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For the bucket, we knew it had to turn 180 degrees at by the end of its path and do so quickly so as to scatter its contents on the table, then quickly reset to its starting position for the next sweep. We first found a complex 8-bar mechanism that would achieve a complex translation and rotation motion, but later simplified it to a simple 6-bar half-rotation motion with a quick return. This allowed us to cut fewer links, minimized our chance of error in joint motion, and minimized the space needed to offset each link. The overlapping motions can be seen running below simultaneously (if the image is not moving, click the image to run the animation).
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