Curvature Constraints Research Articles

A Path Planning Algorithm Based on the CNN Model under the Curvature Constraint

This paper proposes a new path planning algorithm based on the CNN model. The path planning problem is completed with the dynamics of CNN by establishing a relationship between path control points and CNN cells. Based on the analysis of one dimensional space of CNN algorithm, a CNN equation is constructed and the path updating algorithm under the curvature constraint is obtained, then the stability of the algorithm is discussed. Path planning simulation based on two-dimensional space shows that this algorithm can avoid re-planning or falling into the local minimum, which means it can be successfully used in the path planning and maintenance of robots on the ground in dynamic environment.

Stereo, Shading, and Surfaces: Curvature Constraints Couple Neural Computations

Vision problems are inherently ambiguous: Do abrupt brightness changes correspond to object boundaries? Are smooth intensity changes due to shading or material properties? For stereo: Which point in the left image corresponds to which point in the right one? What is the role of color in visual information processing? To answer these (seemingly different) questions we develop an analogy between the role of orientation in organizing visual cortex and tangents in differential geometry. Machine learning experiments suggest using geometry as a surrogate for high-order statistical interactions. The cortical columnar architecture becomes a bundle structure in geometry. Connection forms within these bundles suggest answers to the above questions, and curvatures emerge in key roles. More generally, our path through these questions suggests an overall strategy for solving the inverse problems of vision: decompose the global problems into networks of smaller ones and then seek constraints from these coupled problems to reduce ambiguity. Neural computations thus amount to satisfying constraints rather than seeking uniform approximations. Even when no global formulation exists one may be able to find localized structures on which ambiguity is minimal; these can then anchor an overall approximation.

3D reconstruction of Chinese hickory tree for dynamics analysis

Vibratory harvest is the most appropriate and efficient way to harvest Chinese hickory (Carya cathayensis Sarg.) nuts because of the hilly growth environment and scattered shooting architecture. To develop a mechanical harvester that can be adapted to different Chinese hickory trees, a three-dimensional (3D) model of the tree should be obtained and its dynamics parameters identified. This paper introduces an approach to reconstruct a 3D Chinese hickory tree without foliage using two images with different view angles, and then illustrates a dynamics analysis for the reconstructed tree. Two-dimensional (2D) branches were roughly segmented from the image by the “stripe programming” method, and refined by the ribbon snake model. After extracting the skeleton of 2D branches, a binary-directed tree representing 2D tree structure was established. Under the topology and epipolar constraints, the branch correspondences between two images could be easily found. To improve the accuracy, curve reconstruction under the minimum curvature constraint instead of point-based reconstruction was employed. The diameters of branches from the base to the tip were parameterised as a linear equation. Using the 3D model of the Chinese hickory tree, its dynamics analysis was solved by finite element software (ANSYS Workbench). Compared with the results of the impact tests, a predicted optimal vibration frequency of trunk shaker can be found eventually. It indicates that dynamics analysis using 3D model of tree reconstructed by the proposed method is effective.

The roles of image decomposition and edge curvature in the ‘snake’ lightness illusion

The snake illusion is an effect in which the lightness of target patches is strongly affected by the luminance of remote patches. One explanation is that such images are decomposed into a pattern of illumination and a pattern of reflectance, involving a classification of luminance edges into illumination and reflectance edges. Based on this decomposition, perceived reflectance is determined by discounting the illumination. A problem for this account is that image decomposition is not unique, and that different decompositions may lead to different lightness predictions. One way to rule out alternative decompositions and ensure correct predictions is to postulate that the visual system tends to classify curved luminance edges as reflectance edges rather than illumination edges. We have constructed several variations of the basic snake display in order to test the proposed curvature constraint and the more general image decomposition hypothesis. Although the results from some displays have confirmed previous findings of the effect of curvature, the general pattern of data questions the relevance of the shape of luminance edges for the determination of lightness in this class of displays. The data also argue against an image decomposition mechanism as an explanation of this effect. As an alternative, a tentative neurally based account is sketched.

