搜索结果：wheel

The wheel $W_{k}$ is the graph on $k+1$ vertices consisting of a vertex joined to a cycle of length $k$, and we say that $W_k$ is an even wheel if $k$ is even. Mao, Wang, Magnant, Schiermeyer proved that the Ramsey number of $W_{2n}$ is between $4n+1$ and $12n-2$. We improve both of these bounds, showing that $5n-\frac{1+(-1)^{n}}{2}\leq R(W_{2n})\leq 8n+664$ for all integers $n\geq 2$. The main focus of the paper concerns two general results on the Ramsey numbers of stars versus even wheels and even cycles versus even wheels, from which the above bounds are obtained as a corollary. That is, we asymptotically determine $R(K_{1,m}, W_{2n})$ and $R(C_{2m}, W_{2n})$ for all sufficiently large $m$ and $n$, both of which were open problems for most regimes. As for odd wheels, we note that the analogous values for stars versus odd wheels and odd cycles versus odd wheels were already known exactly, from which it follows that $6n+4=R(K_{1,2n+1}, W_{2n+1})\leq R(W_{2n+1})\leq 2\cdot R(C_{2n+1}, W_{2n+1})=12n+2$. Very recently, Zhang and Chen improved the upper bound to $R(W_{2n+1})\leq \frac{32n}{3}+O(1)$. We are able to refine their proof, further improving the upper bound to $R(W_{2n+1})\

This paper introduces the Mechacnially prOgrammed Radius-adjustable PHysical (MORPH) wheel, a fully passive variable-radius wheel that embeds mechanical behavior logic for torque-responsive transformation. Unlike conventional variable transmission systems relying on actuators, sensors, and active control, the MORPH wheel achieves passive adaptation solely through its geometry and compliant structure. The design integrates a torque-response coupler and spring-loaded connecting struts to mechanically adjust the wheel radius between 80 mm and 45 mm in response to input torque, without any electrical components. The MORPH wheel provides three unique capabilities rarely achieved simultaneously in previous passive designs: (1) bidirectional operation with unlimited rotation through a symmetric coupler; (2) high torque capacity exceeding 10 N with rigid power transmission in drive mode; and (3) precise and repeatable transmission ratio control governed by deterministic kinematics. A comprehensive analytical model was developed to describe the wheel's mechanical behavior logic, establishing threshold conditions for mode switching between direct drive and radius transformation. Experimental

In 1960, G. B. Robison discovered the general equations relating roads and wheels, where either can have an unusual shape (e.g., the square wheel rolls smoothly on a catenary). But he used some inobvious assumptions regarding the meaning of rolling. Here we derive the equations for the road appropriate for a given wheel using only the single assumption that rolling occurs with no slipping. We do not require that the wheel be differentiable, so this allows the construction of a wheel-road pair when the wheel is a continuous nowhere differentiable function.

This paper proposes an energy-recycling rimless wheel with spring-clutch legs. The proposed mechanism uses a lockable clutch to store part of the impact-induced elastic energy after foot contact and reinject it in the next gait cycle. First, we develop a hybrid dynamic model of the energy-recycling rimless wheel. Second, numerical simulations are used to examine the dynamics, local stability of periodic gaits, and the Cost of Transport (CoT) of the proposed mechanism. The simulation results show that the proposed mechanism reduces the CoT by up to 16.13% compared with a benchmark viscoelastic-legged rimless wheel with telescopic spring-damper legs. Compared with the rigid rimless wheel, the viscoelastic-legged and energy-recycling models reduce the CoT by more than 50%. The energy-recycling model also maintains locally stable periodic gaits over the tested slope and stiffness ranges. Finally, prototype experiments on an inclined plane are conducted to examine the feasibility of the proposed mechanism. The experimental results show that the proposed rimless wheel achieves passive walking on a shallow 1° slope, corresponding to a CoT of approximately 0.02. These results suggest that

