Convex Hull Property Research Articles

Drone inspection path planning with self-organizing neural network of elastic Hebbian learning rule

Abstract As the scale of the power grid continues to expand, inspection scenarios become increasingly complex, and inspection and maintenance work in the power industry faces huge challenges. The traditional manual inspection mode has problems such as high cost, low efficiency, and poor safety. Using electric drones to inspect all equipment nodes in the area can effectively reduce operation and maintenance costs and ensure personnel safety. As a key technology to realize autonomous power inspection, UAV path planning needs to overcome difficulties such as long solution time and intersection of inspection paths. Therefore, this article conducts research on UAV path planning algorithms for large-scale power inspections, which is of great significance to improving the efficiency of power inspections. To avoid the occurrence of path intersections and improve the path planning effect, this paper uses the convex hull property in graph theory to design elastic Hebbian learning rules and proposes a self-organizing neural network algorithm based on elastic Hebbian learning rules.

Simulating Thin Shells by Bicubic Hermite Elements

In this study, we present the bicubic Hermite element method (BHEM), a new computational framework devised for the elastodynamic simulation of thin-shell structures. The BHEM is constructed based on quadrilateral Hermite patches, which serve as a unified representation for shell geometry, simulation, collision avoidance, as well as rendering. Compared with the commonly utilized linear FEM, the BHEM offers higher-order solution spaces, enabling the capture of more intricate and smoother geometries while employing significantly fewer finite elements. In comparison to other high-order methods, the BHEM achieves conforming C1 continuity for Kirchhoff–Love (KL) shells with minimal complexity. Furthermore, by leveraging the subdivision and convex hull properties of Hermite patches, we develop an efficient algorithm for ray-patch intersections, facilitating collision handling in simulations and ray tracing in rendering. This eliminates the need for laborious remodeling of the pre-existing surface as the conventional approaches do. We substantiate our claims with comprehensive experiments, which demonstrate the high accuracy and versatility of the proposed method.

Rectilinear convex hull of points in 3D and applications

Let P be a set of n points in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document} in general position, and let RCH(P) be the rectilinear convex hull of P. In this paper we obtain an optimal O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log n)$$\\end{document} time and O(n) space algorithm to compute RCH(P). We also obtain an efficient O(nlog2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log ^2 n)$$\\end{document} time and O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log n)$$\\end{document} space algorithm to compute and maintain the set of vertices of the rectilinear convex hull of P as we rotate R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document} around the Z-axis. We study some combinatorial properties of the rectilinear convex hulls of point sets in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}. Finally, as an application of the obtained results, we show an approximation algorithm to an optimization fitting problem in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}.

Convex hull property for elliptic and parabolic systems of PDE

We study the convex hull property for systems of partial differential equations. This is a generalization of the maximum principle for a single equation. We show that the convex hull property holds for a class of elliptic and parabolic systems of non-linear partial differential equations. In particular, this includes the case of the parabolic p-Laplace system. The coupling conditions for coefficients are demonstrated to be optimal by means of respective counterexamples.

A Surface Subdivision Scheme Based on Four‐Directional S13 Non‐Box Splines

AbstractIn this paper, we propose a novel surface subdivision scheme called non‐box subdivision, which is generalized from four‐directional S13 non‐box splines. The resulting subdivision surfaces achieve C1 continuity with the convex hull property. This scheme can be regarded as either a four‐directional subdivision or a special quadrilateral subdivision. When used as a quadrilateral subdivision, the proposed scheme can control the shape of the limit surface more flexibly than traditional schemes due to the natural introduction of auxiliary face control vertices.

Observer-based adaptive event-triggered control method for input-saturated systems

This paper presents an adaptive event-triggered control method for linear systems subject to input saturation. At the first, using convex hull properties, an observer-based controller is designed and then a generalized event-triggering mechanism is regarded. In order to extend more time interval between any two successive events, an adaptive event-triggered control method is also proposed. The proposed control methodology is based on finite-gain L2 stability. It is noteworthy that the proposed methods are designed in a co-design framework where the observer-based controller design with the event-triggering mechanisms are performed at the same time. Moreover, the presented adaptive approach offers more degrees of freedom as well as preserves the performance criteria high. Besides, Zeno avoidance analysis is studied by introducing a lower bound on inter-event times. Finally, the efficiency of the proposed methods is shown through a case-study numerical example.