QMCF: QMorph Cross Field-driven Quad-dominant Meshing Algorithm

We propose a new method for generation of quadrilateral meshes. Our algorithm takes a triangle mesh generated on a computer-aided design surface together with a size field as input. A novel cross field, defined on vertices of the triangle mesh, is derived from both curvature directions and constraints on the boundary. Our algorithm uses a modified QMorph algorithm for quad dominant meshing, to follow size field and to align edges with the cross field previously computed.

Spline-Based RRT Path Planner for Non-Holonomic Robots

Planning in a cluttered environment under differential constraints is a difficult problem because the planner must satisfy the external constraints that arise from obstacles in the environment and the internal constraints due to the kinematic/dynamic limitations of the robot. This paper proposes a novel Spline-based Rapidly-exploring Random Tree (SRRT) algorithm which treats both the external and internal constraints simultaneously and efficiently. The computationally expensive numerical integration of the system dynamics is replaced by an efficient spline curve parameterization. In addition, the SRRT guarantees continuity of curvature along the path satisfying any upper-bounded curvature constraints. This paper presents the underlying theory to the SRRT algorithm and presents simulation and experiment results of a mobile robot efficiently navigating through cluttered environments.

Holonomy spin foam models: asymptotic geometry of the partition function

Abstract We study the asymptotic geometry of the spin foam partition function for a large class of models, including the models of Barrett and Crane, Engle, Pereira, Rovelli and Livine, and, Freidel and Krasnov. The asymptotics is taken with respect to the boundary spins only, no assumption of large spins is made in the interior. We give a sufficient criterion for the existence of the partition function. We find that geometric boundary data is suppressed unless its interior continuation satisfies certain accidental curvature constraints. This means in particular that most Regge manifolds are suppressed in the asymptotic regime. We discuss this explicitly for the case of the configurations arising in the 3-3 Pachner move. We identify the origin of these accidental curvature constraints as an incorrect twisting of the face amplitude upon introduction of the Immirzi parameter and propose a way to resolve this problem, albeit at the price of losing the connection to the SU(2) boundary Hilbert space. The key methodological innovation that enables these results is the introduction of the notion of wave front sets, and the adaptation of tools for their study from micro local analysis to the case of spin foam partition functions.

Eigenvalue inequalities for the p-Laplacian on a Riemannian manifold and estimates for the heat kernel

In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirichlet p-Laplacian (1<p<∞) obtained by Matei (2000) [19] and Takeuchi (1998) [22], respectively. Moreover, we use this generalized eigenvalue comparison theorem to get estimates for the first eigenvalue of the Dirichlet p-Laplacian of geodesic balls on complete Riemannian manifolds with radial Ricci curvature bounded from below w.r.t. some point. In the rest of this paper, we derive an upper and lower bound for the heat kernel of geodesic balls of complete manifolds with specified curvature constraints, which can supply new ways to prove the most part of two generalized eigenvalue comparison results given by Freitas, Mao and Salavessa (2013) [9].

Component-wise method of smoothing the curvature of spatial paths constructed by noisy measurements in robot motion planning problems

A motion planning problem for a robot is considered. A robot follows a path in the manual mode and path coordinates are measured by a GNSS receiver. Repeating this path in the automatic mode requires constructing a geometric path that satisfies certain smoothness conditions and curvature constraints. This paper suggests the component-wise method for improving the curvature of a 3D path consisting of uniform cubic B-splines. Finally, the results of applying the proposed method to real paths of a motor vehicle and helicopter measured by a GNSS receiver are provided.

Continuous curvature path-smoothing algorithm using cubic Bzier spiral curves for non-holonomic robots

This paper presents a path-smoothing algorithm over the piecewise linear path for non-holonomic robots. Based on the upper-bounded continuous curvature path-smoothing algorithm, three algorithms are proposed to enhance the path smoothing performance. First, an interactive algorithm, which fully utilizes extra distance margins of linear path, is suggested. Second, a bisection algorithm is proposed to relieve the violation of the maximum curvature constraints. Finally, an interpolating path-smoothing algorithm which passes intermediate points is suggested. Simulation results show the validity of the suggested algorithms.

