Kinematic Couplings | Hardware II
Kinematic couplings provide a repeatable, accurate method for locating two parts in relation to one another by constraining the six degrees of motion - x, y, z and pitch, yaw, and roll. Six is the magic number, any less than six and the system will be under constrained, but also any more than six and the system becomes over constrained – both with consequences. An over constrained system risks inducing parasitic stresses and forces and can break down under deformation. Exact constraint of a system ensures that the system will behave deterministically.
Kinematic couplings are strategic in designs that require part of the assembly to be removed and re-attached with high repeatability and precise placement. Example applications could be interchangeable mounted robotic arms for different functionality or interchanging tool couplings accurately for a mill.
What type of materials can one use to create 10 thou precision kinematic couplings? Can you use donuts and candy? To test this I built a Kelvin coupling, shown below, using donuts, peanut butter as the epoxy, Whoppers as the curved mating surface, spice drops as the grooves and a double length piece of spaghetti as the measuring tool.
The top coupling with the whoppers and spaghetti was removed and replaced 13 times and the tip of the spaghetti was marked each time, the markings are shown below. This data was then put into an excel sheet and the angular error was calculated as a function of the length of the spaghetti. Averaging the deviations for X and Y I found that the X average deviation was 0.1042 in and the Y deviation was 0.01446 in.
What contributed to this significant difference in error?
The green coupling significantly impacts the accuracy of the x-degree of freedom as it's two contact points constrain the ball in the x direction. As you can see in the image, the two pieces of candy have a slight draft angle from the top to the bottom. I also observed during testing that the two green pieces of candy tended to slowly move apart in the peanut butter and midway through I pushed them closer again. The compliance and variability of this couple definitely was a factor in the increased error of the x-axis.
A more practical example of a kinematic coupling is hanging artwork. Mounting artwork in a dorm room doesn’t require much precision beyond stepping back to make sure it looks straight, but hanging professional artwork in a museum or gallery is a non-trivial task that takes a significant amount of time and is most often hand measured. Furthermore, a smaller painting can be fairly easily adjusted, but if you’re working with “The Wedding at Cana” the largest piece of art in the Louvre, strong and precise ways to attach the artwork become essential. The wall hanging kinematic coupling is a potential method for accurately hanging artwork. A kinematic coupling is particularly useful for this application, because it would allow a curator to remove the artwork for cleaning or a sale and replace it without needing to re-measure and level the art.
Kinematic Coupling Design
Typically, the installation process for a piece of artwork, especially a larger piece, would be to first mark the top of the desired placement, and calculate how much lower to place the hooks. Then, draw a perfectly horizontal line and bolt two (or more) hooks into the wall. The piece of artwork can then be hung from a horizontal wire attached to two sides of the frame, but needs to be adjusted side to side to ensure that it is centered and hanging evenly from the two hooks. Larger pieces have more secure bolts fastened to the frame.
Initially, the proposed kinematic coupling included a decoupled alignment and attachment system, but the final designs integrates the two functions into a kinematic coupling that provides preload and precision alignment.
To install, when the art is removed and replaced the exact positioning is retained and manual adjustment, alignment and leveling is unnecessary.
For this application, no more than 0.1 degree of tilt / rotation is acceptable, as this offset will be noticeable to the human eye as crooked.
Figures: Dimensioned drawings of the KC vertical, Creating the grooves for the ball connectors and the underlying sketches for the part, Solid model of a vertical kinematic coupling, and final constructed vertical kinematic coupling.
Kinematic Coupling Analysis and Predicted Performance
Unfortunately, the kinematic coupling spreadsheet is optimized for a 3 ball 3 groove horizontal kinematic coupling. I filled out the excel sheet with the relevant data for this vertical coupling to practice and understand the spreadsheet, the analysis is shown below.
The most important degree of motion to accurately constrain for this application is the rotational motion about the center of the coupling triangle. Rotational motion would lead to an offset from parallel of the top of the coupling, which would indicate a tilt in the artwork attached to the coupling. Thus, this was the primary degree of motion tested for error.
The test setup sought to minimize errors by securing the bottom of the coupling to a heavy table in the shop and hot gluing the laser pointer to the ball side of the coupling to make sure there was so slop from the tape shifting during testing. The distance of the laser pointer was 12.5 meters (1249 cm). Ten trials were completed, in which the ball side of the coupling was fully removed and then replaced on the groove side and the laser pointer center was measured. The laser pointer made a large mark on the paper regardless of distance. To maintain consistency of measurement across the trials two crossed lines were drawn on the laser ellipse on the paper, and the center point of these two lines was considered the measurement point. A 1 lb. weight was attached to the front of the ball side of the coupling to emulate the moment created by the weight of a piece of artwork hanging from the coupling.
Shown below is a table, graph and image of the results of testing. The first data point shown in gray in the table was not included in the calculations, because the ball was not fully placed within the groove on the back of the coupling, but the point is included in the data to demonstrate the error of misalignment compared to alignment of all the grooves.
Figure 6: Plotted measurements from laser test
The key questions underlying the theory of this exercise are: Does the coupling deform less than the maximum acceptable amount under the range of loads it will be subjected to, and is it still constrained in the way you want it to be when subjected to load?
Stated in the proposal, the maximum amount of angular error acceptable is 0.1 degrees in the x-y plane. The measured maximum angular error in the y-direction was 0.000411 degrees, significantly less than the acceptable range. This was measured with a load of 1 lb. Future measurements could be conducted with heavier loads to simulate different sized pieces of artwork.
Would this coupling work for “The Marriage Feast at Cana” a huge piece of artwork? There are two reasons why this exact coupling wouldn’t work: the material and fabrication would not hold the weight and this kinematic coupling is too small to resist the large moments from a sizable painting. A much larger coupling or series of couplings could be designed specifically for the largest pieces.
The proposed design of this kinematic coupling initially mentioned that the vertical coupling could be modified to integrate both the hanging and alignment mechanisms, eliminating the need for hook installation - this would be an added bonus to the design. The final coupling presented does integrate these two functionalities by using the principle of reciprocity, placing the top center groove on the back of the main plate to provide a force resisting the moment of the picture frame hanging on the mount.