Adaptive multiple model control for nonlinear constrained systems with a novel barrier Lyapunov function

AbstractThis paper investigates the problem of the multiple model control of nonlinear full state constrained systems with a novel barrier Lyapunov function. To handle the problem of unknown parameters, the identification model set containing identification models is established. The novel barrier Lyapunov functions (BLFs) are employed to guarantee that all system states do not violate their boundaries and can remove the restricted condition associated with traditional BLFs. By using the first order filter technique, the "explosion of complexity" difficulty caused by the repeated differential from the backstepping technique is removed. In addition, the estimation of unknown parameters is obtained by using the convex hull property. Based on the second level adaptation technique, the convex hull parameters are designed by using the identification error dynamics, and this algorithm has better convergence property than other algorithms. Then, a multiple model control strategy is developed, which not only guarantees that the requirement of all state constraints is realized, but also ensures that the boundedness of all signals of closed‐loop systems are not violated. The key advantage is that the proposed strategy does not require enormous identification models, that is, the number of identification model is reduced significantly. Finally, the effectiveness of the presented scheme is illustrated through the simulation results.

Jerk-Optimal Piecewise Planning of Tool Orientation for 5-Axis Ball-End Machining With Linearized Kinematic Constraints

Abstract When planning tool orientations for 5-axis machining from the perspective of kinematics, existing works often tend to optimize the angular velocity or acceleration of the rotary axes rather than the angular jerk due to its calculation sensitivity to discretization variations, though the jerk is fundamentally tied to the tracking errors and residual vibrations of the actuators. In this paper, a study is reported on how to optimize the angular jerk of the rotary axes while comprehensively considering the kinematic constraints, thus achieving a jerk-optimal tool orientation along the tool path, with driving capacity of the rotary axes respected. In this method, the displacements of the rotary axes are continuously represented by two quintic B-spline curves, and then the angular velocity, acceleration, and jerk of the rotary axes, which are the derivatives of the displacements, can be succinctly represented as a B-spline curve. Taking advantage of the convex hull property of B-spline curve, the linear analytical representations of the kinematic constraints of the rotary axes can be successfully derived in form of control coefficient combinations. To prevent the machining interference at the same time, a greedy strategy that incorporates a process of alternately smoothing tool orientation and checking machining interference is employed. Then, the smooth displacement splines of the rotary axes can be obtained by solving a constructed quadratic programming (QP) model that minimizes the angular jerk along the tool path, while satisfying kinematic constraints and without machining interference. Moreover, to generate efficiently tool orientations for long tool paths, a piecewise planning strategy that optimizes the tool orientation from coarse to fine is developed. Finally, the conducted experiments validate the proposed method.

A Safety Planning and Control Architecture Applied to a Quadrotor Autopilot

This letter presents a safety trajectory planning and tracking architecture for a quadrotor autopilot. Motor saturation constraints are explicitly considered in obstacle avoidance mission. Two challenging cases are covered: agile flight with a short task time and stable flight with actuator degradation. By explicitly considering the real physical constraints, the proposed method can make better use of the quadrotor maneuverability. Specifically, from control aspect, a sliding mode observer (SMO) based safety controller is implemented on the autopilot. Subsequently, the actuator saturation problem is analyzed and modeled as axis-coupled constraints. The B-spline curve is used to deal with the constraints due to its convex hull property. As a result, the planning problem can be transformed into a quadratically constrained quadratic programming (QCQP) problem with the proposed minimum energy cost function. Experiments show that the resulting trajectories can yield improvements in tracking accuracy in comparison of conventional planning method.

