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In this paper, we give some characterizations of Frenet curves in 3-dimensional $ \delta $-Lorentzian trans-Sasakian manifolds. We compute the Frenet equations and Frenet elements of these curves. We also obtain the curvatures of non-geodesic Frenet curves on 3-dimensional $ \delta $-Lorentzian trans-Sasakian manifolds. Finally, we give some results for these curves.

In this study, the Serret-Frenet frame and derivative formulas were obtained for all intermediate points of the rational Bezier curves with the algorithm method, and much more general results were computed from the previous studies. In addition, the center and radius of the osculator circle and sphere were calculated.

This paper presents a novel method for the evaluation of three-dimensional blood-flow simulations based, on the decomposition of the velocity field into localized coordinate systems along the vessels centerline. The method is based on the computation of accurate centerlines with the Vascular Modeling Toolkit (VMTK) library, to calculate the localized Frenet-frames along the centerline and the morphological features, namely the curvature and torsion. Using the Frenet-frame unit vectors, the velocity field can be decomposed into axial, circumferential and radial components and visualized in a diagram along the centerline. This paper includes case studies with four idealized geometries resembling the carotid siphon and two patient-specific cases to demonstrate the capability of the method and the connection between morphology and flow. The proposed evaluation method presented in this paper can be easily extended to other derived quantities of the velocity fields, such as the wall shear stress field. Furthermore, it can be used in other fields of engineering with tubular cross-sections with complex torsion and curvature distribution.

Motion-trajectory (MT) curves are used to introduce Frenet oscillations. Time-varying orientation of the motion plane that contains the absolute velocity and acceleration vectors is defined in terms of three Frenet–Euler angles; the curvature, vertical-development, and bank angles, referred to as the Frenet angles for brevity. The Frenet bank angle and the associated Frenet super-elevation of the motion plane, which measure deviation of the centrifugal inertia force from the horizontal plane, can be used to shed light on definition of the balance speed used in practice. The concept of the pre-super-elevated osculating (PSEO) plane is introduced and Rodrigues’ formula is employed to develop an orthogonal rotation matrix that provides a geometric interpretation of the PSEO plane. A new inverse-dynamics problem that utilizes experimentally or simulation recorded motion trajectories (RMT) is used to define the Frenet inertia forces and demonstrate their equivalence to the Cartesian form of the inertia forces. New expressions for the curvature vector in terms of the velocity and acceleration, limit on the magnitude of the tangential acceleration for a given forward velocity, condition required for the centrifugal force to remain horizontal, and condition of curvature singular points are derived. The Frenet bank angle can be used to prove existence of the normal vectors at the curvature singular points. It is shown that the inertia force can assume different forms, depending on the curve parameter used. The results of a simple analytical curve demonstrate Frenet oscillations and importance of distinguishing between the highway-ramp and railroad track bank angles and super-elevations, which are time-invariant, and the Frenet bank angle and super-elevation, which are motion-dependent.

A data-driven science approach, based on integrating nonlinear multibody system (MBS) formulations and new geometric concepts, is used in this paper to compare the performance of two widely used railroad bogies: the three-piece bogie and the Y25 bogie. MBS algorithms are used to solve the bogie nonlinear differential/algebraic equations (DAEs) to determine the bogie motion trajectories. To have a better understanding of the bogie dynamic behavior, a distinction is made between the geometry of actual motion trajectories (AMT) and the track geometry. The AMT curves are described using the motion-dependent Frenet–Euler angles, referred to as Frenet bank, curvature, and vertical development angles, which differ from their counterparts used in the description of the track geometry. In particular, the Frenet bank angle defines the super-elevation of the AMT curve osculating plane, referred to as the motion plane, distinguishing this Frenet super-elevation from the fixed-in-time track super-elevation. The paper explains the difference between the lateral track plane force balance used in practice to determine the balance speed and the Frenet force balance which is based on recorded motion trajectories. Computer simulations of bogies travelling on a track, consisting of tangent, spiral, and curve sections are performed with particular attention given to the deviations of the AMT curves from the track centerline. The results obtained in this study demonstrate the dependence of the AMT curve geometry on the wheelset forward motion, highlighting the limitations of tests performed using roller test rigs which do not allow longitudinal wheelset motion.

In this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space E3. We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal (equivalently, helicoid). We examine the conditions for the curves lying on this surface to be asymptotic curves, geodesics or lines of curvature. Finally, we obtain the Frenet vectors of generalized normal ruled surface and get some relations with helices and slant ruled surfaces and we give some examples for the obtained results.

