Marek Franaszek

Melnikov-Based Open-Loop Control of Escape for a Class of Nonlinear Systems

The performance of certain nonlinear stochastic systems is deemed acceptable if, during a specified time interval, the systems have sufficiently low probabilities of escape from a preferred region of phase space. We propose an open-loop control method for reducing these probabilities. The method is applicable to stochastic systems whose dissipation- and excitation-free counterparts have homoclinic or heteroclinic orbits. The Melnikov relative scale factors are system properties containing information on the frequencies of the random forcing spectral components that are most effective in inducing escapes. Numerical simulations show that substantial advantages can be achieved in some cases by designing control systems that take into account the information contained in the Melnikov scale factors.

Fitting Spheres to Range Data From 3-D Imaging Systems

Two error functions used for nonlinear least squares (LS) fitting of spheres to range data from 3-D imaging systems are discussed: the orthogonal error function and the directional error function. Both functions allow unrestricted gradient-based minimization and were tested on more than 40 data sets collected under different experimental conditions (e.g., different sphere diameters, instruments, data density, and data noise). It was found that the orthogonal error function results in two local minima and that the outcome of the optimization depends on the choice of starting point. The centroid of the data points is commonly used as the starting point for the nonlinear LS solution, but the choice of starting point is sensitive to data segmentation and, for some sparse and noisy data sets, can lead to a spurious minimum that does not correspond to the center of a real sphere. The directional error function has only one minimum; therefore, it is not sensitive to the starting point and is more suitable for applications that require fully automated sphere fitting.

Noise-induced snap-through of a buckled column with continuously distributed mass: A chaotic dynamics approach

Abstract For a spatially-extended dynamical system we illustrate the use of a chaotic dynamics approach to obtain criteria on the occurrence of noise-induced escapes from a preferred region of phase space. Our system is a buckled column with continuous mass, subjected to a transverse continuously distributed load that varies randomly with time. We obtain a stochastic counterpart of the Melnikov necessary condition for chaos—and snap-through—derived by Holmes and Mardsen for the harmonic loading case. Our approach yields a lower bound for the probability that snap-through cannot occur during a specified time interval. In particular, for excitations with finite-tailed marginal distribution, a simple criterion is obtained that guarantees the non-occurrence of snap-through.

Gauging the Repeatability of 3-D Imaging Systems by Sphere Fitting

Multiple scans of the same object acquired with a 3-D imaging system (e.g., a laser scanner) under the same experimental conditions could provide valuable information about the instruments performance (e.g., repeatability, existence of bias). A geometrical primitive may be fitted to multiple data sets and the variances of the fitted objects parameters may be used to evaluate the instruments repeatability. We test this procedure on simulated data as well on data acquired in a laboratory. Two different error functions (orthogonal and directional) are used to fit a sphere of known radius to the data. The spread of the sphere centers fitted with the directional function to simulated unbiased data is in agreement with the theoretically calculated variances of the fitted centers. For sphere centers fitted to simulated data with added bias and to the data acquired in a laboratory, the variances do not agree with the spread. This is interpreted as evidence that the data were not collected under repeatable conditions. The orthogonal fitting yields sphere centers in disagreement with theory both for the simulated and laboratory data sets and therefore, the choice of the error function is important.

Augmenting BIM with 3D Imaging Data to Control Drilling for Embeds into Reinforced Concrete Bridge Decks

Placing embeds into reinforced concrete structures, after concrete is poured, without damaging reinforcement bars (rebar) is an industry-wide challenge encountered across the construction industry. In concrete structures such as containment vessels, bridge decks and post-tensioned concrete floors damaging rebar may compromise structural integrity and result in considerable rework. Although negative impressions for embeds can be made by placing various objects such as wooden dowels or steel rods into the rebar cage prior to pouring the concrete (and removing them once the concrete has partially or fully set), this practice is labor intensive and time consuming. A method of mapping the locations of the rebar free spaces before pouring and controlling the drilling process in real-time could have significant benefits. This paper presents research that investigated and implemented conceptual solutions for processing and incorporating point cloud data obtained from various 3D-imaging technologies into the drilling process. The 3D imaging technologies were used to map the locations of rebar within a replica of a section of a railway bridge deck. Once the point clouds were processed, zones that are safe for drilling are automatically detected and saved as a Building Information Model (BIM) that is then used to provide real-time feedback to the drill operator about whether it is safe to continue drilling based on the position and orientation of the drill. A conceptual method for providing visual feedback about the rebar-free zones to the drill operator using a laser projector was also developed. Finally, a visualization method for comparing the data obtained from the various 3D imaging technologies using the BIM is discussed.

