B-spline Curve Research Articles

Trajectory optimization of B-splines interpolation based on dynamic error adjustment

In this paper, we propose a motion planning algorithm based on the five-fold B-spline curve for multi-axis motion control systems with non-smooth trajectories, acceleration, plus acceleration over the limit, etc. We construct a smooth transition model for the micro-line segment G3 of the continuous five-fold B-spline curve. A derivation of the corresponding theoretical equation is given. Using the transition model, we propose an optimization method for dynamic adjustment of motion trajectory errors based on the velocity forward planning constraints by smoothing the transition after the trajectory of the micro-line segment for the forward planning velocity. Finally, the design flow for the forward planning of the velocity is given. Finally, the above algorithm is used to perform simulation experiments for the optimization algorithm. Experimental results show that the proposed optimization improves the accuracy of fitting motion trajectories under the constraints of acceleration and deceleration addition, which validates the proposed algorithm.

Trajectory planning and simulation of upper limb rehabilitation robot based on NSGA2

In order to increase the stability and comfort of hemiplegic rehabilitation training, this paper integrates B-spline curves with a non-dominated sorting genetic algorithm (NSGA2) for trajectory planning of a 5-DOF upper limb exoskeleton rehabilitation robot, the structural characteristics of the robot are analyzed. Subsequently, the forward kinematics equation is derived by the D-H method, and its correctness is verified by Adams simulation. Subsequently, the robot workspace is calculated, and the outcomes show that it satisfies the patient’s demand for activity. Finally, the trajectory curve is constructed using the 5th-order B-splines. Under kinematic constraints, the NAGA2 is used to discover the ideal time interval sequence for time, impact and energy, and then complete the trajectory planning. The simulation findings demonstrate the suggested trajectory planning method produces continuous, smooth, and high-performance motion trajectories, providing good rehabilitation training for patients.

On Intersections of B-Spline Curves

Bézier and B-spline curves are foundational tools for curve representation in computer graphics and computer-aided geometric design, with their intersection computation presenting a fundamental challenge in geometric modeling. This study introduces an innovative algorithm that quickly and effectively resolves intersections between Bézier and B-spline curves. The number of intersections between the two input curves within a specified region is initially determined by applying the resultant of a polynomial system and Sturm’s theorem. Subsequently, the potential region of the intersection is established through the utilization of the pseudo-curvature-based subdivision scheme and the bounding box detection technique. The projected Gauss-Newton method is ultimately employed to efficiently converge to the intersection. The robustness and efficiency of the proposed algorithm are demonstrated through numerical experiments, demonstrating a speedup of 3 to 150 times over traditional methods.

Newton Geometric Iterative Method for B-Spline Curve and Surface Approximation

We introduce a progressive and iterative method for B-spline curve and surface approximation, incorporating parameter correction based on the Newton iterative method. While parameter corrections have been used in existing Geometric Approximation (GA) methods to enhance approximation quality, they suffer from low computational efficiency. Our approach unifies control point updates and parameter corrections in a progressive and iterative procedure, employing a one-step strategy for parameter correction. We provide a theoretical proof of convergence for the algorithm, demonstrating its superior computational efficiency compared to current GA methods. Furthermore, the provided convergence proof offers a methodology for proving the convergence of existing GA methods with location parameter correction.

Multi-objective optimal trajectory planning for robot manipulator attention to end-effector path limitation

Abstract In the process of trajectory optimization for robot manipulator, the path that is generated may deviate from the intended path because of the adjustment of trajectory parameters, if there is limitation of end-effector path in Cartesian space for specific tasks, this phenomenon is dangerous. This paper proposes a methodology that is based on the Pareto front to address this issue, and the methodology takes into account both the multi-objective optimization of robotic arm and the quality of end-effector path. Based on dung beetle optimizer, this research proposes improved non-dominated sorting dung beetle optimizer. This paper interpolates manipulator trajectory with quintic B-spline curves, achieves multi-objective trajectory optimization that simultaneously optimizes traveling time, energy consumption, and mean jerk, proposes a trajectory selection strategy that is based on Pareto solution set by introducing the concept of Fréchet distance, and the strategy enables the end-effector to approach the desired path in Cartesian space. Simulation and experimental results validate the effectiveness and practicability of the proposed methodology on the Sawyer robot manipulator.

B-spline curve approximation with transformer neural networks

Approximating a curve with a B-spline is a well-known problem with many challenges. Computing parametric values and knot vector that leads to the best approximation of a point sequence is an open problem. Existing methods are usually based on heuristics, genetic algorithms, or meta-heuristics. Nowadays, Deep Neural Networks have demonstrated their usefulness as shown in the use of a Multi-Layer Perceptron in the existing literature. Since its inception, the Transformer architecture has achieved state-of-the-art in multiple domains, like Natural Language Processing and Computer Vision. In this paper, we propose a method for knot placement that focuses on using a Transformer neural network architecture for B-spline approximation. We present and compare the results of our ongoing experimentations with Transformers for B-spline curve approximation. We conclude with possible improvements and modifications to our method for future experiments.