A CaTherine wheel is a surjective continuous map $f:S^1 \to S^2$ such that for every closed interval $I\subset S^1$ the image $f(I)$ is homeomorphic to a disk, and $f(\partial I)$ is contained in the boundary of this disk. CaTherine wheels arise in many areas of low-dimensional geometry and topology, including conformal dynamics (expanding Thurston maps, expanding origamis), probability theory (whole plane ${\rm SLE}_κ$ for $κ\ge 8$, LQG metric trees) and elsewhere. We develop their theory in generality, and explain how CaTherine wheels and their associated structures can serve as a dictionary between these various fields. Our most substantial applications are to the theory of hyperbolic 3-manifolds. If $M$ is a closed hyperbolic 3-manifold and $G=π_1(M)$, we show that there is a canonical bijection between four kinds of structures associated to $M$: 1. orbit-equivalence classes of pseudo-Anosov flows on $M$ without perfect fits; 2. $G$-equivariant CaTherine wheels up to conjugacy; 3. minimal $G$-zippers; and 4. connected components of the space of uniform quasimorphisms on $G$. This generalizes and amplifies the theory of fiberings of hyperbolic 3-manifolds over the circle and the

A CaTherine wheel is a space-filling curve $f : S^1\to S^2$ such that for every closed interval $J\subset S^1$, $f(J)$ is homeomorphic to a closed disk and $f(\partial J)$ is contained in $\partial f(J)$. A CaTherine wheel gives rise to a pair of disjoint, dense topological trees in $S^2$ which roughly speaking lie to the left and right of $f$. We give necessary and sufficient conditions for a topological tree in $S^2$ to arise as one of these trees for some CaTherine wheel $f$. We apply this result to show that there is a unique CaTherine wheel corresponding to the geodesic tree rooted at $\infty$ for the $γ$-Liouville quantum gravity (LQG) metric, for $γ\in (0,2)$. In other words, we construct the space-filling curve which is the contour exploration of the LQG geodesic tree.

We show that the wheel classes in the Kontsevich graph complex $GC_d$ admit representatives supported on graphs with only $3$- and $4$-valent vertices. This verifies that Merkulov's low-valence conjecture holds for the wheel classes. More precisely, for every $m \ge 2$, we prove that the wheel graph $W_{2m+1}$ is homologous to an explicit linear combination of $2^{m-2}$ graphs, each having only $3$- and $4$-valent vertices.

For ensuring vehicle safety, the impact performance of wheels during wheel development must be ensured through a wheel impact test. However, manufacturing and testing a real wheel requires a significant time and money because developing an optimal wheel design requires numerous iterative processes to modify the wheel design and verify the safety performance. Accordingly, wheel impact tests have been replaced by computer simulations such as finite element analysis (FEA); however, it still incurs high computational costs for modeling and analysis, and requires FEA experts. In this study, we present an aluminum road wheel impact performance prediction model based on deep learning that replaces computationally expensive and time-consuming 3D FEA. For this purpose, 2D disk-view wheel image data, 3D wheel voxel data, and barrier mass values used for the wheel impact test were utilized as the inputs to predict the magnitude of the maximum von Mises stress, corresponding location, and the stress distribution of the 2D disk-view. The input data were first compressed into a latent space with a 3D convolutional variational autoencoder (cVAE) and 2D convolutional autoencoder (cAE). Subsequentl

There is this old, eternal question: Why don't animals have wheels? In this perspective we show that they actually do. And they do so in a physically extraordinary way -- by combining incompatible elasticity, differential geometry and dissipative self-organization. Nature's wheel -- the ``wheel-within'' -- has been mysteriously concealed in plain sight, yet it spins in virtually every slender-body organism: in falling cats, crocodilians spinning to subdue their prey, rolling fruit-fly larvae, circumnutating plants and even in some of our own body movements. Flying somehow under the radar of our cognition, in recent years the wheel-within also tacitly entered the field of soft robotics, finally opening our eyes for its ubiquitous role in Nature. We here identify its underlying physical ingredients, namely the existence of a neutrally-stable, shape-invariant and actively driven elastic mode. We then reflect on various man-made realizations of the wheel-within and outline where it could be spinning from here.