Optimal Trajectories of Curvature Constrained Motion in the Hamilton–Jacobi Formulation

We propose a PDE approach for computing time-optimal trajectories of a vehicle which travels under certain curvature constraints. We derive a class of Hamilton---Jacobi equations which models such motions; it unifies two well-known vehicular models, the Dubins' and Reeds---Shepp's cars, and gives further generalizations. Numerical methods (finite difference for the Reeds---Shepp's car and semi-Lagrangian for the Dubins' car) are investigated for two-dimensional domains and surfaces.

Improved Snake Model Algorithm with Application in Liver Image Segmentation

Snake model is one of classical active contour models, and is widely used in the image segmentation. But it has defects of sensitive to the initial contour and lack of curvature constraints. This paper presents an improved Snake model algorithm, which initializes contour by maker control watershed algorithm and adds an external force related with the curve shape into the energy function of Snake model to solve the problem about curvature constraints. Testing on the Magnetic Resonance (MR) images of the abdomen for liver image segmentation proves that the new algorithm can achieve good results.

The Reduced Ostrovsky Equation: Integrability and Breaking

The reduced Ostrovsky equation is a modification of the Korteweg‐de Vries equation, in which the usual linear dispersive term with a third‐order derivative is replaced by a linear nonlocal integral term, which represents the effect of background rotation. This equation is integrable provided a certain curvature constraint is satisfied. We demonstrate, through theoretical analysis and numerical simulations, that when this curvature constraint is not satisfied at the initial time, then wave breaking inevitably occurs.

Reconstruction of missing data in remote sensing images using conditional stochastic optimization with global geometric constraints

This paper addresses the issue of missing data reconstruction for partially sampled, two-dimensional, rectangular grid images of differentiable random fields. We introduce a stochastic gradient–curvature (GC) reconstruction method, which is based on the concept of a random field model defined by means of local interactions (constraints). The GC reconstruction method aims to match the gradient and curvature constraints for the entire grid with those of the sample using conditional Monte Carlo simulations that honor the sample values. The GC reconstruction method does not assume a parametric form for the underlying probability distribution of the data. It is also computationally efficient and requires minimal user input, properties that make it suitable for automated processing of large data sets (e.g. remotely sensed images). The GC reconstruction performance is compared with established classification and interpolation methods for both synthetic and real world data. The impact of various factors such as domain size, degree of thinning, discretization, initialization, correlation properties, and noise on GC reconstruction performance are investigated by means of simulated random field realizations. An assessment of GC reconstruction performance on real data is conducted by removing randomly selected and contiguous groups of points from satellite rainfall data and an image of the lunar surface.

Associative submanifolds of the 7-sphere

Associative submanifolds of the 7-sphere S^7 are 3-dimensional minimal submanifolds which are the links of calibrated 4-dimensional cones in R^8 called Cayley cones. Examples of associative 3-folds are thus given by the links of complex and special Lagrangian cones in C^4, as well as Lagrangian submanifolds of the nearly K\"ahler 6-sphere. By classifying the associative group orbits, we exhibit the first known explicit example of an associative 3-fold in S^7 which does not arise from other geometries. We then study associative 3-folds satisfying the curvature constraint known as Chen's equality, which is equivalent to a natural pointwise condition on the second fundamental form, and describe them using a new family of pseudoholomorphic curves in the Grassmannian of 2-planes in R^8 and isotropic minimal surfaces in S^6. We also prove that associative 3-folds which are ruled by geodesic circles, like minimal surfaces in space forms, admit families of local isometric deformations. Finally, we construct associative 3-folds satisfying Chen's equality which have an S^1-family of global isometric deformations using harmonic 2-spheres in S^6.

リニアな曲率・捩率を持つセグメントによる軌道生成

In two-dimensional(2D) space, the clothoid is a preferred trajectory curve because its curvature varies linearly with its curve length. However, in three-dimensional(3D) space, both curvature and torsion must be considered. This paper deals with path generation using linear curvature and torsion segments which can be considered a 3D extension of the 2D clothoid. In our study, the path segments are generated by solving the Frenet-Serret equation. In every path segment, its curvature and torsion varies linearly with its curve length. In order to obtain more free parameters, plural curve segments are connected in series to make a compound curve. The curve is used to connect two given points which may have given Frenet-Frame, curvature and torsion constraints. These curves are also used to construct a smooth transition passing through an arbitrary point sequence. The resultant path possesses C2 as well as torsion continuity and matches all given Frenet-frame, curvature and torsion constraints at the given points.

