 0 0 0 1 (5.9) To determine the unit vectors of the axes of frame {B}, the vectors E\ and E2 are calculated as ~x2 y\ -y3 , E2 — -y3 _ Z 3 . .Z2 _ Z 3 . where J t i , 2 , 3 , 7 1 , 2 , 3 and z i , 2 , 3 are the position components of the points 1, 2 and 3 with respect to the global coordinate system. The unit vectors are now determined as EX=Y£-> , 4 = ] | l ,E3=ExxE2 (5.11) \E\\ 1-^ 21 The directional cosines are calculated using Equations 5.9 and 5.11 as rn = I • Ex , r]2 = I • E2 , rl3 = I • E3 r2l=J-El , r22=J-E2 , r23=J-E3 (5.12) r3i=K-Ex , r32=K-E2 , r33=K-E3 Linear velocity of a point on a segment is determined as 159 BV _ ± B P _ VE(^ + AQ-VE(0 dt A / ^ Q Ar (5.13) where BVE and B P E are the velocity and position vector of a selected point E on the segment with respect to frame B. Angular velocity of the segment with respect to the global coordinate system, AQB, can be determined at any instant by using the time derivatives of a, B and y as QB=a Ak + J3 AjA. + y Ai, (5.14) where Ak, AjA, and AiA. are the unit vectors for the z, y and x-axes of the frames {A}, {A'} and {A"} with respect to the frame {A}, respectively [96,97]. The time derivatives of a, B and y are determined as <*t &l^>0 d_ dt Ol A/_>0 a(t + At) -a(t) At Pit + At) -Pit) \ At y{t + At) -yit) At (5.15a) (5.15b) (5.15c) Ak, AjA. and AiA. are calculated by using the following equations: 4k = 0 0 1 (5.16a) 160 ca -sa 0 "0" -sa AJA'= sa ca 0 1 = ca 0 0 1 0 0 (5.16b) ca -sa 0" 0 sP "1" cacP sa ca 0 0 1 0 0 = sacP 0 0 1 -sP 0 cp 0 -sP (5.16c) Substituting Equations 5.16 into Equation 5.14, the relation for AQ.B is determined as "0 -sa cacP a AQB = 0 ca sacP P 1 0 -sp t (5.17) 5.2.1.4 Acceleration of a Segment Linear and angular acceleration of body segment can be determined at any instant by using the linear and angular velocity vectors as _d_sv _ g K E ( f + A Q - V E ( Q at A / - . 0 A? (5.18a) dAr, _ . . AnB(t+&tyAnB(t) B - ^B - lim ai A/->O A? (5.18b) where BVE is the linear velocity of point E in the frame B [96]. 161 Linear acceleration— The linear velocity of the vector P E with respect to frame {A} is given as AVE=AVB0RG + ABRBVE+AaBxARBPE (5.19) The linear acceleration of the vector APE is determined by calculating the derivative of Equation 5.19 as AVB=AVBORG+BRBVE +2AnBxARBVz+AnBxARBPE+AQB x(AQBxARBPE) (5.20) hi case b P E is constant, Equation 5.20 simplifies to AVE=AVB0RG+AClBxARBPE+AQBx(AQBxARBPE) (5.21) Angular acceleration— Considering three frames {A}, {B} and {C}, if frame {B} rotates relative to {A} with angular velocity AQB, and {C} rotates relative to frame {B} with BQc, then AQc is determined as AQC=AQ.B + ARBQC (5.22) and by differentiating, we obtain A£lc=A£lB+ABRB£lc+AaBxARB£lc (5.23) When BQc is zero, Equation 5.23 simplifies to 162 Anc=Aa, (5.24) 5.2.2 Kinetics of the Upper Limbs MWP produces repetitive stress on the user's joints. Determining the loads that cause this stress can help researchers to better understand the biomechanics of MWP. It is possible to directly measure moments and forces in the joints by surgically implanting transducers! However, this method is used in special circumstances. One case is with the implantable prostheses. Indirect estimation of joints loads can be made from the measured external loads, kinematic data (trajectory points of the upper limb), and anthropometric data. Knowledge of the time profiles of the joint loads is necessary for an understanding of the cause and implications for any movement [12]. In this research, rigid linkage system model was used to calculate net joint action/reaction forces and net muscle moments using inverse solution with the Newton-Euler method. The lengths of the segments are assumed to remain constant during the motion. The free body diagrams for all segments are similar. 