Search Results (6)

This study explores the dynamics of particle motion in pseudo-Galilean $3-$space $G_3^1$ by considering actions that incorporate both curvature and torsion of trajectories. We consider a general energy functional and formulate Euler–Lagrange equations corresponding to this functional under some boundary conditions in $G_3^1$. By adapting the geometric tools of the Frenet frame to this setting, we analyze the resulting variational equations and provide illustrative solutions that highlight their structural properties. In particular, we examine examples derived from natural Hamiltonian trajectories in $G_3^1$ and extend them to reflect the distinctive geometric features of pseudo-Galilean spaces, offering insight into their foundational behavior and theoretical implications. Full article

Ruled surfaces are considered one of the significant aspects of differential geometry. These surfaces are formed by the motion of a straight line called a generator, and every curve that intersects all the generators is called a directrix. In the present research paper, we explore a family of ruled surfaces constructed from circular helices (W-curve) using the Frenet frame in the Euclidean space ${\mathbb{E}}^3$. We derive the explicit formulas for the second mean curvature and second Gaussian curvature. We present some ruled surfaces, and we describe their properties. In addition, we determine the sufficient conditions for these surfaces to be minimal, flat, II-minimal, and II-flat. Also, we obtain sufficient conditions for the base curve for these ruled surfaces to be a geodesic curve, an asymptotic line, and a principal line. Furthermore, we present an application for a ruled surface whose base curve is a circular helix, we compute some quantities for this surface such as the mean curvature and Gaussian curvatures and we plot the ruled surface with its base curve, and at symmetric points and along a symmetry axis. Full article

“Three straight and two flat” is the inevitable demand when realizing the intelligent mining of a fully mechanized mining face. To address the crucial technical issue of lacking accurate perception of the shape of the scraper conveyor during intelligent coal mining, a three-dimensional curvature sensor involving fiber Bragg grating (FBG) is used as a perceptive tool to conduct curve reconstruction research based on different local motion frames and to reconstruct the shape of the scraper conveyor. Firstly, the formation process of the ‘S’-shaped bending section of the scraper conveyor during the pushing process is determined. Based on the FBG sensing principle, a mathematical model between the variation in the central wavelength and the strain and curvature is established, and the cubic B-spline interpolation method is employed to continuously process the obtained discrete curvature. Secondly, based on differential geometry, a spatial curve reconstruction algorithm based on the Frenet moving frame is derived, and the shape curve prediction interpolation model is built based on a gated recurrent unit (GRU) model, which reduces the impact of the decrease in curve reconstruction accuracy caused by damage to some grating measuring points. Finally, an experimental platform was designed and built, and sensors with curvature radii of 6 m, 7 m, and 8 m were tested. The experimental results showed that the reconstructed curve was essentially consistent with the actual shape, and the absolute error at the end was about 2 mm. The feasibility of this reconstruction algorithm in engineering has been proven, and this is of great significance in achieving shape curve perception and straightness control for scraper conveyors. Full article

Research on the cooperative adaptive cruise control (CACC) algorithm is primarily concerned with the longitudinal control of straight scenes. In contrast, the lateral control involved in certain traffic scenes such as lane changing or turning has rarely been studied. In this paper, we propose an adaptive cooperative cruise control (CACC) algorithm that is based on the Frenet frame. The algorithm decouples vehicle motion from complex motion in two dimensions to simple motion in one dimension, which can simplify the controller design and improve solution efficiency. First, the vehicle dynamics model is established based on the Frenet frame. Through a projection transformation of the vehicles in the platoon, the movement state of the vehicles is decomposed into the longitudinal direction along the reference trajectory and the lateral direction away from the reference trajectory. The second is the design of the longitudinal control law and the lateral control law. In the longitudinal control, vehicles are guaranteed to track the front vehicle and leader by satisfying the exponential convergence condition, and the tracking weight is balanced by a sigmoid function. Laterally, the nonlinear group dynamics equation is converted to a standard chain equation, and the Lyapunov method is used in the design of the control algorithm to ensure that the vehicles in the platoon follow the reference trajectory. The proposed control algorithm is finally verified through simulation, and validation results prove the effectiveness of the proposed algorithm. Full article

We propose a motion planning method for automated vehicles (AVs) to complete driving tasks in dynamic traffic scenes. The proposed method aims to generate motion trajectories for an AV after obtaining the surrounding dynamic information and making a preliminary driving decision. The method generates a reference line by interpolating the original waypoints and generates optional trajectories with costs in a prediction interval containing three dimensions (lateral distance, time, and velocity) in the Frenet frame, and filters the optimal trajectory by a series of threshold checks. When calculating the feasibility of optional trajectories, the cost of all optional trajectories after removing obstacle interference shows obvious axisymmetric regularity concerning the reference line. Based on this regularity, we apply the constrained Simulated Annealing Algorithm (SAA) to improve the process of searching for the optimal trajectories. Experiments in three different simulated driving scenarios (speed maintaining, lane changing, and car following) show that the proposed method can efficiently generate safe and comfortable motion trajectories for AVs in dynamic environments. Compared with the method of traversing sampling points in discrete space, the improved motion planning method saves 70.23% of the computation time, and overcomes the limitation of the spatial sampling interval. Full article

In recent years, control design for unmanned systems, especially a tractor–trailer system, has gained popularity among researchers. The emergence of such interest is caused by the potential reduction in cost and shortage of number of workers and labors. Two industries will benefit from the advancements of these types of systems: agriculture and cargo. By using the unmanned tractor–trailer system, harvesting and cultivating plants will become a safe and easy task. It will also cause a reduction in cost which in turn reduces the price on the end consumers. On the other hand, by using the unmanned tractor–trailer system in the cargo industry, shipping cost and time for the item delivery will be reduced. The work presented in this paper focuses on the development of a path tracking and a cascaded controller to control a tractor–trailer in reverse motion. The path tracking controller utilizes the Frenet–Serret frame to control the kinematics of the tractor–trailer system on a desired path, while the cascade controller’s main objective is to stabilize the system and to perform commands issued by the path tracker. The controlled parameters in this proposed design are the lateral distance to a path, trailer’s heading angel, articulated angel, and articulated angle’s rate. The main goal of such controller is to follow a path while the tractor–trailer system is moving in reverse and controlling the stability of the articulated vehicle to prevent the occurrence of a jackknife incident (uncontrolled state). The proposed controller has been tested in a different scenario where a successful implementation has been shown. Full article