Investigation of Effects of Controllable Variables on System Performance Using the Statistical Quality Tool of Factorial Design
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Zalina Abdul Aziz, Lim Chee Peng, Anwar Hasni, Muhammad Hafiz Kassim,
Wan Muhamad Norhafiz Wan Ahmad.
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i-QUEST Intelligent Quality Engineering Systems Research Group
School of Electrical and Electronics Engineering
Universiti Sains Malaysia , Engineering Campus
14300 Nibong Tebal, Pulau Pinang , Malaysia
Tel: +604-599599, Fax: +604-594102
e-mail(corresponding author) : zalina@eng.usm.my
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Abstract
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This paper presents the application of a statistical quality tool for determining the effects of two controllable variables on the performance of a stereo vision measurement system. The statistical quality tool used is the two-factor factorial design. The latter was chosen for the following reasons: only two controllable variables were involved; it is easy to use; and it provides objective answers regarding which variable(s) is significant, which variable(s) is not significant, and how the variables interact with each other. Only one of the variables of the above system was found to be significant. The interaction between the two variables was also found to be significant. A good understanding of how the measurement system behaves in response to changes in the values of the significant variable at different values of the insignificant variable was obtained using this design. The use of this design for determining the effects of controllable variables on the performance of a wireless robot control system is also proposed in this paper as a sound alternative to the widely-used one-variable-at-a-time design.
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Keywords : stereo vision measurement system, wireless robot control system, statistical quality tool, factorial design, one-variable-at-a-time design
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Introduction
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The development of statistical quality tools represents an important development in the evolution of Quality tools. The tools provide statistical methods for collecting and analyzing data in an objective manner so that objective conclusions can be obtained. Since the beginning of its development in the 1920s, the range of statistical quality tools has expanded rapidly and has been applied in research endeavors across various disciplines [1,2,3,4].
The purpose of our research work is to utilize a statistical quality tool known as factorial design in order to determine the significance of several controllable variables and their interactions on the performance of a wireless robot control system and a stereo vision measurement system developed by [5] and [6] respectively. This paper presents the application of this design to the stereo vision measurement system and the results and conclusions that were obtained. To facilitate easy understanding of this paper, several definitions are given in the next two paragraphs.
Controllable variables' are variables whose values can be easily set and controlled by the experimenter. In the case of the stereo vision system, two controllable variables were selected by [6] and are shown in Table 1. The variables are labeled as A and B. The levels of the two variables at which runs will be made are also shown in the table. A ‘response' is a measured performance of a system. The response for the stereo vision system is the distance of an object from the cameras that is measured by the system.
The ‘(main) effect' of a variable is the change inthe response of a system produced by a change in in the value of the variable. An ‘interaction' is failure of a variable to produce the same change in the response of a system at different values of another variable. In other words, the effect of a variable depends on the value of the other variable. The quantification of this is known as the ‘interaction effect' between the two variables. For the stereo vision system, the main effects of A and B as well as the interaction effect between A and B are presented in the ‘Results and Discussion' section.
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Table 1 : Variables for Stereo Vision System
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Variable
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Levels
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Distance between right and left cameras (A)
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0.08m , 0.20m
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Actual distance between object and cameras (B)
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0.4m,0.6m,0.8m,1.0m,1.2m, 1.4m, 1.6m, 1.8m, 2.0m
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Methods
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Factorial Design and Comparison to One-Variable-At-A-time Design
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Factorial design (FD) provides a strategy for designing experiments in which two or more controllable variables are selected for the experiment. The levels of the variables are simultaneously changed by the experimenter. A factorial design which consists of only two variables is known as a two-factor factorial design.
In contrast to FD, a one-variable-at-a-time design (OVAT) consists of selecting a starting point, or baseline set of levels for the variables. The levels of the variables are changed one variable at a time while the levels of the other variables are held constant at the baseline level.
To illustrate what has been said above, consider an experiment involving only two variables; variables A and B. Suppose there are two levels for each of the variables; the low and the high levels. Therefore, the total number of possible combinations is four.
Column 3 of Table 2 illustrates the combinations for FD while columns 4 and 5 of Table 2 illustrates two possible sets of combinations for OVAT. A comparison of the combinations for FD and OVAT shows the following:
FD consists of all possible combinations of the levels of variables A and B whereas the number of combinations for OVAT is only three. Thus, in OVAT, the response of the system at all possible combinations is unobtainable. Since the latter is needed in order to calculate the interaction effect, this effect cannot be calculated from OVAT. Should interaction exist between variables, the one-at-a-time variable will not yield the correct conclusions.
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Table 2 : Combinations for Factorial Design And One-Variable-At-A-Time Design
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Comb.
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A
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B
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FD
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OVAT
1
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OVAT
2
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1
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Low
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Low
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Yes
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Yes
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Yes
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2
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High
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Low
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Yes
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Yes
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Yes
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3
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Low
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High
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Yes
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Yes
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No
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4
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High
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High
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Yes
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No
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Yes
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Combinations for Stereo Vision Measurement System
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Table 3 shows the combinations for the above system. The total number of possible combinations is 18. The combinations were carried out in a random order. The experiment was carried out twice, resulting in a total of 36 pieces of data for the response (measured distance).
Minitab statistical software(version 14) was used used to analyze the data in Table 3. The results of the analysis are presented and discussed in the next section.
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Table 3 : Combinations and Response for Stereo Vision Measurement System
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Variable(m)
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Response(m)
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Comb.
