Curvature Constraints Research Articles

A robust virtual-camera 3D shape reconstruction of deforming bubbles/droplets with additional physical constraints

Bubbles and droplets in strong turbulence exhibit non-affine deformation, especially near a breakup event when the strong necking and interfacial instability begin to play roles. The virtual-camera (VC) method was designed and implemented to mitigate the limited-angle reconstruction problem for complex bubble/droplet geometries. The VC method incorporates the additional physical constraint of minimum surface curvature into reconstruction. This framework helps to reduce the reconstruction uncertainty for experiments studying multiphase flows that use a limited number of cameras, non-ideal camera positions, or the combination of both. The method was tested with three synthetic geometries– sphere, ellipsoid, and dumbbell–that represent the undeformed, gently-deformed, and severely-deformed bubbles. In addition, a DNS dataset of bubbles with a 2% void fraction in turbulent channel flow was also utilized to generate a synthetic dataset for tests, and the volume overestimation was found to decrease from over 20% to about 10% after implementing the VC method. Finally, the method was applied to the experimental dataset of bubble-turbulence interaction, and the result shows over 20% improvement in volume estimation. An indirect uncertainty quantification that relies on the bubble aspect ratio and orientation has also been developed to estimate the bubble reconstruction uncertainty in each frame. This allows a more selective postprocessing step to obtain the statistics of deformation and breakup based on the quality of reconstruction.

Design of fiber-reinforced variable-stiffness composites for different open-hole geometries with fiber continuity and curvature constraints

In this study, the optimum design of fiber-reinforced composite plates with different hole geometries for various load conditions is investigated. Strain energy/compliance is optimized using three different design methodologies: constant-stiffness (CS), variable-stiffness (VS), and variable-stiffness with least-squares & continuity constraints (LSC). In all the cases, variable-stiffness and LSC designs both reveal better structural performance with reduced compliance and less stress concentrations around the hole in comparison to the optimum constant-stiffness designs. The compliance significantly decreases after optimization especially for the bi-axial loading case. Most importantly, the LSC method allows manufacturable fiber angle distributions that satisfy the maximum curvature limit, making the proposed LSC method applicable to well-established variable-stiffness manufacturing techniques such as 3-D printing and automated fiber placement. The benefits obtained by using variable-stiffness becomes less pronounced as the hole size decreases with respect to the area of the plate. The use of variable-stiffness is found to be more advantageous over shape-optimization methods of constant-stiffness composites. The comparison to the literature findings (Huang et al., 2016) reveals that 13% further reduction in structural compliance with the proposed LSC design method is possible. The proposed design method provides manufacturable variable-stiffness composites with significant improvements in structural performance.

The smooth Riemannian extension problem

Given a metrically complete Riemannian manifold $(M,g)$ with smooth nonempty boundary and assuming that one of its curvatures is subject to a certain bound, we address the problem of whether it is possibile to realize $(M,g)$ as a domain inside a geodesically complete Riemannian manifold $(M',g')$ without boundary, by preserving the same curvature bounds. In this direction we provide three kind of results: (1) a general existence theorem showing that it is always possible to obtain a geodesically complete Riemannian extension without curvature constraints; (2) various topological obstructions to the existence of a complete Riemannian extension with prescribed sectional and Ricci curvature bounds; (3) some existence results of complete Riemannian extensions with sectional and Ricci curvature bounds, mostly in the presence of a convexity condition on the boundary.

Path Planning for Multi-UAV Formation Rendezvous Based on Distributed Cooperative Particle Swarm Optimization

This paper studies the problem of generating cooperative feasible paths for formation rendezvous of unmanned aerial vehicles (UAVs). Cooperative path-planning for multi-UAV formation rendezvous is mostly a complicated multi-objective optimization problem with many coupled constraints. In order to satisfy the kinematic constraints, i.e., the maximum curvature constraint and the requirement of continuous curvature of the UAV path, the Pythagorean hodograph (PH) curve is adopted as the parameterized path because of its curvature continuity and rational intrinsic properties. Inspired by the co-evolutionary theory, a distributed cooperative particle swarm optimization (DCPSO) algorithm with an elite keeping strategy is proposed to generate a flyable and safe path for each UAV. This proposed algorithm can meet the kinematic constraints of UAVs and the cooperation requirements among UAVs. Meanwhile, the optimal or sub-optimal paths can be obtained. Finally, numerical simulations in 2-D and 3-D environments are conducted to demonstrate the feasibility and stability of the proposed algorithm. Simulation results show that the paths generated by the proposed DCPSO can not only meet the kinematic constraints of UAVs and safety requirements, but also achieve the simultaneous arrival and collision avoidance between UAVs for formation rendezvous. Compared with the cooperative co-evolutionary genetic algorithm (CCGA), the proposed DCPSO has better stability and a higher searching success rate.

