It started with this video: https://youtu.be/aLaqMreVj9o?t=56

showing a "Virtual Rolling Contact Joint" - VRCJ:

Mechanical joint with two plates connected with three rods

and then pmesten18 made a model in Fusion 360 here: https://forums.autodesk.com/t5/fusion-360-design-validate/how-would-you-create-the-functional-joints-in-this-spherical/m-p/9072192#M198903

The model it not working. It is unknown if it is because the VRCJ is mathematical impossible!? -or if the software locks when trying to solve the position. Obviously the VRCJ shown in the video is working, but that could be because of slack in the pivot joints.

If the three arms are able to twist, then Fusion 360 can model the VRCJ just fine. It is shown also here: https://forums.autodesk.com/t5/fusion-360-design-validate/how-would-you-create-the-functional-joints-in-this-spherical/m-p/9072192#M198903

Is the ideal mathematical VRCJ possible? Does it need special angles/lengths of the arms?

Edit: After the answer from @Vasily Mitch I modeled one arm, of the VRCJ, and a rod connecting the centers of the two discs. The rod is connected with a ball joint in each end, and is only there to enforce a fixed distance between the two discs (a needed condition for a VRCJ). 3D model of a VRCJ with one arm

The VRCJ can swing to the side in one projection(dimension): Swing in one dimension

and to the side in another:

Swing in another dimension

  • 1
    $\begingroup$ I will try to address this interesting problem later, but from the first glimpse, it looks like that the size of hinges (distance between the axes), the height of the shafts and radius of hinge points locations should be in agreement for the machine to work. Fusion 360 model shows presumably another ratios of those sizes. $\endgroup$ Oct 8, 2019 at 9:09

1 Answer 1


Let's consider one general element of the mechanism.

enter image description here

We have two principal motions here. In one projection there is no doubt that the motion can be depicted as virtual ball rolling. Because two centers of the ball are connected with the constant length.

enter image description here

In other projection, things are not obvious.

enter image description here

So we reformulate this question in a simpler problem: is there 2 points inside 2 balls, so when the balls roll, the distance between the points remain the same?

enter image description here

Let coordinate of this point in ball's frame of reference be $(x,y)$ and two balls of radius $1$ rolled on the angle $\varphi$. Then if point of contact is the origin and $Ox$ goes through centers, coordinates of the points are: $$ P_L=(-1+x\sin\varphi+y\cos\varphi,-x\cos\varphi+y\sin\varphi),\\ P_R=( 1+x\sin\varphi-y\cos\varphi, x\cos\varphi+y\sin\varphi) $$

Square distance between the points are: $$ \frac14 d^2 = \frac14|P_L-P_R|^2 = x^2\cos^2\varphi + (1-y\cos\varphi)^2 = 1 - 2y\cos\varphi + (x^2+y^2)\cos^2\varphi. $$

Unfortunately, this expression doesn't depend on $\varphi$ only when $x=y=0$ (case of the first projection).

So I believe two things are happening on the video. First, there is likely a small slack in pivots allowing for the motion. Second, the motion itself is not mathematically a virtual roll (for, example distance between balls is varying a bit).

  • $\begingroup$ I do not fully understand your answer - see my edit. Can you mark the angles on your 3D model? $\endgroup$ Oct 10, 2019 at 13:54
  • $\begingroup$ To create rolling motion, you need something more, than just conserve distance between centers. You also need this triangle to be isosceles. It's obvious from symmetry: if there is no slipping, axes of the balls should rotate the same amount. $\endgroup$ Oct 10, 2019 at 15:00
  • $\begingroup$ Try to append gears to the disks in the software to represent the rolling contact. $\endgroup$ Oct 10, 2019 at 15:02

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