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A
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B
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Measured Distance
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1
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0.4
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0.08
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.40269
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.40173
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2
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0.6
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0.08
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.59028
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.60139
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3
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0.8
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0.08
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.79746
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.78003
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4
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1.0
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0.08
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.96411
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.97448
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5
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1.2
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0.08
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1.1251
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1.12090
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6
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1.4
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0.08
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1.3235
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1.33690
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7
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1.6
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0.08
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1.4825
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1.51050
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8
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1.8
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0.08
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1.6552
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1.65840
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9
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2.0
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0.08
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1.8421
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1.84700
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10
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0.4
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0.2
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.40423
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.40502
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1
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0.6
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0.2
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.59350
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.60098
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12
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0.8
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0.2
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.76918
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.77790
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13
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1.0
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0.2
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.98199
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.97269
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14
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1.2
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0.2
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1.11650
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1.10350
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15
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1.4
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0.2
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1.30540
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1.23480
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16
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1.6
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0.2
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1.55630
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1.59880
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17
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1.8
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0.2
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1.69170
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1.60710
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18
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2.0
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0.2
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1.72670
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1.78240
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Results and Discussion
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Main and Interaction Effects
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The main and interaction effects generated by Minitab are shown in Table 4.
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Table 4 : Estimated Effects for Measured Distance(m)
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Term
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Effect
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Actual Distance(A)
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1.41238
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Camera Distance(B)
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-0.01031
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Interaction between Actual Distance(A) and Camera Distance(B)
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-0.01811
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The larger the absolute value of the effect, the more likely is the variable to be significant. Variable A appears to be the most significant variable, followed by the interaction between variables A and B, and lastly variable B.The statistical method known as ‘Analysis of Variance' (ANOVA) is used to confirm the above findings. The ANOVA table generated by Minitab is shown in Table 5.
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Table 5 : Two-way ANOVA for Measured Distance (m)
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Source
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DF
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SS
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MS
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F
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P
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A
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8
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7.50192
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.937740
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1777.75
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.000
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B
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1
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0.00096
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.000957
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1.81
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.195
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Interaction
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8
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0.01785
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.002231
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4.23
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.005
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Error
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18
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0.00949
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.000527
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Total
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35
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7.53022
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Table 5 shows that variable A and the interaction between variables A and B are highly significant since their P-values are very small. In fact, their P-values are less than .01.
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Interpretation of Significant Effects
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When an interaction between two variables is significant, it is more meaningful to interpret the significance of the interaction rather than the significance of an individual variable. In the case of the stereo vision system, it is more meaningful to interpret the significance of the interaction between variables A and B rather than the significance of variable A. An interaction plot can be used for this purpose, as shown in Figure 1.
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Figure 1 : Interaction Plot for Variables A and B
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Figure 1 shows that the performance of the stereo vision system depends on the levels of variable A. Table 6 is presented in order to understand Figure 1 better.
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Table 6 : Average Response for Stereo Vision System
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Comb.
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A
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B
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Average Response (m)
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1
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0.4
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0.08
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0.402210
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2
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0.6
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0.08
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0.595835
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3
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0.8
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0.08
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0.788746
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4
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1.0
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0.08
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0.969295
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5
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1.2
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0.08
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1.123000
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6
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1.4
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0.08
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1.330200
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7
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1.6
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0.08
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1.496500
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8
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1.8
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0.08
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1.656800
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9
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2.0
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0.08
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1.844550
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10
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0.4
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0.2
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0.404625
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1
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0.6
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0.2
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0.597240
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12
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0.8
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0.2
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0.773540
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13
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1.0
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0.2
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0.977340
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14
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1.2
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0.2
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1.110000
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15
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1.4
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0.2
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1.270100
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16
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1.6
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0.2
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1.577550
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17
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1.8
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0.2
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1.649400
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18
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2.0
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0.2
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1.754550
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When variable A(actual distance) is at 0.4m, 0.6m, 0.8m, and 1.0m, the distance measured by the system on the average is close to the value of variable A for both values of variable B (camera distance). For instance, when variables A and B are at 0.6m and 0.08m respectively, the average distance measured is 0.595835m. When variables A and B are at 0.6m and 0.2m respectively, the average distance measured is 0.597240m.
However, when variable A exceeds 1.0m, the distance measured by the system on the average varies from the value of variable A for both values of variable B. For instance, when variable A is at 2.0m, the average distance measured is 1.84460m and 1.754550m when variable B is at 0.08m and 2.0m respectively. |
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Conclusions
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A two-factor factorial design was used to design an experiment in order to objectively evaluate the performance of a stereo vision system. The experimental data was easily analyzed using the Minitab statistical software. Based on the effects that were found to be highly significant (the significance level being .01), it can be concluded that, when the distance between the two cameras is set at 0.08m or 2.0 m, the system is capable of measuring the distance of an object accurately when the object is placed 0.4m to 1.0m away from the system. The system is not able to measure the distance of an object accurately when the object is more than 1.0m away.
Based on the successful application of factorial design to the stereo vision system, it is highly recommended that this design be used to evaluate the performance of the wireless robot control system developed by [5]. Potential controllable variables could be the Internet speed, processor speed, and network traffic while a potential response could be data transmission delay time. The use of this design would enable significant interactions between the variables to be detected.
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References
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[4] Mckeighen, R.E. 1996 Optimization of Broadband Transducer Designs by Use of Statistical Design of Experiments, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control , 43(1): 63-70
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