 Alessio Quaglino
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d070b1893e
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Improved SDF objective function.
For two colliding SDFs `A` and `B`, the function is `(A+B)+abs(max(A,B))`. This can also be written as `clearance + abs(intersection)`. This function has the following properties:
- On the penetrating surface, it is equal to `A+B`, i.e. the clearance field which is the penetration depth.
- The object boundary is a set of local minima in 3D space, since `abs(max(A, B))>0` away from the surface.
- Along the penetrating surface, the maximum penetration is a local minimum of the clearance since this field is orthogonal to the midsurface field `A-B`, which acts as a support plane for the contact.
[Level sets of the improved function for two colliding circles](https://www.wolframalpha.com/input?i=plot+sqrt%28x%5E2%2By%5E2%29-1+%2B+sqrt%28%28x-1%29%5E2%2B%28y-1%29%5E2%29-1+%2B+max%28max%28sqrt%28x%5E2%2By%5E2%29-1%2C+sqrt%28%28x-1%29%5E2%2B%28y-1%29%5E2%29-1%29%2C+0%29+-+min%28max%28sqrt%28x%5E2%2By%5E2%29-1%2C+sqrt%28%28x-1%29%5E2%2B%28y-1%29%5E2%29-1%29%2C+0%29+).
PiperOrigin-RevId: 585106580
Change-Id: I5b18a1ef262ceb0a1ab8abcad8acfd6dc1992e2c
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2023-11-24 09:07:17 -08:00 |
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Alessio Quaglino
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dbf44fe2b4
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New SDF objective function.
This solves the jittery behavior observed with the gear example in the case of small applied torques (~0.5).
The new quadratic option has the form max(A, 0)^2/2 + max(B, 0)^2/2 - min(A, 0)*min(B, 0). This function has a minimum in the intersections of two SDFs A and B, while avoiding the flat areas which would be generated if only the clearance field A+B were employed. See for example [the function resulting from two colliding circles](https://www.wolframalpha.com/input?i=minimize+max%28sqrt%28x%5E2%2By%5E2%29-1%2C0%29%5E2+%2B+max%28sqrt%28%28x-1%29%5E2%2B%28y-1%29%5E2%29-1%2C0%29%5E2+-+2*min%28sqrt%28x%5E2%2By%5E2%29-1%2C0%29*min%28sqrt%28%28x-1%29%5E2%2B%28y-1%29%5E2%29-1%2C0%29)
PiperOrigin-RevId: 583992855
Change-Id: I135a1b5931cd136d7d33cc275f8d361a8b7e290c
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2023-11-20 05:06:33 -08:00 |
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