Crisis-induced intermittency and Melnikov scale factor

Abstract We study the post-critical behavior of a perturbed bistable Hamiltonian system to which the Melnikov approach is applicable under the assumption that the perturbation is asymptotically small. We examine the case of perturbations that are sufficiently large to cause chaotic transport between phase space regions associated with the systems potential wells. The main results are: (1) a small additional harmonic excitation can cause substantial changes in the systems mean residence time, and (2) the dependence of the magnitude of these changes on the additional excitations frequency is similar to the dependence on frequency of the systems Melnikov scale factor. We discuss the relevance of these results to the design of efficient, Melnikov-based open loop controls aimed at increasing the mean residence time for the stochastically excited counterpart of the system.

Performance evaluation of consumer-grade 3D sensors for static 6DOF pose estimation systems

Low-cost 3D depth and range sensors are steadily becoming more widely available and affordable, and thus popular for robotics enthusiasts. As basic research tools, however, their accuracy and performance are relatively unknown. In this paper, we describe a framework for performance evaluation and measurement error analysis for 6 degrees of freedom pose estimation systems using traceable ground truth instruments. Characterizing sensor drift and variance, and quantifying range, spatial and angular accuracy, our framework focuses on artifact surface fitting and static pose analysis, reporting testing and environmental conditions in compliance with the upcoming ASTM E57.02 standard.

Variances of Cylinder Parameters Fitted to Range Data.

Industrial pipelines are frequently scanned with 3D imaging systems (e.g., LADAR) and cylinders are fitted to the collected data. Then, the fitted as-built model is compared with the as-designed model. Meaningful comparison between the two models requires estimates of uncertainties of fitted model parameters. In this paper, the formulas for variances of cylinder parameters fitted with Nonlinear Least Squares to a point cloud acquired from one scanning position are derived. Two different error functions used in minimization are discussed: the orthogonal and the directional function. Derived formulas explain how some uncertainty components are propagated from measured ranges to fitted cylinder parameters.

Target Penetration of Laser-Based 3D Imaging Systems

The ASTM E57.02 Test Methods Subcommittee is developing a test method to evaluate the ranging performance of a 3D imaging system. The test method will involve either measuring the distance between two targets or between an instrument and a target. The first option is necessary because some instruments cannot be centered over a point and will require registration of the instrument coordinate frame into the target coordinate frame. The disadvantage of this option is that registration will introduce an additional error into the measurements. The advantage of this option is that this type of measurement, relative measurement, is what is typically used in field applications. A potential target geometry suggested for the test method is a planar target. The ideal target material would be diffuse, have uniform reflectivity for wavelengths between 500 nm to 1600 nm (wavelengths of most commercially-available 3D imaging systems), and have minimal or no penetration of the laser into the material. A possible candidate material for the target is Spectralon1. However, several users have found that there is some penetration into the Spectralon by a laser and this is confirmed by the material manufacturer. The effect of this penetration on the range measurement is unknown. This paper will present an attempt to quantify the laser penetration depth into the Spectralon material for four 3D imaging systems.

Registration of Six Degrees of Freedom Data with Proper Handling of Positional and Rotational Noise.

When two six degrees of freedom (6DOF) datasets are registered, a transformation is sought that minimizes the misalignment between the two datasets. Commonly, the measure of misalignment is the sum of the positional and rotational components. This measure has a dimensional mismatch between the positional component (unbounded and having length units) and the rotational component (bounded and dimensionless). The mismatch can be formally corrected by dividing the positional component by some scale factor with units of length. However, the scale factor is set arbitrarily and, depending on its value, more or less importance is associated with the positional component relative to the rotational component. This may result in a poorer registration. In this paper, a new method is introduced that uses the same form of bounded, dimensionless measure of misalignment for both components. Numerical simulations with a wide range of variances of positional and rotational noise show that the transformation obtained by this method is very close to ground truth. Additionally, knowledge of the contribution of noise to the misalignment from individual components enables the formulation of a rational method to handle noise in 6DOF data.