In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ \mathbb{G} $ with bi-invariant metric. Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ \mathbb{G} $. Especially, we obtain some results for the natural mate and the conjugate mate of a Frenet curve in $ \mathbb{G} $ when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski (constant curvature and non-constant torsion), anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve. Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group., 16 pages

The geometry description plays a central role in many engineering applications and directly influences the quality of the computer simulation results. The geometry of a space curve can be completely defined in terms of two parameters: the horizontal and vertical curvatures, or equivalently, the curve curvature and torsion. In this paper, distinction is made between the track angle and space-curve bank angle, referred to in this paper as the Frenet bank angle. In railroad vehicle systems, the track bank angle measures the track super-elevation required to define a balance speed and achieve a safe vehicle operation. The formulation of the track space-curve differential equations in terms of Euler angles, however, shows the dependence of the Frenet bank angle on two independent parameters, often used as inputs in the definition of the track geometry. This paper develops the general differential equations that govern the track geometry using the Euler angle sequence adopted in practice. It is shown by an example that a curve can be twisted and vertically elevated but not super-elevated while maintaining a constant vertical-development angle. The continuity conditions at the track segment transitions are also examined. As discussed in the paper, imposing curvature continuity does not ensure continuity of the tangent vectors at the curve/spiral intersection. Several curve geometries that include planar and helix curves are used to explain some of the fundamental issues addressed in this study.

In this work, we study the canal surfaces foliated by pseudo hyperbolic spheres $ \mathbb{H}_{0}^{2} $ along a Frenet curve in terms of their Gauss maps in Minkowski 3-space. Such kind of surfaces with pointwise 1-type Gauss maps are classified completely. For example, an oriented canal surface that has proper pointwise 1-type Gauss map of the first kind satisfies $ \Delta \mathbb{G} = -2K\mathbb{G}, $ where $ K $ and $ \mathbb{G} $ is the Gaussian curvature and the Gauss map of the canal surface, respectively.

In this study, we investigate how to construct surfaces using a line of curvature in a 3-dimensional Lie group. Then, by utilizing the Frenet frame, we give the conditions that a curve becomes a line of curvature on a surface when the marching-scale functions are more general expressions. After then, we provide some crucial examples of how efficient our method is on these surfaces.

In this paper, we introduce original definitions of Smarandache ruled surfaces according to Frenet-Serret frame of a curve in E 3 . It concerns TN-Smarandache ruled surface, TB-Smarandache ruled surface, and NB-Smarandache ruled surface. We investigate theorems that give necessary and sufficient conditions for those special ruled surfaces to be developable and minimal. Furthermore, we present examples with illustrations.

There are common features such as the tool path curvature and torsion, and cutter-orientation change in sculptured surface machining, which bring new challenges to the accurate prediction of cutting force. Aiming at the tool path curvature and torsion, and cutter-orientation change, a cutting force prediction method based on the screw theory is proposed in this paper. For the first time, the screw theory is used to describe the cutter spatial motion, which includes the feed motion considering the tool path curvature and torsion, and cutter-orientation change. Combining with Frenet frame, the analytical formula of the screw of cutting edge elements is derived through the homogeneous coordinate transformation. Then, the instantaneous uncut chip thickness (IUCT) of each cutting edge element is calculated by the vector projection method. And the cutting state of the cutting edge element is determined by its IUCT and position relative to the workpiece surface, which is updated by the cutter envelope surface along the machined tool path derived with the screw. To verify the effectiveness of the proposed cutting force prediction method, a milling experiment is conducted on a conical surface workpiece along a tool path with curvature and torsion characteristics, and changing cutter-orientations. Then, the effects of the tool path curvature and torsion, and cutter-orientation change on the cutting force are simulated and analyzed. The results show that the proposed cutting force model based on the screw theory has higher prediction accuracy for the sculpture surface machining with curving and torsional tool paths and changing cutter-orientations.