Simulation of vehicle motion in planetary environments is challenging. This is due to the modeling of complex terrain, optical conditions, and terrain-aware vehicle dynamics. One of the critical issues of typical simulators is that they assume terrain is a rigid body, which limits their ability to render wheel traces and compute the wheel-terrain interactions. This prevents, for example, the use of wheel traces as landmarks for localization, as well as the accurate simulation of motion. In the context of lunar regolith, the surface is not rigid but granular. As such, there are differences in the rover's motion, such as sinkage and slippage, and a clear wheel trace left behind the rover, compared to that on a rigid terrain. This study presents a novel approach to integrating a terramechanics-aware terrain deformation engine to simulate a realistic wheel trace in a digital lunar environment. By leveraging Discrete Element Method simulation results alongside experimental single-wheel test data, we construct a regression model to derive deformation height as a function of contact normal force. The region of interest in a height map is retrieved from the wheel poses. The elevation value

The familiar color wheel is a disk divided into six sectors, colored red, orange, yellow, green, blue, and purple, in circular order. Three of the colors can be obtained by blending the colors in the two neighboring sectors. One might wonder: is there a color wheel in which all six of the sections have this property, without all the sections being the same color? We show that the answer is no, not just for the 6-cycle but for any finite connected graph; indeed, for any finite, strongly connected, edge-weighted digraph. The result generalizes the ``harmonic lemma" for graphs, replacing the well-behaved averaging function by paint blending, about which almost nothing is assumed. Our proof makes use of a Markov chain stopping rule.

A reliable self-contained navigation system is essential for autonomous vehicles. Based on our previous study on Wheel-INS \cite{niu2019}, a wheel-mounted inertial measurement unit (Wheel-IMU)-based dead reckoning (DR) system, in this paper, we propose a multiple IMUs-based DR solution for the wheeled robots. The IMUs are mounted at different places of the wheeled vehicles to acquire various dynamic information. In particular, at least one IMU has to be mounted at the wheel to measure the wheel velocity and take advantages of the rotation modulation. The system is implemented through a distributed extended Kalman filter structure where each subsystem (corresponding to each IMU) retains and updates its own states separately. The relative position constraints between the multiple IMUs are exploited to further limit the error drift and improve the system robustness. Particularly, we present the DR systems using dual Wheel-IMUs, one Wheel-IMU plus one vehicle body-mounted IMU (Body-IMU), and dual Wheel-IMUs plus one Body-IMU as examples for analysis and comparison. Field tests illustrate that the proposed multi-IMU DR system outperforms the single Wheel-INS in terms of both positioning

Localization is a fundamental requirement for an autonomous vehicle system. One of the most often used systems for autonomous vehicle localization is the global positioning system (GPS). Nevertheless, the functionality of GPS is strongly dependent on the availability of satellites, making it unreliable in some situations. As a result, autonomous vehicles must possess autonomous self-localization capabilities to ensure their independent operation. Odometry techniques are employed to achieve vehicle localization by predicting the vehicle position and orientation based on sensor measurements of the vehicle motion. One of the approaches employed in odometry is known as wheel odometry. Wheel odometry has a lower degree of reliance on the surrounding environment than visual odometry and laser odometry. This study aims to evaluate the performance of wheel odometry implementation for an autonomous wheelchair in the context of the localization process. The differential drive kinematic model is employed to determine the predicted pose of a wheelchair. This prediction is derived from the measurement of the linear and angular velocity of the wheelchair. Several experiments have been conducted

An edge e in a matching covered graph G is removable if G-e is matching covered; a pair {e; f} of edges of G is a removable doubleton if G-e-f is matching covered, but neither G-e nor G-f is. Removable edges and removable doubletons are called removable classes, which was introduced by Lovasz and Plummer in connection with ear decompositions of matching covered graphs. A brick is a nonbipartite matching covered graph without nontrivial tight cuts. A brick G is wheel-like if G has a vertex h, such that every removable class of G has an edge incident with h. Lucchesi and Murty conjectured that every planar wheel-like brick is an odd wheel. We present a proof of this conjecture in this paper.