A New Extension of the Cubic Bézier Curve

An extended cubic Bézier curve is constructed, which can realize <italic>FC</italic>³ continuity under the condition of <italic>C</italic>² continuity, and ensure that the shape of both single-segment curve and composite curve can be freely adjusted without changing the control points. For this reason, a new curve with the same structure as the cubic Bézier curve is constructed. Firstly, the preliminary expression of the blending function is given, including undetermined coefficients. Then, according to the preset properties of the curves during joining, the endpoint properties of the blending function are deduced in reverse, so as to establish the equations that the undetermined coefficients in the blending function should satisfy. By solving the equations, a set of quintic polynomial blending functions with four parameters is obtained, which can be reduced to quartic when special parameters are taken. At last, a new curve with four shape parameters determined by four control points is defined by linear combination of the blending function and the control points. It has the basic properties of Bézier curve, such as convex hull property, geometric invariance and affine invariance. Thanks to the introduction of multiple shape parameters, curves of different shapes can be defined by the same control polygon, and the change of each shape parameter will drive the points on the curve to move linearly along a fixed direction. When constructing composite curve, the <italic>FC</italic>³ continuity conditions between adjacent curve segments are the same as the <italic>C</italic>² continuity conditions, and the shape of composite curve can be adjusted freely on the premise of maintaining continuity, control points and parameter segmentation unchanged. The new method can expand the processing ability of CAD system to industrial products, improve the speed of product design, modification and optimization, and save the investment of manpower and material resources in design.

Error-constant estimation under the maximum norm for linear Lagrange interpolation

For the linear Lagrange interpolation over a triangular domain, we propose an efficient algorithm to rigorously evaluate the interpolation error constant under the maximum norm by using the finite-element method (FEM). In solving the optimization problem corresponding to the interpolation error constant, the maximum norm in the constraint condition is the most difficult part to process. To handle this difficulty, a novel method is proposed by combining the orthogonality of the space decomposition using the Fujino–Morley FEM space and the convex-hull property of the Bernstein representation of functions in the FEM space. Numerical results for the lower and upper bounds of the interpolation error constant for triangles of various types are presented to verify the efficiency of the proposed method.

An Efficient Online Trajectory Generation Method Based on Kinodynamic Path Search and Trajectory Optimization for Human-Robot Interaction Safety.

With the rapid development of robot perception and planning technology, robots are gradually getting rid of fixed fences and working closely with humans in shared workspaces. The safety of human-robot coexistence has become critical. Traditional motion planning methods perform poorly in dynamic environments where obstacles motion is highly uncertain. In this paper, we propose an efficient online trajectory generation method to help manipulator autonomous planning in dynamic environments. Our approach starts with an efficient kinodynamic path search algorithm that considers the links constraints and finds a safe and feasible initial trajectory with minimal control effort and time. To increase the clearance between the trajectory and obstacles and improve the smoothness, a trajectory optimization method using the B-spline convex hull property is adopted to minimize the penalty of collision cost, smoothness, and dynamical feasibility. To avoid the collisions between the links and obstacles and the collisions of the links themselves, a constraint-relaxed links collision avoidance method is developed by solving a quadratic programming problem. Compared with the existing state-of-the-art planning method for dynamic environments and advanced trajectory optimization method, our method can generate a smoother, collision-free trajectory in less time with a higher success rate. Detailed simulation comparison experiments, as well as real-world experiments, are reported to verify the effectiveness of our method.

Improved flattening algorithm for NURBS curve based on bisection feedback search algorithm and interval reformation method

Deformation of flattening is one of the difficulties in ship reconstruction, which requires inversion of spline curves to accurately calculate the knots of end points of flattening line segment. To reduce the error of spline knot in the flattening algorithm caused by unstable inversion, the bisection feedback search (BFS) algorithm is proposed to stably achieve the global optimum. The feedback object of feedback operation, which is recorded by a bisection table, is judged using convergence criteria. Considering the strong convex hull property of the spline curve, the IR-FBS algorithm is proposed, which is the BFS algorithm accelerated by the proposed interval reformation (IR) method by reducing the feedback time and the number of iterations. Finally, the flattening algorithm of the spline curve is improved based on the IR-BFS algorithm, considering both stability and accuracy. To verify the effectiveness of the algorithms, comparative experiments are designed. First, the BFS algorithm and Newton–Raphson (NR) method are compared to verify the stability of solutions. Then, the IR-FBS algorithm is compared with FBS and the best existing three compound algorithms to check the acceleration effects and computational efficiency. Next, the improved flattening algorithm is used to process spline curves, and the flattening effect is investigated through the change of curve curvature. The experiments demonstrate that the BFS algorithm can ensure both accuracy and stability of solutions. In addition, the IR-BFS has the optimal performance among compound algorithms by the accelerated effects of IR method, and the improved flattening algorithm can ensure the high smoothness and meets the requirements of practical engineering.

