Construction of periodic adapted orthonormal frames on closed space curves

The construction of continuous adapted orthonormal frames along C1 closed–loop spatial curves is addressed. Such frames are important in the design of periodic spatial rigid–body motions along smooth closed paths. The construction is illustrated through the simplest non–trivial context — namely, C1 closed loops defined by a single Pythagorean–hodograph (PH) quintic space curve of a prescribed total arc length. It is shown that such curves comprise a two–parameter family, dependent on two angular variables, and they degenerate to planar curves when these parameters differ by an integer multiple of π. The desired frame is constructed through a rotation applied to the normal–plane vectors of the Euler–Rodrigues frame, so as to interpolate a given initial/final frame orientation. A general solution for periodic adapted frames of minimal twist on C1 closed–loop PH curves is possible, although this incurs transcendental terms. However, the C1 closed–loop PH quintics admit particularly simple rational periodic adapted frames

On the formulation and uses of SVD-based generalized curvatures

2016 Summer.Includes bibliographical references.In this dissertation we consider the problem of computing generalized curvature values from noisy, discrete data and applications of the provided algorithms. We first establish a connection between the Frenet-Serret Frame, typically defined on an analytical curve, and the vectors from the local Singular Value Decomposition (SVD) of a discretized time-series. Next, we expand upon this connection to relate generalized curvature values, or curvatures, to a scaled ratio of singular values. Initially, the local singular value decomposition is centered on a point of the discretized time-series. This provides for an efficient computation of curvatures when the underlying curve is known. However, when the structure of the curve is not known, for example, when noise is present in the tabulated data, we propose two modifications. The first modification computes the local singular value decomposition on the mean-centered data of a windowed selection of the time-series. We observe that the mean-center version increases the stability of the curvature estimations in the presence of signal noise. The second modification is an adaptive method for selecting the size of the window, or local ball, to use for the singular value decomposition. This allows us to use a large window size when curvatures are small, which reduces the effects of noise thanks to the use of a large number of points in the SVD, and to use a small window size when curvatures are large, thereby best capturing the local curvature. Overall we observe that adapting the window size to the data, enhances the estimates of generalized curvatures. The combination of these two modifications produces a tool for computing generalized curvatures with reasonable precision and accuracy. Finally, we compare our algorithm, with and without modifications, to existing numerical curvature techniques on different types of data such as that from the Microsoft Kinect 2 sensor. To address the topic of action segmentation and recognition, a popular topic within the field of computer vision, we created a new dataset from this sensor showcasing a pose space skeletonized representation of individuals performing continuous human actions as defined by the MSRC-12 challenge. When this data is optimally projected onto a low-dimensional space, we observed each human motion lies on a distinguished line, plane, hyperplane, etc. During transitions between motions, either the dimension of the optimal subspace significantly, or the trajectory of the curve through pose space nearly reverses. We use our methods of computing generalized curvature values to identify these locations, categorized as either high curvatures or changing curvatures. The geometric characterization of the time-series allows us to segment individual,or geometrically distinct, motions. Finally, using these segments, we construct a methodology for selecting motions to conjoin for the task of action classification

Effects of parameterization and knot placement techniques on primal and mixed isogeometric collocation formulations of spatial shear-deformable beams with varying curvature and torsion

We present a displacement-based and a mixed isogeometric collocation (IGA-C) formulation for free-form, three-dimensional, shear-deformable beams with high and rapidly-varying curvature and torsion. When such complex shapes are concerned, the approach used to build the IGA geometric model becomes relevant. Although IGA-C has been so far successfully applied to a wide range of problems, the effects that different parameterization and knot placement techniques may have on the accuracy of collocation-based formulations is still an unexplored field. To fill this gap, primal and mixed formulations are used combining two parameterization methods (chord-length and equally spaced) with two knot placement techniques (uniformly spaced and De Boor). With respect to the space-varying Frenet local frame, we derive the strong form of the governing equations in a compact form through the definition of two matrix operators conveniently used to perform first and second order derivatives of the vector fields involved in the formulations. This approach is very efficient and easy to implement within a collocation-based scheme. Several challenging numerical experiments allow to test the different considered parameterizations and knot placement techniques, revealing in particular that with the primal formulation an equally spaced parameterization is definitively the most recommended choice and it should always be used with an approximation degree of, at least, , although some caution must be adopted when very high Jacobians and small curvatures occur. The same holds for the mixed formulation, with the difference that is enough to yield accurate results

Modeling and Experimental Validation of a Compliant Underactuated Parallel Kinematic Manipulator

© 1996-2012 IEEE. Parallel kinematic manipulators (PKMs) are increasingly used in a wide range of industrial applications due to the characteristics of high accuracy and compact structure. However, most of the existing PKMs are structured with heavy actuators and high stiffness. In this respect, this article proses a simple, yet effective, parallel manipulator that distinguishes itself through the following basis. First, underactuation: it employs only a single motor and a driving cable to actuate its three legs. Second, novel foot location: it uses a smart shape memory alloy clutch-based driving system (SCBDS), which catches/releases the driving cable, thus, making possible the robot underactuation. Finally, adjustable compliance: its double compliant joints on each limb with a stiffness-adjustable section, which renders a safe human-robotic interaction. To support and predict the performance of this underactuated compliant manipulator, a novel kinetostatic model was developed by considering the generalized internal loads (i.e., force and moment) in three compliant limbs and the external loads on the upper platform. Finally, based on the physical prototype, a set of experiments were conducted to validate the model proposed in this article. It was found that the proposed kinetostatic model can be validated with the average deviations of 1.8% in position and 2.8% in orientation, respectively. Furthermore, the workspace of the system (e.g., discrete and continuous workspace) was studied when different actuating strategies were employed, thus, emphasizing the advantages and the limitations of this novel system

Study of the relation between blood flow and the age-dependent localisation of early atherosclerosis

Atherosclerosis develops non-uniformly within the arterial system and the distribution of lesions has been observed to change with age. This thesis investigates the concept that the patchiness of the disease is related to local variations in blood flow. Based on the insights from a systematic literature review, a novel study was designed to analyse the relation between haemodynamic factors and age-dependent atherogenesis in the thoracic aorta of rabbits. Arterial geometries were reconstructed by micro-Computed Tomography of vascular corrosion casts, with particular attention to the anatomical accuracy of the dataset. Blood flow was simulated in these geometries using a spectral/hp element method. Distributions of traditional shear-related metrics were calculated and both qualitatively and quantitatively compared to maps of lesion prevalence. In addition, a time-averaged transverse wall shear stress was introduced. A geometric analysis of the dataset of rabbit thoracic aortas revealed a significant change with age in the degree of aortic taper. The geometric changes could explain age-related differences in flow characteristics, in particular in the extent of Dean-type vortical structures into the descending aorta and the strength of a dorsal streak of high shear. The comparative analysis of shear and lesion distributions did not unequivocally support the theory that lesions occur in regions of low shear. The novel haemodynamic metric, in combination with current metrics, enabled an improved identification of zones of multi-directional disturbed flow. In conclusion, this thesis adds to the understanding of the relation between blood flow and early atherosclerosis, and provides tools for use in future studies