Thus, the error was calculated as the difference between these two vectors:

 error = af sin(0t) ar sin(ot-+) (7-1)

Through the use of vector algebra, the magnitude of this error was calculated as:


 errors= (af(1-AR cos))2 +(af ARsin )2] (7-2)
 (7-2)

where I error I magnitude of the error (pixels) and

 AR amplitude ratio (ar/af).

 To express the error in terms of the maximum allowable phase lag for a given

amplitude ratio, equation 7-2 was solved for 4 as

 Os-1 1 AR error
 A=aos -- +--- --
 S 2 AR (7-3)

Given the maximum allowable error from the established picking envelope (47 pixels) and the

amplitudes of the fruits' motions from the fruit motion tests, equation 7-2 was solved to

represent the maximum allowable phase lag for the end-effector (equation 7-3). This

calculation was made for both cases of fruit motion. The results are presented in Figure 7.7.

 Because the motion of the fruit was much greater in the horizontal direction and the

allowable error for joint 1 was smaller, it was assumed that calculation of the maximum

allowable phase lag for joint 1 provided sufficient margin for both components of the vision

system. A plot of this maximum allowable phase lag versus amplitude ratio is presented in

Figure 7.7. For the smaller, high-frequency fruit motions as in Case II, a larger phase lag of

approximately 10 degrees was allowable. For the large-amplitude/low-frequency fruit motion,

Case I, a smaller allowable phase lag of approximately 2.5 degrees was calculated. Also from

Figure 7.7, necessary amplitude ratios were determined. For Case I, an amplitude ratio of 0.96

or better was required of the vision control system to be able to pick a fruit moving at 0.5 HZ (3.1

rad/sec) with a magnitude of 50 cm. Likewise, the control system was required to have an

amplitude ratio of 0.83 or better for fruit motions of 12.5 cm magnitude and 1.1 HZ (6.9 rad/sec),