Convex hull property for ancient harmonic map heat flows

Convex hull property for ancient harmonic map heat flows

Adaptive multiple model control for a class of nonlinear discrete time systems: second-level adaption design approach

In this study, a multiple model control scheme based on a second-level adaption design approach is established for a class of nonlinear discrete time systems. The nonlinear function of systems is relaxed to zero-order bounded. In the first-level adaption, N identification models are employed to estimate unknown parameters, which can weaken the effect of unknown parameters. The convex hull property based on all identification models is constructed, which ensures that the convex hull identified parameters converge to the real value rapidly. Compared with the existing multiple model controller, the actual controller is developed by a convex hull for the identification of unknown parameters and it can avoid some potential drawbacks of the switch mechanism. In the constructed scheme, the discrete time system is bounded input bounded output (BIBO) and the tracking error is bounded. Finally, a simulation example is presented to validate the proposed scheme.

Flexibility-Enhanced Continuous-Time Scheduling of Power System Under Wind Uncertainties

Flexibility helps accommodate uncertainties from the increasing renewables, such as wind power generation (WPG), but limited to scheduling methods, flexibility potentials of the power system have not been fully realized. This paper proposes the mathematical formulation and novel transformation for flexibility-enhanced continuous-time stochastic scheduling (FE-CTSS) of the power system under wind uncertainties. The Bernstein polynomial (BP) spline-based solution space transformation (SST) for solving continuous-time (CT) scheduling is revisited and the potential capacity and ramping flexibility, which are lost during the SST from function space to algebraic space, are then exposed for the first time. Developed from De Casteljau's algorithm (DCA), the enhancement matrix is derived to tighten the convex hull property and the enhanced SST is proposed to alleviate the feasible region narrowing to activate the potential flexibility in FE-CTSS. Case studies in the modified 6-bus system and the IEEE 118-bus system demonstrate the effectiveness and the promising role of FE-CTSS in the future high-renewable-penetrated and high-flexibility-required power system.

New Cubic Trigonometric Bezier-Like Functions with Shape Parameter: Curvature and Its Spiral Segment

This work presents the new cubic trigonometric Bézier-type functions with shape parameter. Basis functions and the curve satisfy all properties of classical Bézier curve-like partition of unity, symmetric property, linear independent, geometric invariance, and convex hull property and have been proved. The C 3 and G 3 continuity conditions between two curve segments have also been achieved. To check the applicability of proposed functions, different types of open and closed curves have been constructed. The effect of shape parameter and control points has been observed. It is observed that, by decreasing the value of shape parameter, the curve moves toward the control polygon and vice versa. The CT-Bézier curve is closer to the cubic Bézier curve for a fixed value of shape parameter. The proposed CT-Bézier curve can be used to represent ellipse. Using proposed basis functions, we have constructed the spiral segment which is very useful to construct fair curves and desirable to design trajectories of mobile robots, highway, and railway routes’ designing.