In this paper, we get the time evolution equations of the curvature and torsion of the evolving spacelike curves in the Minkowski space. Also, we give inextensible evolutions of timelike ruled surfaces that are produced by the timelike normal and spacelike binormal vector fields of spacelike curve and derive the necessary conditions for an inelastic surface evolution. Then, we compute the coefficients of the first and second fundamental forms, the Gauss and mean curvatures for timelike special ruled surfaces. As a result, we give applications of the evolution equations for the curvatures of the curve in terms of the velocities and get the exact solutions for these new equations., Comment: 12 pages, 4 figures

An ameliorated Frenet trajectory optimization (AFTO) method based on artificial emotion (AE) and an equilibrium optimizer (EO) is proposed for the local trajectory planning of an unmanned ground vehicle (UGV). An artificial emotional potential field (AEPF) model is established to simulate AE. To realize a humanoid driving mode with emotional intelligence, AE is introduced into the Frenet trajectory optimization (FTO) method to determine the optimal trajectory. Based on the optimal discrete goal state of the FTO method, a first-sampling-then-optimization (FSTO) framework combining the FTO method with the EO is designed to obtain the optimal trajectory in a continuous goal state space. With different AEPF levels corresponding to different types of obstacles, simulation results show that the AEPF effectively adjusts the trajectory into different levels of safe distance between the UGV and obstacles corresponding to the humanoid driving mode. From the results of 30 independent experiments based on the AEPF, the FSTO framework in the AFTO method is effective for optimizing the trajectory of the FTO method at a lower cost. Moreover, the effectiveness of the proposed method for different types of roads is verified on a straight road and a curved road with obstacles in simulation. The improvement based on emotional intelligence and trajectory optimization in the AFTO method provides a humanoid driving mode for the UGV in the continuous goal state space.

In this paper, the available formulae for the curvature of plane curve are reviewed not only for the time-like but also for the space-like parameter curve. Two ways to describe the curve are proposed. One is the straight way to obtain the Frenet formula according to the given curve of parameter form. The other is that we can construct the curve by solving the state equation of Frenet formula subject to the initial position, the initial tangent, normal and binormal vectors, and the given radius of curvature and torsion constant. The remainder theorem of the matrix and the Cayley–Hamilton theorem are both employed to solve the Frenet equation. We review the available formulae of the radius of curvature and examine their equivalence. Through the Frenet formula, the relation among different expressions for the radius of curvature formulae can be linked. Therefore, we can integrate the formulae in the engineering mathematics, calculus, mechanics of materials and dynamics. Besides, biproduct of two new and simpler formulae and the available four formulae in the textbook of the radius of curvature yield the same radius of curvature for the plane curve. Linkage of centrifugal force and radius of curvature is also addressed. A demonstrative example of the cycloid is given. Finally, we use the two new formulae to obtain the radius of curvature for four curves, namely a circle. The equivalence is also proved. Animation for 2D and 3D curves is also provided by using the Mathematica software to demonstrate the validity of the present approach.

In this article, we mainly discuss the local differential geometrical properties of the lightlike Killing magnetic curve $ \mathit{\boldsymbol{\gamma }}(s) $ in $ \mathbb{S}^{3}_{1} $ with a magnetic field $ \boldsymbol{ V} $. Here, a new Frenet frame $ \{\mathit{\boldsymbol{\gamma }}, \boldsymbol{ T}, \boldsymbol{ N}, \boldsymbol{ B}\} $ is established, and we obtain the local structure of $ \mathit{\boldsymbol{\gamma }}(s) $. Moreover, the singular properties of the binormal lightlike surface of the $ \mathit{\boldsymbol{\gamma }}(s) $ are given. Finally, an example is used to understand the main results of the paper.

We present a displacement-based and a mixed isogeometric collocation (IGA-C) formulation for free-form, three-dimensional, shear-deformable beams with high and rapidly-varying curvature and torsion. When such complex shapes are concerned, the approach used to build the IGA geometric model becomes relevant. Although IGA-C has been so far successfully applied to a wide range of problems, the effects that different parameterization and knot placement techniques may have on the accuracy of collocation-based formulations is still an unexplored field. To fill this gap, primal and mixed formulations are used combining two parameterization methods (chord-length and equally spaced) with two knot placement techniques (uniformly spaced and De Boor). With respect to the space-varying Frenet local frame, we derive the strong form of the governing equations in a compact form through the definition of two matrix operators conveniently used to perform first and second order derivatives of the vector fields involved in the formulations. This approach is very efficient and easy to implement within a collocation-based scheme. Several challenging numerical experiments allow to test the different considered parameterizations and knot placement techniques, revealing in particular that with the primal formulation an equally spaced parameterization is definitively the most recommended choice and it should always be used with an approximation degree of, at least, p = 6 , although some caution must be adopted when very high Jacobians and small curvatures occur. The same holds for the mixed formulation, with the difference that p = 4 is enough to yield accurate results.

