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Improved SDF objective function.

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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
This commit is contained in:
Alessio Quaglino
2023-11-24 09:06:19 -08:00
committed by Copybara-Service
parent 6a83b86553
commit d070b1893e
3 changed files with 22 additions and 23 deletions
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doc/programming/extension.rst
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@@ -288,11 +288,11 @@ Currently, there are three directories of first-party plugins:
<https://github.com/google-deepmind/mujoco/blob/main/plugin/sdf/README.md>`__. The rest of this section will give more
detail concerning the collision algorithm and the plugin engine interface.
Collision points are found by minimizing a quadratic form of the two colliding SDFs via gradient descent.
Because SDFs are non-convex, multiple starting points are required in order to converge to multiple local minima.
The number of starting points is set using :ref:`sdf_initpoints<option-sdf_initpoints>`, and are
initialized using the Halton sequence inside the intersection of the axis-aligned bounding boxes.
The number of gradient descent iterations is set using :ref:`sdf_iterations<option-sdf_iterations>`.
Collision points are found by minimizing the function A + B + abs(max(A, B)), where A and B are the two colliding
SDFs, via gradient descent. Because SDFs are non-convex, multiple starting points are required in order to converge to
multiple local minima. The number of starting points is set using :ref:`sdf_initpoints<option-sdf_initpoints>`, and
are initialized using the Halton sequence inside the intersection of the axis-aligned bounding boxes. The number of
gradient descent iterations is set using :ref:`sdf_iterations<option-sdf_iterations>`.
While *exact* SDFs---encoding the precise signed distance to the surface---are preferred, collisions are possible with
any function whose value vanishes at the surface and grows monotonically away from it, with a negative sign in the