B-splines Of Degree Research Articles

Optimal trajectory planning of robot energy consumption based on improved sparrow search algorithm

In order to reduce the energy consumption of the welding robot and ensure the cooperative movement of the robot joints, a trajectory planning method with optimal energy consumption based on improved sparrow search algorithm is proposed. Firstly, the trajectory planning model with optimal energy consumption is established based on the joint torque and angular velocity of the robot. To make the velocity, acceleration and jerk of each joint of the robot be bounded and continuous, the joint space trajectory is constructed with seventh degree B-spline curve. The total energy consumption of the robot is calculated by combining kinematic and dynamic parameters. On the basis of improved sparrow search algorithm, the time series corresponding to the optimal energy consumption is solved by using elite reverse learning, non-dominated sorting and Gaussian-Cauchy variation strategy, and then the optimal continuous motion trajectory of energy consumption is planned. The simulation results show that the proposed method can not only achieve continuous smooth control objective, but also effectively reduce energy consumption.

A rapid outliers detection and correction method for external ballistic velocity measurement data

Due to the influence of various sudden and abnormal factors during the tracking environment, equipment working conditions, and operation process, there are inevitable outliers in the tracking measurement process of aircraft such as carrier rockets, artificial satellites, and missiles. Therefore, the prerequisite for ensuring the reliability of processing results is to timely and accurately detect and correct outliers. In this paper, we proposed the sliding window-based variable degree B-spline function method (SWVD B-spline), which can handle isolated outliers and spotted outliers. SWVD B-spline uses variable degree B-spline function to model observed data in sliding windows, which can detect and correct outliers point by point along with window sliding. Then, we propose an initial window data selection method to remove outliers in initial windows to ensure the processing effect. In addition, because there are often inflection points in external ballistic velocity measurement data, differential evolution is used to optimize variable degree B-spline in windows that include inflection points to improve processing accuracy. The experimental results verify that SWVD B-spline can handle various outliers rapidly and efficiently.

Research on the End Effector and Optimal Motion Control Strategy for a Plug Seedling Transplanting Parallel Robot

Due to the phenomenon of holes and inferior seedlings in trays, it is necessary to remove and replenish unqualified seedlings. The traditional operation is labor-intensive, and the degree of mechanization is low. This paper took broccoli seedlings as the research object and developed an image recognition system suitable for seedling health recognition and pose judgement, researched and designed a plug-in end effector that reduces leaf damage, and conducted orthogonal tests to obtain a substrate parameter combination containing the moisture content, seedling age, and transplanting acceleration suitable for culling operations. A parallel robot kinematics and dynamics model was built. The fifth degree B-spline curve was used to construct the joint space motion curve for seven nodes, and the motor speed, torque, and end-effector acceleration were used to construct the joint space motion curves. The end-effector acceleration was the constraint condition to plan the optimal trajectory of the joint space in time, and the optimal time was obtained using the artificial fish swarm–particle swarm hybrid optimization algorithm. A single operation time was greatly reduced; the whole machine was systematically built; the average time of single-time seedling removal was measured; and the transplanting efficiency of the whole machine was high. In the seedling damage rate gap test, the leaf damage rate was low. This research provides a reference for the localized development of greenhouse high-speed and low-loss seedling removal equipment.

A shape preserving C2 non-linear, non-uniform, subdivision scheme with fourth-order accuracy

The objective of this study is to present a shape-preserving non-linear subdivision scheme generalizing the exponential B-spline of degree 3, which is a piecewise exponential polynomial with the same support as the cubic B-spline. The subdivision of the exponential B-spline has a crucial limitation in that it can reproduce at most two exponential polynomials, yielding the approximation order two . Also, finding a best-fitting shape parameter in the exponential B-spline is a challenging and important problem. In this regard, we present a method for selecting an optimal shape parameter and then formulate it in the construction of new refinement rules. As a result, the new scheme provides an improved approximation order four while maintaining the same C 2 smoothness as the (exponential) B-spline of degree 3. Moreover, we show that the proposed method preserves geometrically important characteristics such as monotonicity and convexity, under some suitable conditions. Some numerical examples are provided to demonstrate the ability of the new subdivision scheme.