Adapting the Segment Anything Model for Plant Recognition and Automated Phenotypic Parameter Measurement

Traditional phenotyping relies on experts visually examining plants for physical traits like size, color, or disease presence. Measurements are taken manually using rulers, scales, or color charts, with all data recorded by hand. This labor-intensive and time-consuming process poses a significant obstacle to the efficient breeding of new cultivars. Recent innovations in computer vision and machine learning offer potential solutions for accelerating the development of robust and highly effective plant phenotyping. This study introduces an efficient plant recognition framework that leverages the power of the Segment Anything Model (SAM) guided by Explainable Contrastive Language–Image Pretraining (ECLIP). This approach can be applied to a variety of plant types, eliminating the need for labor-intensive manual phenotyping. To enhance the accuracy of plant phenotype measurements, a B-spline curve is incorporated during the plant component skeleton extraction process. The effectiveness of our approach is demonstrated through experimental results, which show that the proposed framework achieves a mean absolute error (MAE) of less than 0.05 for the majority of test samples. Remarkably, this performance is achieved without the need for model training or labeled data, highlighting the practicality and efficiency of the framework.

Designing gradient coils with the shape derivative and the closed B-spline curves

This study proposes a versatile and efficient optimisation method for discrete coils that induce a magnetic field by their steady currents. The prime target is gradient coils for MRI (Magnetic Resonance Imaging). The derivative (gradient) of the z-component the magnetic field, which is calculated by the Biot–Savart's law, with respect to the z-coordinate in the Cartesian xyz coordinate system is considered as the objective function. Then, the derivative of the objective function with respect to a change of coils in shape is formulated according to the concept of shape optimisation. The resulting shape derivative (as well as the Biot–Savart's law) is smoothly discretised with the closed B-spline curves. In this case, the control points (CPs) of the curves are naturally selected as the design variables. As a consequence, the shape derivative is discretised to the sensitivities of the objective function with respect to the CPs. Those sensitivities are available to solve the present shape-optimisation problem with a certain gradient-based nonlinear-programming solver. The numerical examples exhibit the mathematical reliability, computational efficiency, and engineering applicability of the proposed methodology based on the shape derivative/sensitivities and the closed B-spline curves.

Trajectory Planning and Singularity Avoidance Algorithm for Robotic Arm Obstacle Avoidance Based on an Improved Fast Marching Tree

The quest for efficient and safe trajectory planning in robotic manipulation poses significant challenges, particularly in complex obstacle environments where the risk of encountering singularities and obstacles is high. Addressing this critical issue, our study presents a novel enhancement of the Fast Marching Tree (FMT) algorithm, ingeniously designed to navigate the complex terrain of Cartesian space with an unprecedented level of finesse. At the heart of our approach lies a sophisticated two-stage path point sampling strategy, ingeniously coupled with a singularity avoidance mechanism that leverages geometric perception to assess and mitigate the risk of encountering problematic configurations. This innovative method not only facilitates seamless obstacle navigation but also adeptly circumvents the perilous zones of singularity, ensuring a smooth and uninterrupted path for the robotic arm. To further refine the trajectory, we incorporate a quasi-uniform cubic B-spline curve, optimizing the path for both efficiency and smoothness. Our comprehensive simulation experiments underscore the superiority of our algorithm, showcasing its ability to consistently achieve shorter, more efficient paths while steadfastly avoiding obstacles and singularities. The practical applicability of our method is further corroborated through successful implementation in real-world robotic arm trajectory planning scenarios, highlighting its potential to revolutionize the field with its robustness and adaptability.

Trajectory optimization of unmanned surface vehicle based on improved minimum snap

To solve the problem of path discontinuity and tracking difficulty in unmanned surface vehicle path planning, this paper introduces the Minimum Snap algorithm to optimize the trajectory, and makes some improvements to the Minimum Snap algorithm to address the problem of imperfect paths caused by uneven time allocation. The concept of a safe corridor is introduced to optimize the Minimum Snap algorithm. In response to the problem of excessive deviation from the reference value in some areas of the optimized path, a guidance function is added to make the path closer to the original trajectory. In the simulation, the proposed improved Minimum Snap is compared with Bezier curve and B-spline curve, the results show that the improved Minimum Snap can enable unmanned surface vehicle to complete turns and other actions with minimal energy consumption while driving safely, without experiencing severe acceleration and Jerk, laying the foundation for subsequent path tracking control.

