Terrazyme

The concept of uncertainty describes the degree of goodness of a measurement or experimental results [17]. Kline defines uncertainty as “what we think the error would be if we could and did measure it by calibration” [18]. Uncertainty is thus an estimate of experimental error.

Uncertainty analysis is a necessary and powerful tool, particularly when used in the planning and design of experiments. There are cases in which all of the measurements in an experiment can be made with 1% uncertainty, yet the uncertainty in the final experimental result could be greater than 50% [17].

Uncertainty analysis, used in an experiment’s initial planning phase can identify such situations and save the researcher much time.

4.4.1 General Uncertainty Analysis

In the planning phase of an experimental program, one focuses on the general or overall uncertainties.

Consider a general case in which an experimental result, f, is a function of n measured variables Xi [17]:

f = f(X1, X2,…, Xn) (4.15)

Equation (4.15) is the data reduction equation used for determining f from the measured values of the variables Xi. The overall uncertainty in the result is then given by

(4.16)

where are the uncertainties in the measured variables Xi.

It is assumed that the relationship given by (4.15) is continuous and has continuous derivatives in the domain of interest, that the measured variables Xi are independent of one another, and that the uncer-tainties in the measured variables are also independent of one another.

If the partial derivatives are interpreted as absolute sensitivity coefficients such that

(4.17)

then Equation (4.16) can be written as

(4.18)

In Equations (4.16) and (4.18), all of the absolute uncertainties (UXi) should be expressed with the same odds or level of confidence. In most cases, 95% confidence level (20:1 odds) is used, with the uncertainty in the result also being at the same level [17].

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4.4.2 Experimental Protocol for Uncertainty Analysis

One able-bodied subject was used for the stationary tests. The main idea was to propel the wheelchair, measure the loads, and calculate uncertainties to verify the system rather than acquiring specific information related to the subjects themselves. The subject tried the wheelchair for 5 min per day for one week to become familiar with the experimental set-up. For the main test, the subject propelled the wheelchair for 2 min, increasing his speed as what is conveniently possible and maintaining it steady for 1 min [12]. The data were then collected for the last minute of the test. MATLAB^® software was used for data filtering. A tenth-order linear-phase digital Equiripple-type filter (FIR) from MATLAB^® was used for signal conditioning.

4.4.3 Uncertainty of Preloads

The general uncertainty equation for preloads is obtained by using Equations (4.16) and (4.1) as (4.19) where Up represents the uncertainty for different preloads [N or N.m], Ua and Ub are the primary uncertainties for the constants a and b [N or N.m.] from Equation (4.1), and U_θ is the uncertainty for the wheel’s angular position [radian]. Uncertainty equations for different preload components are then calculated by using Equation (4.19) according to

(4.20)

Tables 4.2, 4.3, and 4.4 show the values for primary uncertainties. These values are determined in static tests and on the basis of parameter resolution as reported by the manufacturer of the transducer.

The values of , and are calculated by using Equations (4.20) and Tables 4.2 and 4.4, with ϕ varied between 0 and 180°, the probable interval for the pushing phase.

4.4.4 Uncertainty of Local Loads

The uncertainties of the local forces and moments are obtained by using Equations 4.2 and 4.20, and Table 4.3 as follows: 0775_C004.fm Page 17 Tuesday, September 4, 2007 9:25 AM

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These uncertainties are shown in Figures 4.15 and 4.16.

The uncertainties for different outputs of the system are determined with respect to the local and global coordinate systems. The uncertainties for the preloads and local loads are calculated with the primary uncertainties for the measured variables and with Equations (4.19), (4.20) and (4.21), and the values presented in Tables 4.2, 4.3, and 4.4. The results show that there is no significant difference between the uncertainties of the preloads and local loads. Therefore, only the uncertainties of the local loads appear in the presented results.

Figure 4.15 shows the uncertainties for the local forces in the interval during which the propulsion phase can occur (from +x to –x direction of the global coordinate system). It indicates that FLx and FLz have their highest uncertainties (about 1.75 N for FLx and about 3.40 N for FLz) in the +y direction of the global coordinate system, and that FLy has its highest uncertainty (about 1.75 N) in the +x and –x directions of the global coordinate system. FLx and FLy are the components of the applied force, which produces the propulsive moments. This information indicates that the highest uncertainties for FLx and FLy are low and acceptable.

Figure 4.16 shows the uncertainties for local moments in the interval during which the propulsion phase can take place (from +x to –x direction of the global coordinate system). It shows that MLy and MLz have their highest uncertainties (about 0.70 [N.m] for MLy and about 0.30 N.m for MLz) in the +y direction of the global coordinate system and that MLx has its highest uncertainty (about 0.70 N.m) in the +x and –x directions of the global coordinate system. MLz is the moment, which produces the propulsion; its highest uncertainty is very low.

4.4.5 Uncertainty of Global Loads

The following relations are obtained by calculating FLx, FLy, FLz, MLx, MLy, MLz, , and using Equations (4.2), (4.16), and (4.21), respectively, and employing equations (4.3) and (4.16):

(4.22)

These uncertainties for the global forces and moments are shown in Figures 4.14 and 4.15. FLx, FLy, MLx, MLy, and θ are the parameters calculated from the data measured in the tests.

