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Research Papers

# Rapid Mapping and Exploration of Configuration Space

[+] Author and Article Information
Saigopal Nelaturi, Mikola Lysenko, Vadim Shapiro

Spatial Automation Laboratory,  University of Wisconsin-Madison, 1513 University Avenue, Madison, WI 53706saigopal@gmail.com

For a set A treated as a subset of a group G, the reflection A−1 is defined in terms of the group inversion operation A−1  = {a−1 , a ∈ A}. In this case, rT ⊂ R3 .

Note that the configuration space is identified with the group SE(3) = $RSO(3)$.

If the input array size for both sets is x3 , the convolution array size will be (2x)3 . For pointwise convolution, the input array is indexed into the octant in an array of zeros (with size (2x)3 ) with coordinates between 0 and x in each dimension if the corresponding indicator function is not reflected, and into the octant with coordinates between x and 2x in each dimension if it is reflected. This positioning also eliminates the array shifting operation on the inverse transform needed on the output when the input functions are centrally positioned.

J. Comput. Inf. Sci. Eng 12(2), 021007 (May 14, 2012) (9 pages) doi:10.1115/1.4005776 History: Received September 05, 2011; Revised November 17, 2011; Published May 14, 2012; Online May 14, 2012

## Abstract

We describe a graphics processing unit (GPU)-based computational platform for six-dimensional configuration mapping, which is the description of the configuration space of rigid motions in terms of collision and contact constraints. The platform supports a wide range of computations in design and manufacturing, including three- and six-dimensional configuration space obstacle computations, Minkowski sums and differences, packaging problems, and sweep computations. We demonstrate dramatic performance improvements in the special case of configuration space operations that determine interference-free or containment-preserving configurations between moving solids. Our approach treats such operations as convolutions in the six-dimensional configuration space that are efficiently computed using the fast Fourier transform (FFT). The inherent parallelism of FFT algorithms facilitates a straightforward implementation of convolution on GPUs with existing and freely available libraries, making all such configuration space computations practical, and often interactive.

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## Figures

Figure 1

Piano T (left) that is moving in the presence of an obstacle S (right)

Figure 2

A single slice of the configuration space obstacle, obtained by computing rT, then (rT)− 1 , and then computing S ⊕ (rT)− 1

Figure 3

The SE(2) configuration space obstacle of the shape T (Fig. 1(left)) moving in the presence of the obstacle S (Fig. 1(right)). The r-slices are shown along the vertical axis and the rotations are indexed over the planar rotational group SO(2). The r-slice at 2π should technically be identified with the r-slice at 0 but we omit the identification to help visualization.

Figure 4

Thickening and erosion of a blade with a sphere as a convolution. The Minkowski sum and difference are visualized as level sets of the convolution.

Figure 5

Minkowski sum (right) of the parts (left) computed as a convolution

Figure 6

Two views of the angle axis representation of SO(3) sampled using code from Ref. [18]

Figure 7

Left: Minkowski sum of a cone and torus with improper positioning of the input sets, leading to positioning error due to periodicity of the Fourier transform. Right: Correct Minkowski sum obtained by proper positioning of the input sets.

Figure 8

A typical work part and weld gun in the presence of fixtures

Figure 9

Infinitesimal gun movement causes fragmentation of the freespace because of interference with the work part corner. The configurations shown correspond to rotations about the part surface normal and the free(light) and obstacle(dark) spaces form a partition of SO(2). Each point on the unit circle boundary corresponds to a planar rotation.

Figure 10

Example one-parameter sweep

## Errata

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