Computational modeling of impact and deformation

This thesis tackles several problems arising in robotics and mechanics: analysis and computation of two- and muti-body impacts, planning a contact velocity for robotic batting, impact of an elastic rod onto a fixed foundation, robotic pickup of soft three-dimensional objects, and recovery of their gravity-free shapes. Impact is an event that lasts a very short period of time but generates a very large interaction force. Assuming Stronge’s energy-based restitution, a formal impulse-based analysis is presented for the collision of two rigid bodies at single contact point under Coulomb friction in three dimensions (3D). Based on this analysis, we describe a complete algorithm to take advantage of fast numerical integration and closed-form evaluation. For a simultaneous collision involving more than two bodies, we describe a general computational model for predicting its outcome. Based on the impact model, we then look into the task of planning an initial contact velocity between a bat and an in-flight object to send the latter to a target. In certain situations, a closed-form solution can be found, while in others, a bounding triangle algorithm of iterative nature can be employed. An alternative way of modeling impact is to consider the engaged objects to be elastic rather than rigid. A damped one-dimensional wave equation can model an elastic rod bouncing off the ground at a given initial velocity, under the influence of gravity. We derive an explicit solution based on the Method of Descent and D’Alembert’s formula. We also obtain formulas for the time of contact and analyze the dependence of the energetic coefficient of restitution on the physical constants. I conclude the thesis with two pieces of work involving deformable objects. First, an algorithm for picking up a 3D object is introduced. Homotopy continuation method is applied to solve a non-linear system for slips between objects and fingers. Some simulation and experimental results are compared. Second, I discuss an iterative fixed-point method for recovering the gravity-free shape of an object. An experiment shows that the resulting stiffness matrix gives better predictions on deformations than the conventional stiffness matrix influenced by gravity

Image-based goal-oriented adaptive isogeometric analysis with application to the micro-mechanical modeling of trabecular bone

Isogeometric analysis (IGA) of geometrically complex three-dimensional objects is possible when used in combination with the Finite Cell method (FCM). In this contribution we propose a computational methodology to automatically analyze the effective elastic properties of scan-based volumetric objects of arbitrary geometric and topological complexity. The first step is the reconstruction of a smooth geometry from scan-based voxel data using a B-spline level set function. The second step is a goal-oriented adaptive isogeometric linear elastic analysis. Elements are selected for refinement using dual-weighted residual shape function indicators, and hierarchical splines are employed to construct locally refined spline spaces. The proposed methodology is studied in detail for various numerical test cases, including the computation of the effective Young's modulus of a trabecular bone micro-structure reproduced from μCT-scan data

Morphological analysis of cells by means of an elastic metric in the shape space

Shape analysis is of great importance in many fields, such as computer vision, medical imaging, and computational biology. This analysis can be performed considering shapes as closed planar curves in the shape space. This approach has been used for the first time to obtain the morphological classification of erythrocytes in digital images of sickle cell disease considering the shape space S1, which has the property of being isometric to an infinite-dimensional Grassmann manifold of two-dimensional subspaces (Younes et al., 2008), without taking advantage of all the features offered by the elastic metric related to the possibility of stretching and bending of the curves. In this paper, we study this deformation in the shape space, S2, which is based on the representation of closed planar curves by means of the square-root velocity function (SRVF) (Srivastava et al., 2011), using the elastic metric of this space to obtain more efficient geodesics and geodesic lengths between planar curves. Supervised classification with this approach achieved an accuracy of 94.3%, classification using templates achieved 94.2% and unsupervised clustering in three groups achieved 94.7%, considering three classes of erythrocytes: normal, sickle, and with other deformations. These results are better than those previously achieved in the morphological analysis of erythrocytes and the method can be used in different applications related to the treatment of sickle cell disease, even in cases where it is necessary to study the process of evolution of the deformation, something that can not be done in a natural way in the feature space

Alignment Theory of Parallel-beam CT Image Reconstruction for Elastic-type Objects using Virtual Focusing Method

X-ray tomography has been studied in various fields. Although a great deal of effort has been directed at reconstructing the projection image set from a rigid-type specimen, little attention has been addressed to the reconstruction of projected images from an object showing elastic motion. Here, we present a mathematical solution to reconstruct the projection image set obtained from an object with specific elastic motions: periodically, regularly, and elliptically expanded or contracted specimens. To reconstruct the projection image set from expanded or contracted specimens, we introduce new methods; detection of sample's motion modes, mathematical re-scaling pixel values and converting projection angle for a common layerComment: 30 pages, 11 figure