Another rotation problem


Another rotation problem
#1
This one's really just for fun; I was able to figure it out by eye to within half a degree, and I think it's good enough. But I'm still curious how you'd go about finding the "right" answer…

   

So our task here is simply to have our spaceman get a grip on his control stick (you can see he's already grabbed it in the right hand, so we just have to figure out the left). To do this, we need to figure out the angle of four different rotations. The control stick has to be tilted upward (1) and, along with its base, rotated inward (2), while the spaceman's arm has to pivot at the shoulder (3) to meet it. Finally, the hand has to rotate at the wrist (4) so that it's aligned with the control stick. (This last one is tricky because the hand portion—the actual "clip", or claw—is not aligned with any axis of the part itself, and the part's origin isn't located within the hand but rather at the wrist.)

Because we are perfectionists, the ball of the control stick should be centered longitudinally within the minifig's hand, but flush with its top edge. Since it's an 8-LDU sphere, the center of the sphere should be 4 LDU below the top edge (and again, that's relative to the hand itself, not the part overall).

I was able to get close on the first 3 steps (see below) by using successive approximation along with my intersecting circles method, then made some final small tweaks to account for the wrist rotation—again, good enough, but necessarily imprecise by nature. Guess I'll have to break out the trigonometry again to find the truth!

   
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RE: Another rotation problem
#2
(2019-12-06, 6:41)N. W. Perry Wrote: I was able to get close on the first 3 steps (see below) by using successive approximation along with my intersecting circles method, then made some final small tweaks to account for the wrist rotation—again, good enough, but necessarily imprecise by nature. Guess I'll have to break out the trigonometry again to find the truth!

Smile

You could try to find the exact intersection of the arm disc with the lever's rotational sphere.

Not sure how to that though.

Afterwards set the arm using a helper part at the found location, then use closed triangle info to position the lever in 2 steps.

And last rotate the hand to match the lever direction (could probably do that with rel grid, reset orientation).
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RE: Another rotation problem
#3
(2019-12-06, 19:00)Roland Melkert Wrote: Smile

You could try to find the exact intersection of the arm disc with the lever's rotational sphere.

Not sure how to that though.

Afterwards set the arm using a helper part at the found location, then use closed triangle info to position the lever in 2 steps.

And last rotate the hand to match the lever direction (could probably do that with rel grid, reset orientation).

I think the trouble with that will be that when the hand is rotated, so does the point where the control stick is supposed to sit, because that point doesn't lie on the part's Z axis. I think I'd have to find the rotational disc of this point and compound it with the disc of the arm, which I guess would form, what, some kind of donut? Or I guess it would be a cylinder?
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RE: Another rotation problem
#4
(2019-12-06, 21:09)N. W. Perry Wrote: I think the trouble with that will be that when the hand is rotated, so does the point where the control stick is supposed to sit, because that point doesn't lie on the part's Z axis. I think I'd have to find the rotational disc of this point and compound it with the disc of the arm, which I guess would form, what, some kind of donut? Or I guess it would be a cylinder?

Follow upNo, after messing around with it, it appears to be a donut after all. The combined rotation of the arm and the control point in the hand seems to describe an irregular torus with an elliptical cross section, so its intersection with the sphere would form a tiny ellipse on the surface, and you'd have to figure out which point on that ellipse represents the hand aligned with the stick.

I have no idea how you'd calculate such an intersection, so if we remove the requirement of rotating the hand, we're just left with figuring the intersection points of the arm disc with the lever's sphere—there must be a formula or online calculator out there that can do this, but I haven't found it yet…
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