Bu çalışmada, verilen bir eğriden geçen ve bu eğri boyunca Gauss eğriliği sabit olan yüzeyler elde edildi. Verilen eğrinin Frenet vektör alanları kullanılarak bu eğriden geçen yüzeyler parametrik olarak ifade edildi. Ayrıca, verilen eğriden geçen ve Gauss eğriliği sabit regle yüzeyler için yeterli şartlar verildi. Bazı örnekler verilerek elde edilen yöntem görsel hale getirildi. In this study, we find surfaces with constant Gaussian curvature along a given curve. The parametric representation of the surfaces possessing the given curve expressed using the Frenet vector fields of the curve. Also, we give conditions for ruled surfaces passing through the given curve and having constant Gaussian curvature. We present some illustrative examples validating the presented method.

The experimental results of the processes of complex loading along helical strain trajectories are used to find out that the response to the helical strain trajectory following the simple loading after exhaustion of some trace takes a certain shape of the limit mode, that is, there is a correspondence between the deformation trajectory geometry and the form of response. A new variant of constitutive equations for describing complex loading processes with strain trajectories of arbitrary geometry and dimension is considered. The vector constitutive equations and the system of differential equations for the four angles from the Frenet decomposition are obtained. It is proved that the stress vector is represented in the form of sum of three terms: rapidly decaying plastic traces of elastic states, instantaneous responses to the deformation process, and irreversible stresses accumulated along the deformation trajectory. A new method for mathematical modeling of five-dimensional processes of complex loading is constructed and tested on two- and three-dimensional processes.

A spacelike constant angle surface in $E_1^3$ (Minkowski 3-space) is defined by its unit normal vector field forms a constant angle with some fixed vector direction. In this paper, we use the Frenet frame to express the surface and construct a family of spacelike constant angle surfaces which possess the given curve as isoparametric curve. The normal vector of each surface forms a constant angle with a fixed vector lying in the spacelike cone and in the timelike cone respectively. Finally, we give some representative examples.

In this study, the characterization of position vectors belonging to non-lightlike Bertrand W curve mate with constant curvature are obtained depending on differentiable functions. The position vector of Bertrand W curve is stated by a linear combination of its Frenet frame with differentiable functions. There exist also different cases for the curve depending on the value of curvature and torsion. The relationships between Frenet apparatuas of these curves are stated separately for each case. Finally, the timelike and spacelike Bertrand curve mate visualized of given curves as examples, separately.

This paper presents a trajectory planning framework to deal with the highly dynamic environments for on-road driving. The trajectory optimization problem with parameterized curvature control was formulated to reach the goal state with the vehicle model and its dynamic constraints considered. This in contrast to existing curve fitting techniques guarantees the dynamic feasibility of the planned trajectory. With generation of multiple trajectory candidates along the Frenet frame, the vehicle is reactive to other road users or obstacles encountered. Additionally, to deal with more complex driving scenarios, its seamless interaction with an upper behavior planning layer was considered by having longitudinal motion planning responsive to the desired goal state. The trajectory evaluation and selection methodologies, along with the low-level tracking control, were also developed under this framework. The potential of the proposed trajectory planning framework was demonstrated under different dynamic driving scenarios such as lane-changing or merging with surrounding vehicles with its computation efficiency proven in real-time simulations.

In this study we have examined, involute of the cubic Bezier curve based on the control points with matrix form in E3. Frenet vector fields and also curvatures of involute of the cubic Bezier curve are examined based on the Frenet apparatus of the first cubic Bezier curve in E3. Â

We deal with a notion of weak binormal and weak principal normal for non-smooth curves of the Euclidean space with finite total curvature and total absolute torsion. By means of piecewise linear methods, we first introduce the analogous notation for polygonal curves, where the polarity property is exploited, and then make use of a density argument. Both our weak binormal and normal are rectifiable curves which naturally live in the projective plane. In particular, the length of the weak binormal agrees with the total absolute torsion of the given curve. Moreover, the weak normal is the vector product of suitable parameterizations of the tangent indicatrix and of the weak binormal. In the case of smooth curves with positive curvature, the weak binormal and normal yield (up to a lifting) the classical notions of binormal and normal., Comment: 18 pages, 2 figures

For motion of a material point along a space curve, due to Siacci [1], a resolution of the acceleration vector is well known. In this resolution, the acceleration vector is stated as the sum of two special oblique components in the osculating plane to the curve. In this paper, we have studied the Siacci’s theorem for non-relativistic particles moving along the Frenet curves at very low speeds relative to the speed of light in Minkowski 3-space. Moreover, an illustrative example is given to show how the aforesaid theorem works. This theorem is a new contribution to the field and it may be useful for some specific applications in mathematical and computational physics.