Improving the Efficiency of Calculating B-splines in parallel Programming Tasks

The use of parallel computing tools can significantly reduce the execution time of calculations in many engineer­ing tasks. One of the main difficulties in the development of multithreaded programs remains the organization of simultaneous access from different threads to shared data. The most common solution to this problem is to use locking facilities when accessing shared data. There are a number of tasks where data sharing is not needed, but you need to synchronize access to a limited resource, such as a temporary buffer. In such tasks, there is no data exchange between different threads, but there is an object that at a given time can be used by the code of only one thread. One such task is calculating the value of a B-spline. The software implementation of the functions for calculating B-splines, performed according to classical algorithms, requires the use of blocking objects when accessing the common array of intermediate data from different threads. This reduces the degree of parallelism and reduces the efficiency of computational programs using B-splines running on multiprocessor computing systems. The article discusses a way to improve the efficiency of calculating B-splines in parallel programming tasks by eliminating locks when accessing general modified data. A soft­ware implementation is presented in the form of a C++ class template, which provides placement of a temporary array used for calculating a B-spline into a local buffer of a given size with the possibility of increasing it if necessary. Using the developed template in conjunction with the threadlocal qualifier reduces the number of requests for increasing the buffer for high degree B-splines (larger than the initially specified buffer size). It is also possible to implement this scheme using the std::vector template of the C++ STL Standard Library. The results of the application of the developed class when calculating the values of B-splines in a multithreaded environment, showing a reduction in the calculation time in proportion to an increase in the number of computational processors, are presented. The methods of specifying arrays for storing intermediate calculation results considered in this article can be used in other parallel programming tasks.

Solving the Modified Regularized Long Wave Equations via Higher Degree B-Spline Algorithm

The current article considers the sextic B-spline collocation methods (SBCM1 and SBCM2) to approximate the solution of the modified regularized long wave ( MRLW ) equation. In view of this, we will study the solitary wave motion and interaction of higher (two and three) solitary waves. Also, the modified Maxwellian initial condition into solitary waves is studied. Moreover, the stability analysis of the methods has been discussed, and these will be unconditionally stable. Moreover, we have calculated the numerical conserved laws and error norms L 2 and L ∞ to demonstrate the efficiency and accuracy of the method. The numerical examples are presented to illustrate the applications of the methods and to compare the computed results with the other methods. The results show that our proposed methods are more accurate than the other methods.

Numerical Solution of Fifth Order Boundary Value Problems by Collocation Method with Sixth Order B-Splines

Collocation method with sixth degree B-splines as basis functions has been developed to solve a fifth order special case boundary value problem. To get an accurate solution by the collocation method with sixth degree B-splines, the original sixth degree B-splines are redefined into a new set of basis functions which in number match with the number of collocation points. The method is tested for solving both linear and nonlinear boundary value problems. The proposed method is giving better results when compared with the methods available in literature.

Numerical Solution of the Seventh Order Boundary Value Problems Using B-Spline Method

We develop a numerical method for solving the boundary value problem of The Linear Seventh Ordinary Boundary Value Problem by using seventh degree B-Spline function. Formulation is based on particular terms of order of seventh order boundary value problem. We obtain Septic B-Spline formulation and the Collocation B-spline Method is formulated as an approximation solution. We apply the presented method to solve an example of seventh-order boundary value problem which the results show that there is an agreement between approximate solutions and exact solutions. Resulting low absolute errors show that the presented numerical method is effective for solving high order boundary value problems. Finally, a general conclusion has been included.