Enhanced path smoothing based on conjugate gradient descent for firefighting robots in petrochemical complexes

The firefighting robot system (FFRS) comprises several autonomous robots that can be deployed to fire disasters in petrochemical complexes. For autonomous navigation, the path planner should consider the robot constraints and characteristics. Specifically, three requirements should be satisfied for a path to be suitable for the FFRS. First, the path must satisfy the maximum curvature constraint. Second, it must be smooth for robots to easily execute the trajectory. Third, it must allow reaching the target location in a specific heading. We propose a path planner that provides smooth paths, satisfy the maximum curvature constraint, and allows a suitable robot heading. The path smoother is based on the conjugate gradient descent, and three approaches are proposed for this path planner to meet all the FFRS requirements. The effectiveness of these approaches is qualitatively and quantitatively evaluated by examining the generated paths. Finally, the path planner is applied to an actual robot to verify the suitability of the generated paths for the FFRS, and planning is applied to another type of robot to demonstrate the wide applicability of the proposed planner.

Optimum ramp design in open pit mines

The problem studied in this paper is that of designing the optimal open pit haulage ramp that, for a given ramp width and a maximum ramp gradient, connects two points of the mine, minimising construction and operational costs. Because in-pit ramps require the removal of a considerable amount of non-valuable material (stripping), we discuss two different problems: high stripping (or in-pit) ramp design and low stripping (or ex-pit) road design. For the first situation, we present an integer programming model; in the second case, a shortest path approach is undertaken. In both cases, the models can be solved exactly, and include gradient and curvature constraints. The proposed formulations have been tested on real mine data, showing a significant reduction in cost compared to the previous mine design.

Minimum curvature trajectory planning and control for an autonomous race car

ABSTRACT This paper shows a software stack capable of planning a minimum curvature trajectory for an autonomous race car on the basis of an occupancy grid map and introduces a controller design that allows to follow the trajectory at the handling limits. The minimum curvature path is generated using a quadratic optimisation problem (QP) formulation. The key contributions of this paper are the extension of the QP for an improved accuracy of the curvature approximation, the introduction of curvature constraints and the iterative invocation of the QP to significantly reduce linearisation errors in corners. On the basis of the resulting raceline, a velocity profile is calculated using a forward-backward-solver that considers the velocity dependent longitudinal and lateral acceleration limits of the car. The advantages and disadvantages of the proposed trajectory planning approach are discussed critically with respect to practical experience from various racetracks. The software stack showed to be robust in a real world environment as it ran successfully on the Roborace DevBot during the Berlin Formula E event in May 2018. The lap time achieved was within a tenth of a second of a human driver and the car reached about and 80% of its acceleration limits.

Curvature-Bounded Lengthening and Shortening for Restricted Vehicle Path Planning

In this paper, the traditional shortest path planning problem for vehicle is advanced to length-targeted path planning problem, i.e., to plan path with its length being as close to a specified value as possible. Lengthening and shortening of given initial paths are used to solve this problem. Based on an operation set consisting of three basic path homotopies, we build a comprehensive and systematic framework to plan paths with target length, which is a generalization of the existing related studies. Thereby, the expected paths can be independently searched through such deformation processes within topological classes. The proposed framework can produce the largest length coverage in different scenarios and under different conditions, and it can also generate expected paths with arbitrary topological classification in terms of the curvature constraint and the obstacle constraint. Examples show that our lengthening and shortening method can effectively solve the length-targeted path planning problem in environment without or with obstacles. Note to Practitioners —This paper presents a curvature-bounded lengthening and shortening approach for path planning of vehicles. The curvature and obstacle constraints in the path planning problem are handled using the topological technique. In addition, a rigorous classification system is presented to systematically categorize the potential paths. Within a path class, any path can be deformed to other paths while keeping the curvature bounded. Then, the path deformation process is conducted via a series of basic homotopy operations to approach the target length. The length coverage range of this process is reduced when some endpoint conditions are meet or the vehicle is trapped into a closed region.