To improve the accuracy and robustness of the inertial navigation systems (INS) for wheeled robots without adding additional component cost, we propose Wheel-INS, a complete dead reckoning solution based on a wheel-mounted microelectromechanical system (MEMS) inertial measurement unit (IMU). There are two major advantages by mounting an IMU to the center of a non-steering wheel of the ground vehicle. Firstly, the gyroscope outputs can be used to calculate the wheel speed, so as to replace the traditional odometer to mitigate the error drift of INS. Secondly, with the rotation of the wheel, the constant bias error of the inertial sensor can be canceled to some extent. The installation scheme of the wheel-mounted IMU (Wheel-IMU), the system characteristics, and the dead reckoning error analysis are described. Experimental results show that the maximum position drift of Wheel-INS in the horizontal plane is less than 1.8% of the total traveled distance, reduced by 23% compared to the conventional odometer-aided INS (ODO/INS). In addition, Wheel-INS outperforms ODO/INS because of its inherent immunity to constant bias error of gyroscopes. The source code and experimental datasets used i

The Ramsey number for the pair of graphs $\mathbb{K}_{1,n}$ (star) versus $W_{m}$ (wheel) has been extensively studied. In contrast, the Ramsey number of $\mathbb{K}_{2,n}$ versus the wheel is not yet explored due to the bit more structural complexity of $\mathbb{K}_{2,n}$ compared to the star. In this article, we have established an exact value of $\mathbb{K}_{2,n}$ versus $W_{m}$ for large $n$ and $m$. In particular, we have proved \begin{equation*} R(\mathbb{K}_{2,n}, W_{m})=3n+4, \end{equation*} whenever $n$ and $m$ are sufficiently large integers satisfying $n\geq4m$ and $m$ is an odd integer. This proves the $W_{m}$-goodness of $\mathbb{K}_{2,n}$. Our proof combines probabilistic methods with an analysis of structural dependencies. As part of the argument, we resolve a structural rigidity question concerning highly dependent neighbourhoods (Lemma 3.12).

We consider the problem of observer design for a nonholonomic car (more generally a wheeled robot) equipped with wheel speeds with unknown wheel radius, and whose position is measured via a GNSS antenna placed at an unknown position in the car. In a tutorial and unified exposition, we recall the recent theory of two-frame systems within the field of invariant Kalman filtering. We then show how to adapt it geometrically to address the considered problem, although it seems at first sight out of its scope. This yields an invariant extended Kalman filter having autonomous error equations, and state-independent Jacobians, which is shown to work remarkably well in simulations. The proposed novel construction thus extends the application scope of invariant filtering.

LiDAR-inertial odometry (LIO), which fuses complementary information of a LiDAR and an Inertial Measurement Unit (IMU), is an attractive solution for state estimation. In LIO, both pose and velocity are regarded as state variables that need to be solved. However, the widely-used Iterative Closest Point (ICP) algorithm can only provide constraint for pose, while the velocity can only be constrained by IMU pre-integration. As a result, the velocity estimates inclined to be updated accordingly with the pose results. In this paper, we propose LIWO, an accurate and robust LiDAR-inertialwheel (LIW) odometry, which fuses the measurements from LiDAR, IMU and wheel encoder in a bundle adjustment (BA) based optimization framework. The involvement of a wheel encoder could provide velocity measurement as an important observation, which assists LIO to provide a more accurate state prediction. In addition, constraining the velocity variable by the observation from wheel encoder in optimization can further improve the accuracy of state estimation. Experiment results on two public datasets demonstrate that our system outperforms all state-of-the-art LIO systems in terms of smaller absolute traject

Understanding the wheel-terrain interaction is of great importance to improve the maneuverability and traversability of the rovers. A well-developed sensing device carried by the rover would greatly facilitate the complex risk-reducing operations on sandy terrains. In this paper, an instrumented wheel-on-limb (WOL) system of planetary rovers for wheel-terrain interaction characterization is presented. Assuming the function of a passive suspension of the wheel, the WOL system allows itself to follow the terrain contour, and keep the wheel remain lowered onto the ground during rover motion including climbing and descending, as well as deploy and place the wheel on the ground before a drive commanding. The system concept, functional requirements, and pre-design work, as well as the system integration are presented.