A quartic trigonometric tension b-spline algorithm for nonlinear partial differential equation system

Purpose The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation. Design/methodology/approach The finite elements method (FEM) is a numerical method for obtaining an approximate solution of partial differential equations (PDEs). The development of high-speed computers enables to development FEM to solve PDEs on both complex domain and complicated boundary conditions. It also provides higher-order approximation which consists of a vector of coefficients multiplied by a set of basis functions. FEM with the B-splines is efficient due both to giving a smaller system of algebraic equations that has lower computational complexity and providing higher-order continuous approximation depending on using the B-splines of high degree. Findings The result of the test problems indicates the reliability of the method to get solutions to the CBE. QTT B-spline collocation approach has convergence order 3 in space and order 1 in time. So that nonpolynomial splines provide smooth solutions during the run of the program. Originality/value There are few numerical methods build-up using the trigonometric tension spline for solving differential equations. The tension B-spline collocation method is used for finding the solution of Coupled Burgers’ equation.

Matrix representations for multi-degree B-splines

The paper is concerned with computing the B-spline basis of a multi-degree spline space, namely a space of piecewise functions comprised of polynomial segments of different degrees. To this aim, we provide a general method to work out a matrix representation relating the sought basis with another one easier to compute. This will allow us, for example, to calculate a multi degree B-spline basis starting from local Bernstein bases of different degrees or from the B-spline basis of a spline space where all sections have the same degree. This change of basis can be translated into a conceptually simple and computationally efficient algorithm for the evaluation of multi-degree B-splines.

Spectral analysis of finite-dimensional approximations of 1d waves in non-uniform grids

We study the gap of discrete spectra of the Laplace operator in 1d for non-uniform meshes by analyzing the corresponding spectral symbol, which allows to show how to design the discretization grid for improving the gap behavior. The main tool is the study of a univariate monotonic version of the spectral symbol, obtained by employing a proper rearrangement via the GLT theory. We treat in detail the case of basic finite-difference approximations. In a second step, we pass to precise approximation schemes, coming from the celebrated Galerkin isogeometric analysis based on B-splines of degree p and global regularity $$C^{p-1}$$ , and finally we address the case of finite-elements with global regularity $$C^0$$ and local polynomial degree p. The surprising result is that the GLT approach allows a unified spectral treatment of the various schemes also in terms of the preservation of the average gap property, which is necessary for the uniform gap property. The analytical results are illustrated by a number of numerical experiments. We conclude by discussing some open problems.

Genetic Algorithm for Mobile Robot Route Planning with Obstacle Avoidance

Abstract Nowadays many public and private institutions begin space studies projects. Among many problems to solve there is a planet exploration. Now rovers are controlled directly from the Earth, e.g. Opportunity. Missions must be planned on the Earth using simulators. Much better will be when the mission planner could set the target area and work to do and the rover will perform it independently. The solution is to make it autonomous. Without need of external path planning the rover can cover a much longer distance. To make autonomous rovers real it is necessary to implement a target leaded obstacle avoidance algorithm. Solutions based on graph algorithms use a lot of computing power. The others use intelligent methods such as neural networks or fuzzy logic but their efficiency in a very complex environment is quite low. This work presents an obstacle avoidance algorithm which uses the genetic path finding algorithm. The actual version is based on the 2D map which is built by the robot and the 2nd degree B-spline is used for the path model. The performance in the most cases is high using only one processor thread. The GA can be also easily multithreaded. Another feature of the algorithm is that, due to the GA random nature, the chosen path can differ each time on the same map. The paper shows the results of the simulation tests. The maps have the various complexity levels. On every map one hundred tests were carried out. The algorithm brought the robot to the target successfully in the majority of runs.

Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines

We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations in arbitrary space dimension [Formula: see text]. We employ hierarchical B-splines of arbitrary degree and different order of smoothness. We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (respectively, the sum of energy error plus data oscillations) with optimal algebraic rates. Numerical experiments underpin the theoretical findings.

