# Mathematical and Computational Sciences Division Projects

## Building and Fire Research Laboratory

### Mathematical Problems in Construction Metrology

The Building and Fire Research Laboratory (BFRL) at NIST is currently testing the effectiveness of LADAR (Laser Direction and Ranging) technology for modeling and locating equipment at construction sites. The Mathematical and Computational Sciences Division (MCSD) of the Information Technology Laboratory (ITL) at NIST has provided modeling and computational support to this effort in several areas. This support involved four major tasks.

• The first task was to determine how well LADAR scan data could be used to recognize objects at a construction site. A principal motivation for automated object recognition at a construction site involves the use of autonomous (robot) cranes to position themselves to pick-and-place objects. Two approaches to object recognition have been tried, each with some level of success, but both methods requiring further research and technology progress.
• In the first method, bar codes, using traffic signage material, were placed on panels and scanned by a LADAR at different distances. It was felt that objects could be identified by bar code configurations. The bar codes were of significant size so that a sufficient number of scanned data points could be acquired. D. Gilsinn (MCSD), G. Cheok(BFRL) and D. O’Leary (MCSD) developed a mathematical model using methods from image processing that was applied to the resulting scanned data to determine at what distances the bar code configurations could be reconstructed. For a particular LADAR used it was found that reconstruction was feasible in the 10 - 20 m range.
• A second approach by D. Gilsinn (MCSD), G. Cheok (BFRL) and C. Witzgall (MCSD) involved examining the raw scanned data and grouping points into those “objects” that exhibited a three dimensional shape. Theoretical polygons were built around these objects and comparisons were made with polygons of ideal objects in a database. A close comparison indicated likely object identification. The position and angle relative to the scanner could be determined in order to align an autonomous crane with the object.
• A second major task involved characterizing terrain roughness for autonomous vehicle traversal. D. Gilsinn (MCSD) developed a pitch and roll algorithm that could be easily applied to terrain data in order to evaluate the locations of critical pitch and roll angles that would be experienced by an autonomous vehicle during a path traversal. He also began developing potential measures of terrain surface roughness based on local terrain slope analysis.
• In a third task related to problems associated with construction sites is the ability to model structures with sharp edges. D. E. Gilsinn, M. McClain, and C. Witzgall, all of MCSD) developed a surface modeling algorithm for fitting surfaces modeled by continuously differentiable Hseieh-Clough-Tocher elements over a triangulated surface that included objects with sharp edges. The minimization algorithm combined a relaxation technique plus a re-weighting of the surface energies of the elements in the fitting process. The fits to objects with sharp edges were extraordinarily good.
• In a fourth construction sight related project, D. Gilsinn is working with BFRL to compute the dielectric constants of materials typically used in the construction industry. BFRL has conducted experiments on ultra-wideband synthetic aperture radar transmission through composite walls to develop non-line-of-sight spatial position detection (e.g. firefighter location within buildings). The dielectric constants of wall materials are needed to correct for transmission delays through the construction materials.

## Manufacturing Engineering Laboratory

### Optimization Problems in Smart Machining Systems

Smart Machining Systems (SMS) aim at producing the first and every product correct; improving the response of the production system changes in demands; realizing rapid manufacturing; and, providing data on an as needed basis. An SMS is envisioned to contain a dynamic process optimizer with the capability of assessing the quality of the work and outputs of the SMS as well as improve itself over time. The dynamic optimizer builds and satisfies objective functions using machining models. It includes constraints from design such as dimensional and geometrical tolerances, surface integrity and surface quality. A general machining optimization problem consists in determining some decision variables, such as feed, depth of cut, spindle speed, in such a way that a set of constraints is satisfied and a desired objective function is optimized. F. Potra and D. Gilsinn are investigating the use of robust optimization methods. These methods aim at determining the decision variables such that the objective function is minimized and constraints are satisfied for all possible values of the machining parameter ranges. An objective function and constraints were defined for a preliminary SMS machining model. The objective function was solved by a linear program. The code was incorporated into a GUI for the machine operators by MEL collaborators and was used to machine a test part

## Research and Development in Mathematical Methods and Software

### Approximating Solutions of Delay Differential Equations

D. Gilsinn continues research on approximation of periodic solutions for delay differential equations (DDEs). This class of equations occurs in the stability analysis of machine tool vibrations, an area of metrology interest in the Manufacturing Engineering Laboratory (MEL) at NIST. A discrete Fourier representation algorithm has been developed that lends itself to efficient matrix-vector operations. The stability of solutions for DDEs with periodic coefficients is determined by certain eigenvalues that are called the characteristic multipliers. One method for developing characteristic multipliers for periodic solutions of delay equations involves solving an eigenvalue problem for an integral operator. The eigenvalues are estimated by those of an associated discretized linear operator. A pseudospectral collocation algorithm has been developed to approximate the fundamental matrix of the variational equation about the approximate solution. This algorithm has been included as an alternative to using a numerical delay solver for the fundamental matrix. In collaboration with F. Potra, an article has been submitted to the Journal of Integral Equations and Applications that analyses the convergence properties of the eigenvalues of the discretized integral operator to those of the integral operator.

### Volume Estimation of Molded Industrial Artifacts by B-Splines

D. Gilsinn of ITL, B. Borchardt of MEL, and A. Tebbe of St. Mary's College of Maryland have investigated a method of estimating volumes of some semi-spherical molded artifacts produced for the Food and Drug Administration (FDA) by an industrial contractor. These artifacts are simulated lung cancer nodules, called phantoms, used to compare software on computed tomography (CT) scanners. MEL produced surface coordinate points on the molded artifacts by a coordinate measuring machine (CMM). This data was converted to spherical coordinates and modeled by tensor products of B-Splines. These were used since they have a small support set and lead to sparse matrices in the fitting process. Once B-Spline models were fit to the spherical coordinates the functions were used to extrapolate the data onto uniform grids on the surfaces that were then used along with the classic Divergence Theorem to estimate the artifacts' volumes. A bootstrap mehtod was used to estimate uncertainties. Straightforward least squares fits of the B-Spline models produced oscillations at the semi-sphere poles. Regularization methods are being studied to smooth the oscillations.

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