Geometric Design With Multiple Realizable Motion Tasks in the Vicinity of a Planar Mechanism–Environment Contact

Abstract This paper applies geometric design principles to planar mechanical systems, attempting to explain complex motion in mechanism–object/environment interaction for realizing multiple kinematic tasks in the vicinity of a contact location. The latter is a critical feature not fully captured in existing design methodologies. This is achieved through the development of a general planar geometric model that allows the derivation of multiple velocity and acceleration specifications compatible with mechanism–object curvature constraints in the vicinity of the contact. By incorporating these higher-order kinematic specifications into the design task formulation, contemporary planar kinematic synthesis is generalized allowing robust designs. The results are illustrated by the kinematic synthesis of two planar revolute–revolute (RR) linkages that are able to push and roll-slide along the object’s curvature in the vicinity of a contact location, a hybrid hand that incorporates a four-bar prosthetic finger that is able to grasp objects in two different modes, and a six-bar orthotic wheelchair for disabled canines that integrates body tilt and climbing motions.

Curvature-Controlled Wrinkling Surfaces for Friction.

Topographical patterns endow material surfaces with unique and intriguing physical and chemical properties. Spontaneously formed wrinkling has been harnessed to generate surface topography for various functionalities. Despite promising applications in biomedical devices and robot engineering, the friction behavior of wrinkling on curved surfaces remains unclear. Herein, wrinkled surfaces are induced by sputtering metals on soft polymer microspheres. The wrinkle morphologies and length scales can be controlled precisely by tailoring the microsphere radius (substrate curvature) and film thickness. The wrinkled surfaces exhibit controlled friction property, depending on the wrinkling patterns and length scales. An increase in friction force with increasing surface roughness is generally found for dimple patterns and labyrinth patterns. The dimple patterns show the lowest friction due to strong curvature constraint. The herringbone patterns exhibit apparent friction anisotropy with respect to topographic orientation. These results will guide future design of wrinkled surfaces for friction by harnessing substrate curvature.

Vibration reduction of car body based on 2D dual-base locally resonant phononic crystal

Due to the coating feature and bandgap adjustability of dual-base locally resonant phononic crystal (LRPC), the purpose behind this paper is to explore the application of dual-base LRPC in vibration reduction of car body. In previous research, the band structure and its influencing factors of dual-base LRPC have been investigated theoretically. As an artificial periodic structure, the key factors of application, including the number of unit cells, the curvature of base panel and the boundary constraint, have been taken account into analyzing bandgap characteristics of dual-base LRPC by finite element method (FEM). The results indicate that the non-infinite dual-base LRPC structure with small curvature and boundary constraint can still produce the expected bandgap. Then, according to the acoustic contribution analysis, the floor has been determined as the target panel for vibration reduction, on which the dual-base LRPC has been conducted. In order to obtain the wider bandgap with less added mass, the lattice constant and stiffness have been matched and optimized. With the final arrangement, the total body vibration level sharply reduces by 15 dB, and the interior noise reduces by 9 dB as well. It provides a new method for the vibration reduction of car body.

Brown–York mass and positive scalar curvature II: Besse’s conjecture and related problems

The Besse’s conjecture was posed on the well-known book Einstein manifolds by Arthur L. Besse, which describes critical points of Hilbert–Einstein functional with constraint of unit volume and constant scalar curvature. In this article, we show that there is an interesting connection between Besse’s conjecture and positive mass theorem for Brown–York mass. With the aid of positive mass theorem, we investigate the geometric structure of CPE manifolds and this provides us further understandings about Besse’s conjecture. As a related topic, we also discuss corresponding results for V-static metrics.

