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We used Matlab to find the proper link lengths for this linkage. The requirements were as follows:
The mechanism must be Grashof (because the linkage works with a single continuously-rotating motor).
The ground link must be no more than 12 inches long (to fit within device bounds).
The limits limit of the joint motion must be 180 degrees apart.
By using the last requirement, the Pythagorean theorem can give the relationship between three lengths: the ground link, output link, and the sum of the coupler and driving link. The Grashof condition puts limits on the link lengths but does not dictate the lengths, so this is a bounded system of equations that has infinite solutions. We chose an input link length of 4 inches and output length of 5 inches to maintain a reasonable mechanical advantage.
Toggle Position
Our mechanism had trouble making it through its toggle position, as shown in this video:Before we put the arms on the tower, we placed a steel mallet on the rotating tower base to simulate the rotational inertia of the mechanisms. This allowed the tower to rotate through the toggle position and oscillate as designed. Unfortunately, the weight of the mallet did not accurately simulate the opposing force from the arm to the bevel gear at the tower base. As a result of the power train, the tower resisted travel through the toggle point and instead oscillated on one side of the linkage, as seen in the following video.
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We solved this issue by creating slots in the motor mount which allowed it to slide forward and back. This meant that the linkage a) had more play, and b) could build up more momentum to power through the toggle position. Given the amount of issues that we had with this particular portion of the design, we strongly suggest against designing a linkage to pass through a toggle position.
Gear Train
The gear train includes two bevel gears, one of which is attached to the coupler of the 4-bar mechanism. The second bevel gear connects to a drive shaft which
drives a 3:1 spur gear. The spur gear turns the driving link of the 6-bar mechanism, which is designed with gear teeth on the back end.
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To compensate for the vertical motion being less than 90 degrees, the parallelogram link lengths were increased to 11.31 inches from their original 10 inches. This caused our robot to travel slightly outside of the 2' cube that we initially set as bounds, but we decided this was an acceptable compromise.
Joint Design
Most of the joints in the assembly consist of a single bearingnylon bushing, 1/4" machine screw, and 1/4" locknut, with and washers if as necessary. This joint design allows the bolt (the joint pin) to freely rotate while still preventing the pin from falling out. The bearing bushing reduces friction even further, but it is not strictly necessary - the hole in the second link could simply be made smaller.
The main tower rotation uses a 4-inch ball-bearing turntable purchased from McMaster. Because of the weight on this joint, it was critical that it have low friction, which the ball bearings provided. The turntable we used was designed for thrust loading, but not for tension loading, so we experienced some friction due to the moment exerted by the arm. This did not necessitate any change to the design, but we recommend researching bearings which can better withstand a bending moment, especially for robots with longer life spans.