Saigopal Nelaturi

Group morphology with convolution algebras

Group morphology is an extension of mathematical morphology with classical Minkowski sum and difference operations generalized respectively to Minkowski product and quotient operations over arbitrary groups. We show that group morphology is a proper setting for unifying, formulating and solving a number of important problems, including translational and rotational configuration space problems, mechanism workspace computation, and symmetry detection. The proposed computational approach is based on group convolution algebras, which extend classical convolutions and the Fourier transform to non-commutative groups. In particular, we show that all Minkowski product and quotient operations may be represented implicitly as sublevel sets of the same real-valued convolution function.

Automated fixture configuration for rapid manufacturing planning

The wide adoption of agile manufacturing systems has necessitated the design and use of fixtures or work holding devices that have in-built flexibility to rapidly respond to part design changes. Despite the availability of reconfigurable fixtures, practical fixture configuration largely remains an experience driven manual activity to enable customization for varying workpiece geometry, and most automated solutions do not scale well to accommodate such variation. In this paper, we address the problem of rapidly synthesizing a realistic fixture that will guarantee stability and immobility of a specified polyhedral work-part. We propose that the problem of automated fixture layout may be approached in two distinct stages. First, we determine the spatial locations of clamping points on the work piece boundary using the principles of force and form closure, to ensure immobility of the fixtured part under external perturbation. In particular, we show that the candidate restraints mapped to the six dimensional vector space of wrenches (force-moment pairs) may be hashed in a straightforward manner to efficiently generate force closure configurations that restrain part movement against large external wrenches. When clamps are allowed to exert arbitrarily high reaction forces on the part, the spatial arrangement of the clamping locations ensures the part is in form closure. On generating force/form closure configurations, the chosen locations are matched against a user-specified library of reconfigurable clamps to synthesize a valid fixture layout comprising clamps that are accessible and collision free with each other and the part. Additionally, in the case of determining machining setups the clamps are chosen to avoid collisions with the moving cutting tool. We demonstrate fast algorithms to perform both location selection and fixture matching, and show several results that underscore the practical application of our solution in automated manufacturing process planning.

Representation and analysis of additively manufactured parts

Representations of solid models were initially formulated partially in response to the need to support automation for numerically controlled machining processes. The assumed equivalence between shape, topology, and material properties of manufactured components and their computer representations led to the practice of modeling and simulating the behavior of physical parts before manufacture. In particular, representations of shape and material properties are treated in distinct nominal models for most unit manufacturing processes. Additively manufactured parts usually exhibit deviations from their nominal geometry in the form of stair-stepping artifacts and topological irregularities in the vicinity of small features. Furthermore, structural properties of additively manufactured parts have experimentally been shown to be dependent on the build orientation defining the cross sections where material is accumulated. Therefore geometric models of additively manufactured parts cannot be decoupled from the manufacturing process plan.In this paper we show that as-manufactured shapes may be represented in terms of the convolution operation to capture the additive deposition of material, measure the conformance to nominal geometry in terms of overlap volume, and model uncertainties involved in material flow and process control. We then demonstrate a novel interoperable approach to physical analysis on as-manufactured part geometry represented as a collection of machine-specific cross sections augmented with boundary conditions defined on the nominal geometry. The analysis only relies on fundamental queries of point membership classification and distance to boundary and therefore does not involve the overhead of model preparation required in approaches such as finite element analysis. Results are shown for non-trivial geometries to validate the proposed approach. Representation of as-manufactured models using convolution.Incorporating manufacturing uncertainty into representation.Computing as-manufactured models using planar morphological operations.Interoperable analysis using query based simulation on sliced as-manufacturable models.

Configuration products in geometric modeling

The six-dimensional space SE(3) is traditionally associated with the space of configurations of a rigid solid (a subset of Euclidean three-dimensional space E3). But a solid can be also considered to be a set of configurations, and therefore a subset of SE(3). This observation removes the artificial distinction between shapes and their configurations, and allows formulation and solution of a large class of problems in mechanical design and manufacturing. In particular, the configuration product of two subsets of configuration space is the set of all configurations obtained when one of the sets is transformed by all configurations of the other. The usual definitions of various sweeps, Minkowski sum, and other motion related operations are then realized as projections of the configuration product into E3. Similarly, the dual operation of configuration quotient subsumes the more common operations of unsweep and Minkowski difference. We identify the formal properties of these operations that are instrumental in formulating and solving both direct and inverse problems in computer aided design and manufacturing. Finally, we show that all required computations may be implemented using a fast parallel sampling method on a GPU.

