  Haptic-assisted virtual assembly and prototyping has seen significant
attention over the past two decades. However, in spite of the appealing
prospects, its adoption has been slower than expected. We identify the main
roadblocks as the inherent geometric complexities faced when assembling objects
of arbitrary shape, and the computation time limitation imposed by the
notorious 1 kHz haptic refresh rate. We addressed the first problem in a recent
work by introducing a generic energy model for geometric guidance and
constraints between features of arbitrary shape. In the present work, we
address the second challenge by leveraging Fourier transforms to compute the
constraint forces and torques. Our new concept of 'geometric energy' field is
computed automatically from a cross-correlation of 'skeletal densities' in the
frequency domain, and serves as a generalization of the manually specified
virtual fixtures or heuristically identified mating constraints proposed in the
literature. The formulation of the energy field as a convolution enables
efficient computation using fast Fourier transforms (FFT) on the graphics
processing unit (GPU). We show that our method is effective for low-clearance
assembly of objects of arbitrary geometric and syntactic complexity.