B-spine: Learning B-spline Curve Representation for Robust and Interpretable Spinal Curvature Estimation

Spinal curvature estimation is important to the diagnosis and treatment of the scoliosis. Existing methods face several issues such as the need of expensive annotations on the vertebral landmarks and being sensitive to the image quality. It is challenging to achieve robust estimation and obtain interpretable results, especially for low-quality images which are blurry and hazy. In this paper, we propose B-Spine, a novel deep learning pipeline to learn B-spline curve representation of the spine and estimate the Cobb angles for spinal curvature estimation from low-quality X-ray images. Given a low quality input, a novel SegRefine network which employs the unpaired image-to-image translation is proposed to generate a high quality spine mask from the initial segmentation result. Next, a novel mask-based B-spline prediction model is proposed to predict the B-spline curve for the spine centerline. Finally, the Cobb angles are estimated by a hybrid approach which combines the curve slope analysis and a curve based regression model. We conduct quantitative and qualitative comparisons with the representative and SOTA learning-based methods on the public AASCE2019 dataset and our new proposed JLU-CJUH dataset which contains more challenging low-quality images. The superior performance on both datasets shows our method can achieve both robustness and interpretability for spinal curvature estimation.

Research on NURBS Curve Interpolation Algorithm Based on Mline-Gear

Aiming at the problem of speed fluctuation caused by the large error between the ideal interpolation step and the actual interpolation step in the traditional non-uniform rational B-spline ( NURBS ) curve interpolation, a NURBS curve interpolation algorithm based on Mline-Gear is proposed. The differential and derivative definitions are derived to transform the Mline and Gear formulas respectively, and the prediction and correction formulas of the interpolation parameters are obtained. Then, it is judged whether the interpolation parameter values obtained by the correction formula meet the speed fluctuation rate requirements, and the interpolation can be continued. Otherwise, the Steffensen descent formula is used to iteratively correct the interpolation parameter values. In order to ensure the calculation accuracy and avoid solving high-order derivatives, the fourth-order Runge-Kutta method is used to solve the initial two interpolation parameters. Finally, the algorithm is simulated. The simulation results show that the algorithm reduces the feed speed fluctuation rate while ensuring a small number of iterations.

Randomized progressive iterative approximation for B-spline curve and surface fittings

For large-scale data fitting, the least-squares progressive iterative approximation is a widely used method in many applied domains because of its intuitive geometric meaning and efficiency. In this work, we present a randomized progressive iterative approximation (RPIA) for the B-spline curve and surface fittings. In each iteration, RPIA locally adjusts the control points according to a random criterion of index selections. The difference for each control point is computed concerning the randomized block coordinate descent method. From geometric and algebraic aspects, the illustrations of RPIA are provided. We prove that RPIA constructs a series of fitting curves (resp., surfaces), whose limit curve (resp., surface) can converge in expectation to the least-squares fitting result of the given data points. Numerical experiments are given to confirm our results and show the benefits of RPIA.

Shape Modeling with a Cubic Trigonometric Beta B-Spline with Local Support Basis Functions

The modeling capabilities of trigonometric B-spline curves with form parameters are comparable to those of classical B-spline curves in terms of their significance and use. Through the application of basic functions and shape parameters, this method presents the cubic trigonometric B-spline curves. Constructed by these functions, the concept of designing curves and surfaces possesses all the ideal geometric qualities, such as the partition of unity, convex hull, affine invariance, and variation decreasing. The parameters that are used in the creation of the recommended approach are helpful for several important form characteristics. These vital form aspects include local, global, and biased tension qualities, which provide good control over the curve and allow for shape change as required. In addition, the [Formula: see text] strategy is the one that is suggested for use.

An Improved A-Star Path Planning Algorithm Based on Mobile Robots in Medical Testing Laboratories.

In the blood sample management pipeline environment, we have innovatively improved the traditional A-star algorithm to enhance the efficiency of mobile robots. This study employs a grid environmental modeling approach to accurately simulate medical testing laboratories. On the grid map, we utilize an 8-neighbor search method for path planning to accommodate the complex structure within the laboratory. By introducing an improved evaluation function and a bidirectional search strategy, we have successfully reduced the number of search nodes and significantly improved path search efficiency. Additionally, we eliminate redundant nodes in the path, smooth the path using cubic uniform B-spline curves, remove unnecessary inflection points, and further optimize the motion trajectory of the robot. The experimental results of the path planning simulation under different scenarios and specifications show that the improved A-star algorithm has higher search efficiency and traverses fewer nodes compared to the traditional A-star algorithm and the bidirectional A-star algorithm. Overall, the simulation experiments verify the feasibility of the improved A-star algorithm, which can better meet the needs of actual mobile robots in real medical testing laboratories.

