Haptic Assembly Using Skeletal Densities and Fourier Transforms

Abstract

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.

Implementation and evaluation of a model processing pipeline for assembly simulation

“…Recently, a novel method for solving virtual assembly problems in general (and for haptic rendering, in particular) was proposed by Behandish and Ilies (2015, 2016). The developed concept – “geometric energy field” – is very interesting and could represent a new type of solution to a complex problem.…”

Analytic methods for geometric modeling via spherical decomposition

“…As usual, we assume solids to be 'r-sets', defined by Requicha [31] as compact (i.e., bounded and closed) regular semianalytic subsets of the Euclidean 3−space S ⊂ P(R 3 ). 4 The regularity condition (i.e., S = rS) ensures continuous homogeneity, 5 while the semianalytic requirement guarantees finite describability of the set [31], as well as its medial axis (MA) and medial axis transform (MAT) [7]. The latter is defined as an embedding of the MA in the 4D space with the radius of the maximal ball conceptualized as a new coordinate, and contains enough information to reconstruct the solid S, hence can be used to develop a discretization/sampling scheme in Section 2.2.…”

“…For example, Minkowski operations [35] that are central to mathematical morphology are formalized as convolutions of constituent functions and computed efficiently in the Fourier domain [24]. Minkowski operations have been used extensively to formulate important problems in robot path planning [22], mechanism workspace design [26], virtual reality (graphics/haptics) [5,4], protein docking [1], packaging and nesting [9], and more. Unfortunately, their combinatorial computation even for 3D polyhedral objects quickly becomes impractical with increasing number of polygons [24].…”

Peg-in-Hole Revisited: A Generic Force Model for Haptic Assembly

“…However, this approach goes against the philosophy of avoiding the multiphase approach and manual specifications, from which we set off to pursue this method. We recently presented an alternative implementation in [63,64] that uses GPU-accelerated fast Fourier transforms (FFT) to enable real-time computations, the discussion of which is beyond the scope of this paper.…”

“…The score gradient in ( 9) and (10), needed for the guidance forces and torques in (12) and (13), can be discretized in a similar fashion. Alternatively, one could approximate the gradient using the finite 5 We report on a similar implementation on the Graphics Processing Unit (GPU) in [63]. difference method (FDM) by multiple computations of ( 15) after applying small translational and rotational variations, along each of the 3 coordinate axes one at a time, to the relative transformation T = T −1 1 T 2 ∈ SE(3).…”