Ninth Degree B-spline Collocation Method for Numerical Solution of the Ninth Order Linear Differential Equation

Ninth Degree B-spline Collocation Method for Numerical Solution of the Ninth Order Linear Differential Equation

Reconstruction of Low Degree B-spline Surfaces with Arbitrary Topology Using Inverse Subdivision Scheme

Multivariate B-spline surfaces over triangular parametric domain have many interesting properties in the construction of smooth free-form surfaces. This paper introduces a novel approach to reconstruct triangular B-splines from a set of data points using inverse subdivision scheme. Our proposed method consists of two major steps. First, a control polyhedron of the triangular B-spline surface is created by applying the inverse subdivision scheme on an initial triangular mesh. Second, all control points of this B-spline surface, as well as knotclouds of its parametric domain are iteratively adjusted locally by a simple geometric fitting algorithm to increase the accuracy of the obtained B-spline. The reconstructed B-spline having the low degree along with arbitrary topology is interpolative to most of the given data points after some fitting steps without solving any linear system. Some concrete experimental examples are also provided to demonstrate the effectiveness of the proposed method. Results show that this approach is simple, fast, flexible and can be successfully applied to a variety of surface shapes.

Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis

We develop a multi-degree polar spline framework with applications to both geometric modeling and isogeometric analysis. First, multi-degree splines are introduced as piecewise non-uniform rational B-splines (NURBS) of non-uniform or variable polynomial degree, and a simple algorithm for their construction is presented. Then, an extension to two-dimensional polar configurations is provided by means of a tensor-product construction with a collapsed edge. Suitable combinations of these basis functions, encoded in a so-called isogeometric analysis suitable extraction operator, yield Ck smooth polar splines for any k≥0. We show that it is always possible to construct a set of smooth polar spline basis functions that form a convex partition of unity and possess locality. Explicit constructions for k∈{0,1,2} are presented. Optimal approximation behavior is observed numerically, and examples of applications to free-form design, smooth hole-filling, and high-order partial differential equations demonstrate the applicability of the developed framework.

Numerical approximation of GB-splines by a convolutional approach

Generalized splines are smooth functions belonging piecewisely to spaces which are a natural generalization of algebraic polynomials. GB-splines are a B-spline-like basis for generalized splines, and they are usually defined by means of an integral recurrence relation which makes their evaluation quite cumbersome and computationally expensive. We present a simple strategy for approximating the values of a cardinal GB-spline of arbitrary degree p, with a particular focus on hyperbolic and trigonometric GB-splines due to their interest in applications. The proposed strategy is based on the Fourier properties of cardinal GB-splines. The approximant is expressed as a linear combination of scaled and dilated versions of (polynomial) cardinal B-splines of degree p, whose coefficients can be efficiently computed via discrete convolution. Sharp error estimates are provided and illustrated with some numerical examples.

Hierarchical spline spaces: quasi-interpolants and local approximation estimates

A local approximation study is presented for hierarchical spline spaces. Such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement in any dimensionality. We provide approximation estimates for general hierarchical quasi-interpolants expressed in terms of the truncated hierarchical basis. Under some mild assumptions, we prove that such hierarchical quasi-interpolants and their derivatives possess optimal local approximation power in the general q-norm with 1≤q≤ź$1\leq q\leq \infty $. In addition, we detail a specific family of hierarchical quasi-interpolants defined on uniform hierarchical meshes in any dimensionality. The construction is based on cardinal B-splines of degree p and central factorial numbers of the first kind. It guarantees polynomial reproduction of degree p and it requires only function evaluations at grid points (odd p) or half-grid points (even p). This results in good approximation properties at a very low cost, and is illustrated with some numerical experiments.

Computation of Hermite interpolation in terms of B-spline basis using polar forms

The aim of this paper is to present an Hermite interpolation problem with B-splines of high degree of smoothness. More precisely, we use polar forms to find the B-spline control points for Hermite interpolation. The resulting formula is used to give Hermite basis functions. In particular, Quadratic C1 and cubic C2 interpolations with sharp parameters are analyzed.

Employment Higher Degree B-Spline Function for Solving Higher Order Differential Equations

As the B-spline method was developed for solving higher order differential equations, we present a brief survey to construct a higher degree B-spline. The new technique has been given in this field, accordingly a numerical illustration used to solve boundary value problems by employ quintic B-spline function. An example has been given for calculating maximum absolute error through n nodes.