The role played by non-inertial frames in physics is one of the most interesting subjects that we can study when dealing with a physical theory. It does not matter whether we are studying classical theories such as special relativity or quantum theory, the idea of an accelerated frame is always one of the first ideas to come to our minds. In the case of special relativity, a problem with the concept of rigidity emerged as soon as Max Born gave a reasonable definition of rigid motion: the Herglotz-Noether theorem imposes a strong restriction on the possible rigid motions. In this paper, the equivalence of this theorem with another one that is formulated with the help of Frenet-Serret formalism is proved, showing the connection between the rigid motion and the curvatures of the observer's trajectory in spacetime. In addition, the Dirac equation in the Frenet-Serret frame for an arbitrary observer is obtained and applied to the rotating observers. The solution in the rotating frame is given in terms of that of an inertial one.

Considering the spatial position and shape detection properties of the fiber Bragg grating (FBG) curve sensor used in the human body, the positioning accuracy of the FBG curve sensor plays a major role in the pre-diagnosis and treatment of diseases. We present a new type of shape-sensing catheter (diameter of 2.0 mm and length of 810 mm) that is integrated with an array of four optical fibers, where each contains five nodes, to track the shape. Firstly, the distribution of the four orthogonal fiber gratings is wound around a nitinol wire using novel packaging technology, and the spatial curve shape is rebuilt based on the positioning of discrete points in space. An experimental platform is built, and then a reconstruction algorithm for coordinate point fitting of the Frenet frame is used to perform the reconstruction experiment on millimeter paper. The results show that, compared with those in previous studies, in 2D test, the maximum relative error for the end position is reduced to 2.74%, and in 3D reconstruction experiment, the maximum shape error is 3.43%, which verifies both the applicability of the sensor and the feasibility of the proposed method. The results reported here will provide an academic foundation and the key technologies required for navigation and positioning of non-invasive and minimally invasive surgical robots, intelligent structural health detection, and search and rescue operations in debris.

On-road trajectory planning is a direct reflection of an autonomous vehicle’s intelligence level when traveling on an urban road. The prevalent on-road trajectory planners include the spline-based, sample-and-search-based, and optimal-control-based methods. Path-velocity decomposition and Frenet frame have been widely adopted in the aforementioned methods, which, nonetheless, largely degrade the trajectory planning quality when the road curvature is large and/or the scenario is complex. This paper aims to plan precise and high-quality on-road trajectories, thus we choose to describe the concerned scheme as an optimal control problem, wherein the urban road scenario is described completely in the Cartesian frame rather than in the Frenet frame. The formulated optimal control problem should be numerically solved in real-time. To that end, a light-weighted iterative computation architecture is built. In each iteration, a tunnel construction strategy tractablely models the collision-avoidance constraints, and a constraint softening strategy helps to find an intermediate trajectory for constructing the tunnels in the next iteration. Efficacy of the proposed on-road trajectory planner is validated by simulations on a high-curvature urban road wherein the ego vehicle is surrounded by multiple social vehicles at various speeds.

Based on the Frenet frame, this paper proposes a general mathematical spiral model with an arbitrary smooth space curve as the center path, which can accurately build complex skeleton lines of wire rope strands. From the aspect of geometry, all the wires are spatial cylinders and must meet the actual geometric requirements: 1. The center cylinder is tangent or separated from the spiral cylinder; 2. The adjacent spiral cylinders do not overlap each other. For requirement 1, Costello’ conclusion is referenced and extended universally to suit an arbitrary smooth space central curve case with rigorous proofs. For requirement 2, the overlapping problem is described as obtaining the minimum distance between the two adjacent spatial path curves, which is deduced by a novel cross section method (SCM) with rigorous proofs and solved by the General Particle Swarm Optimization (PSO) algorithm. Based on the above models, the geometric modeling of wire rope strands procedure is proposed and implemented on the platforms of MATLAB and SolidWorks. Validations are conducted through geometric graphical representations, compared with those from some previous researches. For the simple straight strand case, when the number of spiral cylinders and spiral radius are given, the critical relationship between the ratio of spiral wire radius to spiral radius and the spiral angle is firstly obtained, which can be a precise dimension design reference of simple straight strand for eliminating initial geometric overlap. Further, to show the advance, some precise graphical examples of complex wire rope strands like independent wire rope core (IWRC) and multilayered rope are presented. The wire rope strands geometric modeling method proposed in this paper is precise enough averting initial geometric overlap between the wires for the benefit of subsequent mechanical computation accuracy and efficiency.