A UGV Path Planning Algorithm Based on Improved A* with Improved Artificial Potential Field

Aiming at the problem of difficult obstacle avoidance for unmanned ground vehicles (UGVs) in complex dynamic environments, an improved A*-APF algorithm (BA*-MAPF algorithm) is proposed in this paper. Addressing the A* algorithm’s challenges of lengthy paths, excess nodes, and lack of smoothness, the BA*-MAPF algorithm integrates a bidirectional search strategy, applies interpolation to remove redundant nodes, and uses cubic B-spline curves for path smoothing. To rectify the traditional APF algorithm’s issues with local optimization and ineffective dynamic obstacle avoidance, the BA*-MAPF algorithm revises the gravitational field function by incorporating a distance factor, and fine-tunes the repulsive field function to vary with distance. This adjustment ensures a reduction in gravitational force as distance increases and moderates the repulsive force near obstacles, facilitating more effective local path planning and dynamic obstacle navigation. Through our experimental analysis, the BA*-MAPF algorithm has been validated to significantly outperform existing methods in achieving optimal path planning and dynamic obstacle avoidance, thereby markedly boosting path planning efficiency in varied scenarios.

Multi-objective aerodynamic optimization of expansion–deflection nozzle based on B-spline curves

An expansion–deflection nozzle (EDN) is an altitude-compensated nozzle that can accommodate a wide range of nozzle pressure ratio (NPR) variations. An EDN design with good thrust performance under both high- and low-NPR conditions can effectively improve the launch capability of the vehicle. In this study, a multi-objective aerodynamic optimization design is developed considering the performance of the EDN in both open and closed wake modes. The design method of the outer wall and pintle of the EDN is developed using B-spline curves. The design variables are the coordinates of the control points of the curves, and the objective functions are the thrust efficiencies of the EDN at NPRs of 30 and 400. The nozzle thrust is obtained using a Reynolds Averaged Navier–Stokes (RANS) equation solver and validated by experimental results. A radial basis function neural network and a non-dominated sorting genetic algorithm II (NSGA-II) are used to construct the surrogate model and search for the Pareto front, respectively. The relationship between the nozzle contour geometry and the nozzle thrust characteristics of the four optimised individuals on the Pareto front is discussed and analysed. The results demonstrate the effectiveness of the multi-objective optimization design for an EDN. One of the optimized designs improved the thrust efficiency at NPR = 30 by 3.5 % while improving the thrust efficiency at NPR = 400 by 0.6 %. Moreover, to ensure superior performance of the EDN in both operating modes, the nozzle length and pintle ending angle must be designed as a compromise, whereas a larger nozzle exit angle is favoured for both conditions. This study demonstrates the potential of the B-spline method for nozzle optimization design.

Optimization of Buoy Shape for Wave Energy Converter Based on Particle Swarm Algorithm

In order to improve the wave energy capture rate of the buoy of a wave energy generation device, this paper proposes a multi-degree of freedom method to optimize the shape of the buoy with maximum wave energy capture. Firstly, a multi-degree of freedom wave energy converter was designed, and the buoy shape was defined using a B-spline curve to generate the shape vector; then, a numerical model of the multi-degree of freedom wave energy converter was established and numerical calculations were carried out using AQWA/WEC-Sim software; on this basis, the particle swarm optimization algorithm was introduced to find the buoy shape corresponding to the maximum wave energy capture. Finally, the optimization of the buoy shape was in irregular waves. The results show that as the wave energy capture increased, the buoy shape tended to be flatter, with a smaller taper, and the optimal buoy shape had a better motion response than the conventional cone buoy. Eventually, the correctness of the buoy shape optimization method was verified through experimental testing.

Antenna Shape Modeling based on Histogram of Oriented Gradients Feature

A least square support vector machine (SVM) model is proposed for shape modeling of slot antennas. The slot image is mapped into the electromagnetic response by the SVM model. A modified shape-changing technique is also proposed to describe the antenna geometry by the quadratic uniform B-spline curve and generate the slot images. In the model, the histogram of oriented gradients feature is extracted from the slot images to show the appearance and shape of the slot. The relationship between the histogram of oriented gradient features and the electromagnetic responses is preliminarily built on SVM and the transfer function. Then a radial basis function network is used for error correction. The effectiveness of the proposed model is confirmed with an example of a tri-band microstrip-fed slot antenna. Compared with the convolutional neural network (CNN), the feature extracted by CNN is substituted by the histogram of oriented gradients feature, and the proposed model shows the same accuracy and the improvement of training efficiency.

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.