Optimal curvature-constrained paths for general directional-cost functions

This paper studies the problem of finding the minimum cost curvature-constrained path between two directed points where the cost at every point along the path depends on the instantaneous direction. This generalises the results obtained by Dubins for curvature-constrained paths of minimum length, commonly referred to as Dubins paths. We show that there always exists a path of the form \(\mathcal {C}\mathcal {S}\mathcal {C}\mathcal {S}\mathcal {C}\) or a degeneracy which is optimal, where \(\mathcal {C}\) represents an arc of maximum curvature, and \(\mathcal {S}\) represents a straight line. This result is also extended to the case where there is not only a directional-cost, but the cost of curved sections are scaled up by a factor w C≥1. The results obtained can be applied to optimising the development of underground mine networks, where the paths need to satisfy a curvature constraint, the cost of development of the tunnel depends on the direction due to the geological characteristics of the ground, and curved sections may incur more cost due to additional support and ventilation.

Proper biharmonic submanifolds in a sphere

In this paper, we obtain a constraint of the mean curvature for proper biharmonic submanifolds in a sphere. We give some characterizations of some proper biharmonic submanifolds with parallel mean curvature vector in a sphere. We also construct some new examples of proper biharmonic submanifolds in a sphere.

Generating realistic laminate fiber angle distributions for optimal variable stiffness laminates

The advent of advanced fiber placement technology has made it possible, through the use of fiber steering, to exploit the anisotropic properties of composite materials to a larger extent than was previously possible. Spatial variation of stiffness can be induced by steering composite fibers in curvilinear paths to give beneficial load and stiffness distribution patterns. Buckling of composite panels is one area where fiber steering has been proven to be very effective. Fiber angles and predefined fiber angle variations are used in most of the research on fiber steered composites reported in the literature, however, from an optimization point of view it is attractive to design such variable stiffness (VS) structures in terms of lamination parameters (LPs). This results in a two-step design approach. In the first step a VS composite is designed in terms of LPs, and in the second step the LPs are converted into fiber angle distributions for each layer in the laminate. A methodology is proposed to convert a known LP distribution for a VS composite laminate into a realistic design in terms of fiber angles, with minimum loss of structural performance, whilst satisfying a constraint on in-plane fiber angle curvature. The proposed conversion process is formulated as an optimization problem and can be used for any number of equi-thickness plies. The methodology was tested by converting a known optimal LP design for a sample structure, a square plate under bi-axial compression into a fiber angle design. The effect of the in-plane curvature constraint, the number of layers in the laminate, and the choice of objective function for the conversion process were studied for a balanced symmetric lay-up.

A Path Planning Method for Robotic Belt Surface Grinding

The flexible contact and machining with wide strip are two prominent advantages for the robotic belt grinding system, which can be widely used to improve the surface quality and machining efficiency while finishing the workpieces with sculptured surfaces. There lacks research on grinding path planning with the constraint of curvature. With complicated contact between the contact wheel and the workpiece, the grinding paths for robot can be obtained by the theory of contact kinematics. The grinding process must satisfy the universal demands of the belt grinding technologies, and the most important thing is to make the contact wheel conform to the local geometrical features on the contact area. For the local surfaces with small curvature, the curve length between the neighboring cutting locations becomes longer to ensure processing efficiency. Otherwise, for the local areas with large curvature, the curve length becomes shorter to ensure machining accuracy. A series of planes are created to intersect with the target surface to be ground, and the corresponding sectional profile curves are obtained. For each curve, the curve length between the neighboring cutting points is optimized by inserting a cutter location at the local area with large curvatures. A method of generating the grinding paths including curve length spacing optimization is set up. The validity is completely approved by the off-line simulation, and during the grinding experiments with the method, the quality of surface is improved. The path planning method provides a theoretical support for the smooth and accuracy path of robotic surface grinding.