In this study, the spatial quaternionic curve and the relationship between Frenet frames of involute curve of spatial quaternionic curve are expressed by using the angle between the Darboux vector and binormal vector of the basic curve. Secondly, the Frenet vectors of involute curve are taken as position vector and curvature and torsion of obtained Smarandache curves are calculated. The calculated curvatures and torsions are given depending on Frenet apparatus of basic curve. Finally, an example is given and the shapes of these curves are drawn by using Mapple program.

Associated with a Frenet curve $$\alpha $$ in Euclidean 3-space $$\mathbb {E} ^{3} $$ , there exists the notion of natural mate $$\beta $$ of $$\alpha $$ . In this article, we extend the natural mate $$\beta $$ to sequential natural mates $$\{ \alpha _{1},\alpha _{2},\ldots ,\alpha _{n_{\alpha }}\}$$ with $$ \alpha _{1}=\beta $$ . We call each curve $$\alpha _{i},i\in \{1,2,\ldots ,n_{\alpha }\},$$ the i-th natural mate. The main purpose of this article is to study the relationships between the given Frenet curve $$\alpha $$ with its sequential natural mates $$\{ \alpha _{1},\alpha _{2},\ldots ,\alpha _{n_{\alpha }}\}$$ .

We first prove that, unlike the biharmonic case, there exist triharmonic curves with nonconstant curvature in a suitable Riemannian manifold of arbitrary dimension. We then give the complete classification of triharmonic curves in surfaces with constant Gaussian curvature. Next, restricting to curves in a 3-dimensional Riemannian manifold, we study the family of triharmonic curves with constant curvature, showing that they are Frenet helices. In the last part, we give the full classification of triharmonic Frenet helices in space forms and in Bianchi-Cartan-Vranceanu spaces., Comment: To appear in Mediterranean Journal of Mathematics

Tactical decision-making and strategic motion planning for autonomous highway driving are challenging due to predicting other road users' behaviors, diversity of environments, and complexity of the traffic interactions. This paper presents a novel end-to-end continuous deep reinforcement learning approach towards autonomous cars' decision-making and motion planning. For the first time, we define both states and action spaces on the Frenet space to make the driving behavior less variant to the road curvatures than the surrounding actors' dynamics and traffic interactions. The agent receives time-series data of past trajectories of the surrounding vehicles and applies convolutional neural networks along the time channels to extract features in the backbone. The algorithm generates continuous spatiotemporal trajectories on the Frenet frame for the feedback controller to track. Extensive high-fidelity highway simulations on CARLA show the superiority of the presented approach compared with commonly used baselines and discrete reinforcement learning on various traffic scenarios. Furthermore, the proposed method's advantage is confirmed with a more comprehensive performance evaluation against 1000 randomly generated test scenarios. Code: https://github.com/MajidMoghadam2006/RL-frenet-trajectory-planning-in-CARLA

In this paper, we present a hierarchical framework for decision-making and planning on highway driving tasks. We utilized intelligent driving models (IDM and MOBIL) to generate long-term decisions based on the traffic situation flowing around the ego. The decisions both maximize ego performance while respecting other vehicles' objectives. Short-term trajectory optimization is performed on the Frenet space to make the calculations invariant to the road's three-dimensional curvatures. A novel obstacle avoidance approach is introduced on the Frenet frame for the moving obstacles. The optimization explores the driving corridors to generate spatiotemporal polynomial trajectories to navigate through the traffic safely and obey the BP commands. The framework also introduces a heuristic supervisor that identifies unexpected situations and recalculates each module in case of a potential emergency. Experiments in CARLA simulation have shown the potential and the scalability of the framework in implementing various driving styles that match human behavior., Submitted to International Conference on Robotics and Automation (ICRA 2021)

In this paper, we define and characterize pseudo null isophotic curves lying on a non-degenerate surface in Minkowski 3-space. We find the relation between Darboux frame’s Darboux vector (angular velocity vector, centrode) $${\bar{D}}$$ of such curves and Frenet frame’s Darboux vector D. We prove that D spans their axes if and only if it coincides with $${\bar{D}}$$ . In particular, we show that the only pseudo null isophotic curves whose axes are spanned by D are pseudo null helices. Finally, we provide the related examples.