Solving inverse configuration space problems by adaptive sampling

Given two shapes in relative motion, an important class of inverse configuration problems are solved by determining relative configurations that maintain set-inclusion relationships (non-interference, containment, or contact) between the shapes. This class of inverse problems includes the well-known problem of constructing a configuration space obstacle, as well as many other problems in computational design such as sweep decomposition, accessibility analysis, and dynamic packaging. We show that solutions to such problems may be efficiently approximated directly in the 6D configuration space SE(3) of relative motions by adaptive sampling. The proposed method relies on a well-known fact that the manifold of the group SE(3) is a Cartesian product of two manifold subgroups: the group of rotations SO(3) and the group of translations R^3. This property allows generating desired configurations by combining samples that are generated in these subgroups independently and adaptively. We demonstrate the effectiveness of the proposed approach on several inverse problems including the problem of sweep decomposition that arises in reverse engineering applications.

Configuration products and quotients in geometric modeling

The six-dimensional space SE(3) is traditionally associated with the space of configurations of a rigid solid (a subset of Euclidean three-dimensional space R^3). But a solid itself can be also considered to be a set of configurations, and therefore a subset of SE(3). This observation removes the artificial distinction between shapes and their configurations, and allows formulation and solution of a large class of problems in mechanical design and manufacturing. In particular, the configuration product of two subsets of configuration space is the set of all configurations obtained when one of the sets is transformed by all configurations of the other. The usual definitions of various sweeps, Minkowski sum, and other motion related operations are then realized as projections of the configuration product into R^3. Similarly, the dual operation of configuration quotient subsumes the more common operations of unsweep and Minkowski difference. We identify the formal properties of these operations that are instrumental in formulating and solving both direct and inverse problems in computer aided design and manufacturing. Finally, we show that all required computations may be approximated using a fast parallel sampling method on a GPU and provide error estimates for the approximation.

Non-commutative morphology: Shapes, filters, and convolutions

Group morphology is a generalization of mathematical morphology which makes an explicit distinction between shapes and filters. Shapes are modeled as point sets, for example binary images or 3D solid objects, while filters are collections of transformations (such as translations, rotations or scalings). The action of a filter on a shape generalizes the basic morphological operations of dilation and erosion. This shift in perspective allows us to compose filters independent of shapes, and leads to a non-commutative generalization of the Minkowski sum and difference which we call the Minkowski product and quotient respectively. We show that these operators are useful for unifying, formulating and solving a number of important problems, including translational and rotational configuration space problems, mechanism workspace computation, and symmetry detection. To compute these new operators, we propose the use of group convolution algebras, which extend classical convolution and the Fourier transform to non-commutative groups. In particular, we show that all Minkowski product and quotient operations may be represented implicitly as sublevel sets of the same real-valued convolution function.

Rapid Mapping and Exploration of Configuration Space

We describe a 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 six dimensional configuration space computations practical, and often interactive.Copyright

Feasible spaces in weld gun selection

An important requirement of any robot-assisted welding process is to ensure that a set of weld locations on a work assembly can be reached by the robot-weld gun assembly without colliding with the work assembly and surrounding tooling. A weld gun that maintains a valid contact at the a weld location without interference is said to be feasible at that location. An important class of problems in welding process planning and optimization reduce to the problems of verification and synthesis of feasible weld guns for a given set of weld locations. We formulate these problems using standard geometric modeling operations and show how they can be solved in a commercial CAD system. Our approach exemplifies a more general strategy for planning in reconfigurable manufacturing in terms of configuration spaces. A prototype implementation in Unigraphics NX4 with realistic industrial models demonstrates the effectiveness of the proposed approach to weld process planning.

Manufacturability Feedback and Model Correction for Additive Manufacturing

Additive manufacturing, or 3d printing, is the process of building three dimensional solid shapes by accumulating material laid out in sectional layers. Additive manufacturing has been recognized for enabling production of complex custom parts that are difficult to manufacture otherwise. However, the dependence on build orientation and physical limitations of printing processes invariably lead to geometric deviations between manufactured and designed shapes that are usually evaluated after manufacture. In this paper, we formalize the measurement of such deviations in terms of a printability map that simulates the printing process and partitions each printed layer into disjoint regions with distinct local measures of size. We show that manufacturing capabilities such as printing resolution, and material specific design recommendations such as minimal feature sizes may be coupled in the printability map to evaluate expected deviations before manufacture. Furthermore, we demonstrate how partitions with size measures below required resolutions may be modified using properties of the medial axis transform, and use the corrected printability map to construct a representation of the manufactured model. We conclude by discussing several applications of the printability map for additive manufacturing.Copyright