I find that you often don’t need to know the exact center of mass of a robot. What tends to be more important is the weight balance on the wheels - and you can figure that out quickly enough with a bathroom scale. Just pick up one corner of the robot and slide the scale under that wheel. Record the number, and repeat with each corner. From there, it’s easy to figure out which corner or edge to add the weight on, and how much - if your readings are 25, 25, 25, 15, then adding 10 lbs near the 15 wheel should bring you as close to center as you need!
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Yes. The 2-D center of GRAVITY with respect to the floor can be diagrammed and located from those four points (assuming it’s a rectangle). I’d add that to be more accurate, the remaining three wheels should be blocked up so that the machine is level when each wheel location is weighed.
Then, the “balance point” or “seesaw fulcrum point” along each side can be calculated using the weights on each corner, like this:
I’d use graph paper (I’m old) to scale off the frame and then the location of the “balance point” on each side using the fractional distance calculated.
Then, connect the two points along the E-W sides with a line, and then connect the two points along N-S sides with another line. Where they cross is the “center of gravity”.
Calculation of the 3-D center of gravity (a point away from the floor) is way beyond my ability to understand physics.
Yes and no. Every 3D modeling package has a button you can push to get calculated mass properties. With the level of CAD I’ve seen in FRC, including our own, those results will be wrong. Mass properties calculations are the classic example of “garbage in/garbage out”. For them to be accurate, EVERY component of your finished robot, including all fasteners, COTS parts, and wiring must be accurately modeled. Then, you need to weigh and input by hand all the masses of COTS parts. Same for 3D printed parts, because infill percentage changes the actual masses from calculated masses. Then you need to have tight quality assurance to be sure the hardware is fabricated to exactly match your CAD model. (No substituting for different screw lengths, for example.) This level of effort is incredibly time consuming. We do it at NASA (to a degree) because mass properties are very important to trajectory planning and payload weight predictions. Even with having people whose full time job is tracking weight and mass properties, the measured mass props and the calculated mass props STILL don’t match.
So, yes it’s easy to push the “calculate mass properties” button. But, no it’s not easy to get an accurate answer. It’s WAY easier and quicker to just measure your robot.
For those who care to, do this fun little experiment. Have everyone make a prediction of how much weight is in the fasteners and in the wiring of your robot. Then, next year when you disassemble it, drop every fastener in a 5 gallon bucket, and all the wiring in a different bucket. Measure the contents and see who guessed closest. You’ll be surprised how much each of these categories of hard to predict stuff contributes to your overall robot weight. Steel is heavy, and so is copper.
Bonus questions: How much internal volume do you have in your pneumatic system? How much weight gain do you get when you pressurize it to 120 psi? It’s probably insignificant, but it’s not zero, either.
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In the past, we have found our center of mass along a given axis by taking spare wheels and setting them between the floor and the bottom robot frame. If you’re careful to keep the wheels well aligned with each other and parallel to the axis in which you want to measure, this can work really well. While keeping the frame level, roll the frame forward/back or side to side until it is balanced atop the wheels. Your Center of mass is directly above the imaginary line segment that connects the contact points of the spare support wheels on the frame.
This could be done in multiple directions (x or y) if needed, but for example we just did it fore-aft in the orientation we planned to climb in for 2013, and achieved remarkably level lifts.
Hopefully I described that well enough.
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I think it depends on why (and when) you need to know the center of mass. If it’s for some part of your design process, you probably don’t need extreme accuracy, and you do need to know before it’s built. For example, if you’re trying to assess whether you need to mitigate tipping risk, or find the moment-arm of a rotating mechanism, it’s often acceptable to find it in CAD and consider bolts and wires negligible. Or even to add a fudge factor of “the CAD CoM is here, but there’ll be motors at the end that weigh ~1.5 lb each, so actually a little farther to the left”. If you need to know it with very high accuracy, and can wait until after the robot is built (I assume for software calculations), then I agree it’s better to find it empirically.
It’s level of significance is directly proportional to the proximity of a prankster to the scale 
Ambient air masses ~1kg/m³ or ~1g/l. 120 psi is roughly 8 atm, so the extra weight for a 120psi system is about 8 g/l, or about an ounce per gallon. The weight of ambient air doesn’t matter because of buoyancy.
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