Material particles interact with the working moving surfaces of machines in various technological processes. Mechanics considers a technique to describe the movement of a point and decompose the speed and acceleration into single unit vectors of the accompanying trajectory trihedron for simple movement. The shape of the spatial curve uniquely sets the movement of the accompanying Frenet trihedral as a solid body. This paper has considered the relative movement of a material particle in the static plane of the accompanying Frenet trihedron, which moves along a flat curve with variable curvature. Frenet formulas were used to build a system of differential equations of relative particle movement. In contrast to the conventional approach, the chosen independent variable was not the time but the length of the arc of the guide curve along which the trihedron moves. The system of equations has been built in the projections onto the unit vectors of the moving trihedron; it has been solved by numerical methods. The use of the accompanying curve trihedron as a moving coordinate system makes it possible to solve the problems of the complex movement of a point. The shape of the curve guide assigned by parametric equations in its length function determines the portable movement of the trihedron and makes it possible to use Frenet formulas to describe the relative movement of a point in the trihedron system. This approach enables setting the portable movement of the trihedron osculating plane along a curve with variable curvature, thereby revealing additional possibilities for solving problems on a complex movement of a point at which rotational motion around a fixed axis is a partial case. The proposed approach has been considered using an example of the relative movement of cargo in the body of a truck moving along the road with a curvilinear axis of variable curvature. The charts of the relative trajectory of cargo slip and the relative speed for the predefined speed of the truck have been constructed

In this paper, we study some properties in Minkowski 3-space $$\mathrm {R}_1^3$$ of spacelike Smarandache curves $$\psi $$ regard to Bishop frame of a spacelike curve $$\mu $$ . Form that time, we calculate the Frenet apparatus of $$\psi $$ in $$\mathrm {R}_1^3$$ and that we study some properties of those curves when $$\mu $$ have $${\tau }=0$$ . Finally, we introduced two examples for instance these curves.

In this study, we focus on Smarandache curves which is a special class of curves. These curves have previously been studied by many authors in different spaces. We will re-characterize these curves with the help of an alternative frame different from Frenet frame. Also, we will obtain frame elements, curvature and torsion of these curves.

The behavior of electromagnetic waves in chirally twisted structures is a topic of enduring interest, dating back at least to the invention in the 1940s of the microwave travelling wave tube amplifier and culminating in contemporary studies of chiral metamaterials, metasurfaces, and photonic crystal fibers (PCFs). Optical fibers with chiral microstructures, drawn from a spinning preform, have many useful properties, exhibiting for example circular birefringence and circular dichroism. It has recently been shown that chiral fibers with N fold rotationally symmetric (symmetry group CN) transverse microstructures support families of helical Bloch modes (HBMs), each of which consists of a superposition of azimuthal Bloch harmonics (or optical vortices). An example is a fiber with N coupled cores arranged in a ring around its central axis (N core single ring fiber). Although this type of fiber can be readily modelled using scalar coupled mode theory, a full description of its optical properties requires a vectorial analysis that takes account of the polarization state of the light particularly important in studies of circular and vortical birefringence. In this paper we develop, using an orthogonal two dimensional helicoidal coordinate system embedded in a cylindrical surface at constant radius, a rigorous vector coupled mode description of the fields using local Frenet Serret frames that rotate and twist with each of the N cores. The analysis places on a firm theoretical footing a previous HBM theory in which a heuristic approach was taken, based on physical intuition of the properties of Bloch waves. We believe this study provides clarity in what can sometimes be a rather difficult field, and will facilitate further exploration of real-world applications of these fascinating waveguiding systems., Comment: 20 pages, 10 figures

Generating a comfortable and safe trajectory in the dynamic scenario is an essential issue for autonomous driving. This paper presents an improved theta*-based trajectory planning framework to deal with the obstacle avoidance problem. The road is simplified in the Frenet frame, and an ameliorated Theta* is implemented along with the B-spline curve to generate an optimal path. Then, the velocity of the self-driving car is determined in an ST-graph. Two challenging scenarios with static and moving obstacles are designed, and the simulation results verify the effectiveness of the proposed method.