Continuous penetration depth

• The resulting configuration causes the two objects to just touch.
• The penetration depth magnitude and direction continuously change for a continuous relative motion in configuration space.
• The derivatives of the continuous penetration depth are well defined if the contact space is smooth and the motion is continuous.
• Applicability to convex and non-convex contact spaces that are homeomorphic to a sphere.

We present a new measure for computing the continuous penetration depth between two intersecting rigid objects. We generate a set of samples in the configuration space, precompute an approximation of the contact space for two intersecting objects using binary classification techniques, and construct a bijective mapping between the spherical space and the precomputed contact space. For a given in-collision configuration, we search the spherical space for the nearest neighbor, and find the corresponding image in the contact space based on the predefined spherical parameterization. The resulting image is a witness equivalent to the nearest configuration, and it is used to formulate the penetration depth direction based on our measure. Unlike prior algorithms, our algorithm guarantees that both the penetration depth magnitude and direction are continuous with respect to the motion parameters. Our algorithm is approximate in the sense that we approximate the exact contact space. We have applied our algorithm to complex rigid models composed of tens or hundreds of thousands of triangles, and the runtime query takes only around 0.01 ms.