EDEM Investigation and Experimental Evaluation of Abrasive Wear Resistance Performance of Bionic Micro-Thorn and Convex Hull Geometrically Coupled Structured Surface

Procambarus clarkii was found to have excellent anti-wear performance against abrasive materials. To improve the wear resistance performance of the soil-engaging component of agricultural machinery, in this study, the micro-thorn and convex hull coupled geometrical structured surfaces inspired from the cephalothorax exoskeleton of the Procambarus clarkii was selected as the bionic prototype. By adopting bionic engineering techniques, three types of novel geometrical structured surfaces were proposed, which were bionic single, double and triple micro-thorn coupled convex hull surfaces (Bionic Type 2, 3 and 4, respectively). The anti-abrasive wear properties of these proposed geometrical surfaces were compared with a conventional bionic convex hull structured surface (Bionic Type 1) and a surface without any structures (smooth). Abrasive wear tests were conducted by using a rotational abrasive wear testing system. The accumulative test time was 80 h and the total wear distance was 6.09 × 105 m. By adopting the EDEM software (discrete element modeling), the Archard Wear model was selected to simulate the wear behavior of five different surfaces. In addition, the wear mechanisms of different surfaces were investigated. The results showed that the smooth surface suffered the most severe abrasive were, the abrasion loss reached 194.1 mg. The anti-abrasive properties of bionic geometric structured non-smooth surfaces were greatly improved; the reduction in terms of abrasion losses ranged between 20.4% and 94.1%, as compared with the smooth surface. The wear resistance property of micro-thorn and convex hull coupled structured surfaces were greatly improved as compared with convex hull and smooth surface. Bionic Type 3 was found to have the best anti-abrasive wear property: the abrasion loss was 11.5 mg. The wear morphology was observed by a scanning electron microscope. Smooth surface was characterized with wide, large size of grinding debris, while the bionic non-smooth surface featured narrow and small size abrasive dust. The results obtained from EDEM simulation agreed well with those of the aforementioned real scenario tests. It was revealed that the wear areas of the micro-thorn and convex hull coupled structured surface were mainly concentrated on the edge of convex hull and micro-thorn that faced the coming direction of particle flow. The geometric structure of the convex hull had beneficial effects on changing the movement behavior of particles, which means the stream of particle flow could be altered from a sliding to rolling state. Consequently, the ploughing and cutting phenomena of particles that act on the surfaces were greatly mitigated. Moreover, after being coupled with micro-thorns, the anti-abrasive wear preparty of the bionic convex hull geometrical structured surface was further improved. The rebound angle of particle flow that contacted the bionic micro-thorn coupled convex hull structured surface was greater than that of the conventional convex hull surface. Therefore, the dispersion effect of particle flow was further enhanced, since the movement behavior of the subsequent impact particle flow was altered. As a result, the wear of the bionic non-smooth surface was further reduced. This biconically inspired novel micro-thorn and convex hull coupled structured surface could provide theatrical and technical references to enhance the wear resistance performance of the soil-engaging component of agricultural machinery and mitigate the problem of abrasive wear failure.

Some properties of convex hulls generated by Poisson point process

Some properties of convex hulls generated by Poisson point process

Collision-free and dynamically feasible trajectory planning for omnidirectional mobile robots using a novel B-spline based rapidly exploring random tree

Generating a collision-free and dynamically feasible trajectory with a better clearance in a cluttered environment is still a challenge. We propose two dynamically feasible B-spline based rapidly exploring random tree (RRT) approaches, which are named DB-RRT and FMDB-RRT, to achieve path planning and trajectory planning simultaneously for omnidirectional mobile robots. DB-RRT combines the convex hull property of the B-spline and RRT’s rapid expansion capability to generate a safe and dynamically feasible trajectory. Firstly, we analyze the tree’s sustainable growth ability and put forward the dynamically feasible region. A geometric method is proposed to judge whether finding a dynamically feasible trajectory quickly. Secondly, we design two steer functions to guide the tree’s growth, improve efficiency, and decrease the number of iterations. To further increase the clearance and reduce the randomness of the trajectory, we propose FMDB-RRT, which uses the path of fast marching to guide the rapid growth of DB-RRT. Then, assuming that the number of sampled points is sufficient to represent the dynamically feasible region, the DB-RRT is proved to be probabilistically complete. Finally, by conducting experimental comparisons with other algorithms in different environments and deploying the proposed algorithm to an omnidirectional mobile robot, the effectiveness and good performance of the algorithm have been verified.