Collision-Free Path Planning for Cooperative Aerial Manipulators Under Velocity and Curvature Constraints

This paper presents a path planning problem with velocity and curvature constraints for cooperative aerial manipulators in obstacle environments. While potential-flow-based path planning approaches can help generate smooth paths in which cooperative aerial manipulators can avoid obstacles, they do not facilitate the consideration of velocity and curvature constraints. By contrast, while such constraints can be considered in the optimization approach, the approach often requires a heavy computational load. To overcome these drawbacks, we propose a simple approach based on curvature-constrained harmonic potential flow and a streamline-changing algorithm. We also handle the initial and final velocity problem by a smooth activation function, which is crucial for the tracking performance at the initial and final position. The proposed approach can be used to generate a smooth path for cooperative robots in real time. For the validation of the approach, the simulation results obtained with the proposed algorithm were compared with those acquired with an existing approach including dynamic movement primitives(DMPs). Furthermore, we performed load-carrying experiment using custom-made aerial manipulators.

A Gridmap-Path Reshaping Algorithm for Path Planning

Path planning is a vital function of robotic systems. Different from existing roadmap algorithms which first determine the free space and then determine the collision-free path, researchers recently proposed several convex relaxation based smoothing algorithms which first select an initial path to link the starting configuration and the goal configuration and then reshape this path to meet other requirements (e.g., collision-free conditions) by using convex relaxation. However, convex relaxation based smoothing algorithms often fail to give a satisfactory path, since the initial paths are selected improperly. Moreover, the curvature constraints were not considered in many existing convex relaxation based smoothing algorithms. In this paper, we show that we can first grid the whole configuration space to pick a candidate path and reshape the shortest path to meet our goal. This new algorithm inherits the merit of roadmap algorithms and a convex feasible set algorithm. We further discuss how to meet the curvature constraints by using both the Beamlet algorithm to select a better initial path and an iterative optimization algorithm to adjust the curvature of a path. Theoretical analysis and numerical testing results show that our algorithm can almost surely find a feasible path and require a short computation time.

Scalar curvature estimates of constant mean curvature hypersurfaces in locally symmetric spaces

We derive sharp estimates for the infimum and supremum of the scalar curvature of a hypersurface immersed with constant mean curvature in a locally symmetric space obeying standard curvature constraints. Our approach is based on the well known Omori–Yau maximum principle and on the weak version of it.

The distance sum rule from strong lensing systems and quasars – test of cosmic curvature and beyond

Testing the distance-sum-rule in strong lensing systems provides an interesting method to determine the curvature parameter $\Omega_k$ using more local objects. In this paper, we apply this method to a quite recent data set of strong lensing systems in combination with intermediate-luminosity quasars calibrated as standard rulers. In the framework of three types of lens models extensively used in strong lensing studies (SIS model, power-law spherical model, and extended power-law lens model), we show that the assumed lens model has a considerable impact on the cosmic curvature constraint, which is found to be compatible or marginally compatible with the flat case (depending on the lens model adopted). Analysis of low, intermediate and high-mass sub-samples defined according to the lens velocity dispersion demonstrates that, although it is not reasonable to characterize all lenses with a uniform model, such division has little impact on cosmic curvature inferred. Finally, thinking about future when massive surveys will provide their yields, we simulated a mock catalog of strong lensing systems expected to be seen by the LSST, together with a realistic catalog of quasars. We found that with about 16000 such systems, combined with the distance information provided by 500 compact milliarcsecond radio sources seen in future radio astronomical surveys, one would be able to constrain the cosmic curvature with an accuracy of $\Delta \Omega_k\simeq 10^{-3}$, which is comparable to the precision of \textit{Planck} 2015 results.

A G3-Continuous Extend Procedure for Path Planning of Mobile Robots with Limited Motion Curvature and State Constraints