5.2.3 Mass Distribution Each segment is assumed as a rigid body, which can move in three dimensions. Inertia tensor is required to describe the moment of the segment. The inertia tensor with respect to {A} is expressed as the 3><3 matrix: 163 -L -I*y -/» (5.25) The elements / « , lyy and lzz are called the mass moments of inertia. The elements with mixed indices are called mass products of inertia. These six independent quantities depend on the position and orientation of the frame in which they are defined. If we consider the axes of calculating the moment of inertia to coincide with the principal axes, the products of inertia will be zero, and the corresponding mass moments will be the principal moments of inertia. To determine the change of inertia tensor under translation of the reference coordinate system, the parallel axes theorem is used, which relates the inertia tensor in a frame with origin at C O M to the inertia tensor with respect to another reference frame. Following equations present this theorem [96]: AI„=CIzz+m(Xc+y2c) (5.26a) 'I^l^+mx^ (5.26b) where xc, yc, and zc are the coordinates of the center of mass with respect to {A}. The remaining moments and products of inertia are computed from permutation of x, y, and z in Equations 5.26. In this study, the shape of the upper arm and forearm were assumed as a frustum of cone, and that of the hand as a rectangular prism. The local frame of each segment is 164 placed at its own center of mass. Figure 5.4 shows a frustum of conic segment and its local frame {C}. H and h are the height of the frustum and its center of mass in the x direction. The inertia tensors with respect to the local frame at the center of mass, which is the principal frame, is determined as 0 0 0 0 yy 0 0 (5.27) Figure 5.4 Frustum of conic rigid body with local frame on its center of mass. where R and ro are the radius of the proximal (larger) and distal (smaller) ends of the frustum, respectively. The position of C O M for the frustum is given by 165 h = H(R2 +2Rr0+3r02) 4(R2+Rr0+r02) (5.28) The inertia tensor for the frustum is determined using Equations 5.29 as CI =CI = / / 2 (2/? 2 +6/?r 0 +12r 0 2 )-3r 0 2 ^ 2 5H2(R2 + 2Rr0 +3r 0 2 ) 2 , „ n 2 , ^ R'+Rr0+r0 4(R2+Rr0+r02)2 cj Jm(R5-r05) " 10(/? 3 -r 0 3 ) (5.29a) (5.29b) where m is the mass of the segment [98]. Figure 5.5 shows a rectangular prism segment, which represent the rigid body model for the hand. The local frame {C} is located at C O M . Hh, Wh and Lh are the parameters corresponding to the thickness, width and the length of the hand. Figure 5.5 Rectangular prism rigid body with local frame at its center of mass. 166 The components of the inertia tensor for the rectangular prism are determined using Equations 5.30 as :I« =^(K +Hl), % =^iWl +Ll), CIZZ =f2(Hl +Ll) (5.30) 5.2.4 M o d e l i n g In linkage system model, four basic groups of loads act: 1- Gravitational forces (weights), 2- External forces and moments (reactions on the hand of the user), 3- Muscle and ligament forces (net muscle moments and joint forces), 4- Inertial loads. To calculate the net muscle moments and joint forces, three free body diagrams in sagittal (Figure 5.6), frontal (Figure 5.7), and transverse (Figure 5.8) planes, and the following force and moment balance equations are used. 2X=>"a" YaFy=mCXyi HF:=mCl: (5-31a) Z M , = i ^ , 2 X = ^ (5.31b) ^ * dt y dt ^ 2 dt where ax,y Fpy + Ffy -m 8 = m y com _ ^ com _ F p 2 + F d 2 _ 0 _^ com _ Having the external force (Fd), mass of the segment (ni), gravitational acceleration ig), and linear acceleration of COM (CICOM), one can calculate the unknown force on the proximal end of the segment (Fp). 169 XCOM = m yCOM +s -F, ZCOM (5.33) Also, the moment balance in Equation 5.31b, is expressed in the vector notation as IX IX =Ai coy 0), + d{AI) dt 0)r OJ., CO, (5.34) Using Equations 5.12, and 5.29-30, one can determine / as AI=ARcIAR~l = ARcIART (5.35) where £R is the rotation matrix, which describes {C}in C O M of the segment with respect to the global frame A, GR~l is inverse of £R, ARTis transpose of AR, and CI is the inertia tensor with respect to {C}. Equation 5.34 then becomes ^M=AI-d>+__