TABLE 4.2 Primary Uncertainties for Measured Variables Variable uncertainties U_θ [radian] [m] U_Δz [m]

Value used 0.001745 0.001 0.001

TABLE 4.3 Primary Uncertainties for Measured Loads Load uncertainties

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TABLE 4.4 Primary Uncertainties for Constants Constant

uncertainties

Value used 1 2 0.4 0.2

FIGURE 4.15 Uncertainties for local force components during propulsion phase.

FIGURE 4.16 Uncertainties for local moment components during possible range of propulsion phase.

U U

The interval during which the propulsion phase can happen (degree)

Uncertainty of local forces (N)

x

The interval during which the propulsion phase can happen (degree)

Uncertainty of local moments (N.m)

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Figure 4.17 reflects the uncertainties for the local and global forces. These uncertainties were calculated for the normal propulsion phase of 80°, covering a range from 75 to 155° of the possible propulsion phase. The local uncertainties were compared with the global uncertainties in the same graph and for the same period. This figure shows that the global uncertainty of Fz is the same as its local uncertainty.

The global uncertainty of Fx shows a small increase compared with the local value, but its highest value of about 1.60 N is not near the end of the propulsion phase. It reaches its highest point at about 60° into the propulsion phase. The global uncertainty for Fy shows a small decrease compared with the local value.

Its highest value of about 1.70 N is near 10° after the beginning of the contact between the hand and the handrim. It decreases to a minimum at about 60°.

Figure 4.18 shows the uncertainties for the local and global moments. These uncertainties were also calculated for the normal propulsion phase. The local uncertainties were compared with the global uncertainties in the same graph and for the same period. This figure shows that the global uncertainty of Mz is the same as its local uncertainty. The global uncertainty of My shows a modest increase compared with the local values. It starts to decrease after its peak of about 0.63 N.m at about 60° after the beginning of the contact between the hand and the handrim. The global uncertainties for Mx decrease to some extent compared with the local value, whose peak value of about 0.70 N.m occurs near 10° after the beginning of the contact between the hand and the handrim. It drops to a minimum at about 60°.

4.4.6 Uncertainty of Angular Position of Hand on Handrim (U_ϕ)

The uncertainty of the angular position of the hand on the handrim (ϕ) is obtained by using Equations (4.12) and (4.16), as

(4.23) FIGURE 4.17 Uncertainties for local and global force components during propulsion phase.

0 10 20 30 40 50 60 70 80

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The derivatives of ϕ with respect to different variables are calculated as

(4.24)

The lumped parameter C in equation (4.24) is used to simplify some repetitive terms, which is given as

(4.25) The term U_ϕ is determined on the basis of equations (4.23) and (4.24), as

(4.26)

Using these relations and data from measured tests, one can obtain the time-dependent uncertainty for ϕ, as shown in Figure 4.19.

The uncertainty for the hand contact angular position for the reliable region of ϕ was calculated to be less than 10° for much of the region (Figure 4.19). The uncertainty for ϕ at the middle span of the reliable region is about 5°. Compared with the reported results by other researchers [19], and given that we did not use cameras in our measurements, this level of maximum uncertainty for ϕ is good.

FIGURE 4.18 Uncertainties for local and global moment components during propulsion phase.

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4.4.7 Uncertainty for Hand Contact Loads

Equations (4.6) show that the forces are the same in both the second local and global coordinate systems for the hand contact with the handrim. Using equation (4.16), the uncertainties for forces in the second local coordinate system are determined as

(4.27)

Knowing the uncertainties for rh and Δz, and calculating the other required uncertainties, one can obtain the uncertainties for the moments with respect to the local hand coordinate system as follows:

(4.28)

Among the moments in the hand coordinate system, Mhz is of considerable interest because it acts against the propulsion moment, which is produced by the applied forces Fgx and Fgy [4,16]. The uncer-tainty for Mhz is shown in Figure 4.20.

This figure shows the uncertainty for propulsive hand moment with respect to the local hand coor-dinate system. Mhz is the last parameter that is calculated by using the present hierarchy of equations for uncertainty propagation. Given the cumulative nature of the uncertainty of the equations, it is not FIGURE 4.19 Uncertainty for angular position of hand during propulsion.

0 50 100 150 200 250 300

Uncertainty for angular position of hand - phi (degree)

U U

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surprising to see a high uncertainty for Mhz. This figure shows that the uncertainty varies between 0.50 and 3 N.m. For most of the reliable region of the propulsion phase, the uncertainty is less than 2 N.m.

In this section, a general uncertainty analysis is 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. The uncertainty values for these parameters are calculated from the respective equations. The results provide an estimation for the errors and uncertainty in the output of the instrumented wheel. The uncertainties are 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 are about the same as the uncertainties for the local values, but the pattern of variation is different. The uncertainties are found to vary from 5 to 10° and from 0.5 to 3 N.m for ϕ and Mhz, respectively, for about 65% of the propulsion period. Uncertainties determined by Cooper et al. [13]

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 uncer-tainty levels 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.

In the next section, the instrumented wheel system is verified by using an experimental technique, and system specifications are determined by applying statistical methods.