Provably correct and computationally efficient path planning in the presence of various constraints is essential for autonomous driving and agile maneuvering of mobile robots. In this paper, we consider the planning of G 3-continuous planar paths with continuous and limited curvature in a motion environment that is bounded and contains obstacles modeled by a set of (non-convex) polygons. In practice, the curvature constraints often arise from mechanical limitations for the robot, such as limited steering and articulation angles in wheeled robots, or aerodynamic constraints in unmanned aerial vehicles. To solve the planning problem under those stringent constraints, we improve upon known path primitives, such as Reeds–Shepp (RS) and CC-steer (curvature-continuous) paths. Given the initial and final robot configuration, we developed extend-procedure computing paths that can approximate RS paths with arbitrary precision, but guaranteeing G 3-continuity. We show that satisfaction of all stated path constraints is guaranteed and, contrary to many other methods known from the literature, the method of checking for collisions between the planned path and obstacles is given by a closed-form analytic expression. Furthermore, we demonstrate that our approach is not conservative, i.e., it allows for precise maneuvers in tight environments under the assumption of a rectangular robot footprint. The presented extend procedure can be integrated into various motion-planning algorithms available in the literature. In particular, we utilized the Rapidly exploring Random Trees (RRT*) algorithm in conjunction with our extend procedure to demonstrate its feasibility in motion environments of nontrivial complexity and low computational cost in comparison to a G 3-continuous extend procedure based on η 3-splines.

A Real-Time Hydrodynamic-Based Obstacle Avoidance System for Non-holonomic Mobile Robots with Curvature Constraints

The harmonic potential field of an incompressible nonviscous fluid governed by the Laplace’s Equation has shown its potential for being beneficial to autonomous unmanned vehicles to generate smooth, natural-looking, and predictable paths for obstacle avoidance. The streamlines generated by the boundary value problem of the Laplace’s Equation have explicit, easily computable, or analytic vector fields as the path tangent or robot heading specification without the waypoints and higher order path characteristics. We implemented an obstacle avoidance approach with a focus on curvature constraint for a non-holonomic mobile robot regarded as a particle using curvature-constrained streamlines and streamline changing via pure pursuit. First, we use the potential flow field around a circle to derive three primitive curvature-constrained paths to avoid single obstacles. Furthermore, the pure pursuit controller is implemented to achieve a smooth transition between the streamline paths in the environment with multiple obstacles. In addition to comparative simulations, a proof of concept experiment implemented on a two-wheel driving mobile robot with range sensors validates the practical usefulness of the integrated system that is able to navigate smoothly and safely among multiple cylinder obstacles. The computational requirement of the obstacle avoidance system takes advantage of an a priori selection of fast computing primitive streamline paths, thus, making the system able to generate online a feasible path with a lower maximum curvature that does not violate the curvature constraint.

The Competition of Roughness and Curvature in Area-Constrained Polymer Models

The competition between local Brownian roughness and global parabolic curvature experienced in many random interface models reflects an important aspect of the KPZ universality class. It may be summarised by an exponent triple $(1/2,1/3,2/3)$ representing local interface fluctuation, local roughness (or inward deviation) and convex hull facet length. The three effects arise, for example, in droplets in planar Ising models (Alexander, '01, Hammond, '11,'12). In this article, we offer a new perspective on this phenomenon. We consider directed last passage percolation model in the plane, a paradigmatic example in the KPZ universality class, and constrain the maximizing path under the additional requirement of enclosing an atypically large area. The interface suffers a constraint of parabolic curvature as before, but now its local structure is the KPZ fixed point polymer's rather than Brownian. The local interface fluctuation exponent is thus two-thirds rather than one-half. We prove that the facet lengths of the constrained path's convex hull are governed by an exponent of $3/4$, and inward deviation by an exponent of $1/2$. That is, the exponent triple is now $(2/3,1/2,3/4)$ in place of $(1/2,1/3,2/3)$. This phenomenon appears to be shared among various isoperimetrically extremal circuits in local randomness. Indeed, we formulate a conjecture to this effect concerning such circuits in supercritical percolation, whose Wulff-like first-order behaviour was recently established (Biskup, Louidor, Procaccia and Rosenthal, '12).

Median Statistics Analysis of Deuterium Abundance Measurements and Spatial Curvature Constraints

Zavarygin et al. compiled a list of 15 deuterium abundance measurements; two were discarded because the remaining 13 measurements are then consistent with Gaussianity. Those authors found that the weighted-mean baryon density (Ωbh2) determined from the 13 measurements is mildly discrepant (1.6σ) with that determined from the Planck 2015 cosmic microwave background anisotropy data in a flat cosmogony. We find that a median statistic central estimate of Ωbh2 from all 15 deuterium abundance measurements is a more accurate estimate; is very consistent with Ωbh2 estimated from Planck 2015 data in a flat cosmogony, but is about 2σ lower than that found in a closed cosmogonical model from the Planck 2015 data.