Binormal Research Articles

Trajectory Planning of Autonomous Vehicles Based on Parameterized Control Optimization in Dynamic on-Road Environments

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.

Adaptive leader–follower formation control for swarms of unmanned aerial vehicles with motion constraints and unknown disturbances

In this paper, the 3D leader–follower formation control problem, which focuses on swarms of fixed-wing Unmanned Aerial Vehicles (UAVs) with motion constraints and disturbances, has been investigated. Original formation errors of the follower UAVs have been transformed into the Frenet-Serret frame. Formation control laws satisfying five motion constraints (i.e., linear velocity, linear acceleration, heading rate, climb rate and climb angle) have been designed. The convergence of the control laws has been discussed via the Lyapunov stability tool. In addition, to address the unknown disturbances, an adaptive disturbance observer is exploited. Furthermore, formation control laws involving estimated disturbances are presented as well. The collision avoidance between UAVs is achieved with the artificial potential method. Simulation results obtained using four scenarios verify the effectiveness of the proposed method in situations with constant disturbances and varying disturbances, as well as without disturbances.

Null (Lightlike) $f$-Rectifying Curves in the Three Dimensional Minkowski Space $\mathbb{E}^3_1$

A rectifying curve $\gamma$ in the Euclidean $3$-space $\mathbb{E}^3$ is defined as a space curve whose position vector always lies in its rectifying plane (i.e., the plane spanned by the unit tangent vector field $T_\gamma$ and the unit binormal vector field $B_\gamma$ of the curve $\gamma$), and an $f$-rectifying curve $\gamma$ in the Euclidean $3$-space $\mathbb{E}^3$ is defined as a space curve whose $f$-position vector $\gamma_f$, defined by $\gamma_f(s) = \int f(s) d\gamma$, always lies in its rectifying plane, where $f$ is a nowhere vanishing real-valued integrable function in arc-length parameter $s$ of the curve $\gamma$. In this paper, we introduce the notion of $f$-rectifying curves which are null (lightlike) in the Minkowski $3$-space $\mathbb{E}^3_1$. Our main aim is to characterize and classify such null (lightlike) $f$-rectifying curves having spacelike or timelike rectifying plane in the Minkowski $3$-Space $\mathbb{E}^3_1$.

Normal Partner Curves of a Pseudo Null Curve on Dual Space Forms

In this work, a kind of normal partner curves of a pseudo null curve on dual space forms is defined and studied. The Frenet frames and curvatures of a pseudo null curve and its associate normal curve on de-Sitter space, its associate normal curve on hyperbolic space, are related by some particular function and the angles between their tangent vector fields, respectively. Meanwhile, the relationships between the normal partner curves of a pseudo null curve are revealed. Last but not least, some examples are given and their graphs are plotted by the aid of a software programme.

Some Characterizations of Curves in n-dimensional Euclidean Space

In this work, we deal with a curve whose position vector can be expressed with the help of Frenet Frame in  n dimensional Euclidean space n IE . We classify this type of curve with regards to curvature functions and get certain consequences for T constant,  N constant and constant ratio curves in n IE .

Generalized involute and evolute curves of equiform spacelike curves with a timelike equiform principal normal in E_{1}^{3}

Equiform geometry is considered as a generalization of the other geometries. In this paper, involute and evolute curves are studied in the case of the curve α is an equiform spacelike with a timelike equiform principal normal vector N. Furthermore, the equiform frames of the involute and evolute curves are obtained. Also, the equiform curvatures of the involute and evolute curves are obtained in Minkowski 3-space.

F−BIHARMONIC CURVES WITH TIMELIKE NORMAL VECTOR ON LORENTZIAN SPHERE

In this paper, we study $f-$biharmonic curves as the critical points of the $f-$bienergy functional $E_{2}(\psi )=\int_{M}f\mid \tau (\psi )^{2}\mid \vartheta _{g}$, on a Lorentzian para-Sasakian manifold $M$. We give necessary and sufficient conditions for a curve such that has a timelike principal normal vector on lying a $4$-dimensional conformally flat, quasi-conformally flat and conformally symmetric Lorentzian para-Sasakian manifold to be an $f-$biharmonic curve. Moreover, we introduce proper $f-$biharmonic curves on the Lorentzian sphere $S_{1}^{4}.$

Integral vector field control for three-dimensional path following of autonomous underwater vehicle

A new control method for autonomous underwater vehicle (AUV) is developed to achieve three-dimensional (3D) path following under ocean current disturbances when AUV lacks lateral and vertical driving forces. To make the control system more convenient for practical application and reduce the energy consumption of AUV, the controller is designed with relative velocities. To avoid the singularity problem of curve path following, the Serret–Frenet frame is introduced as virtual target and the error model is defined. The adaptive law of the virtual target and an integral vector field (IVF) guidance law are designed in the kinematic controller. The back-stepping method and adaptive dynamical sliding mode control (BADSMC) technology are applied in the design of dynamic controller. The Lyapunov theory and nonlinear cascade system theory are applied to prove the closed-loop stability. Finally, the performances of the IVF + BADSMC and ILOS + PID algorithm are compared through simulation. Simulation results demonstrate that the proposed controller can realize the path following of AUV under disturbances of ocean current and can improve the following quality.

On Bishop frame of a pseudo null curve in Minkowski space-time

In this paper, we introduce the Bishop frame of a pseudo null curve $\alpha$ in Minkowski space-time. We obtain the Bishop frame's equations and the relation between the Frenet frame and the Bishop frame. We find the third order nonlinear differential equation whose particular solutions determine the form of the Bishop curvatures. By using space-time geometric algebra, we derive the Darboux bivectors $D$ and $\tilde{D}$ of the Frenet and the Bishop frame of $\alpha$, respectively. We give geometric interpretations of the Frenet and the Bishop curvatures of $\alpha$ in terms of areas of the projections of the corresponding Darboux bivectors onto the planes spanned by the frame vector's fields.

Bertrand and Mannheim curves of framed curves in the3-dimensional Euclidean space

A Bertrand curve is a space curve whose principal normal line is the same as the principal normal line of another curve. On the other hand, a Mannheim curve is a space curve whose principal normal line is the same as the binormal line of another curve. By definitions, another curve is a parallel curve with respect to the direction of the principal normal vector. Even if that is the regular case, the existence conditions of the Bertrand and Mannheim curves seem to be wrong in some previous research. Moreover, parallel curves may have singular points. As smooth curves with singular points, we consider framed curves in the Euclidean space. Then we define and investigate Bertrand and Mannheim curves of framed curves. We clarify that the Bertrand and Mannheim curves of framed curves are dependent on the moving frame.

Classification of rotational surfaces with constant skew curvature in 3-space forms

In this paper, we classify the rotational surfaces with constant skew curvature in 3-space forms. We also give a variational characterization of the profile curves of these surfaces as critical points of a curvature energy involving the exponential of the curvature of the curve. Finally, we provide a converse process to produce all rotational surfaces with constant skew curvature based on the evolution of a critical curve under the flow of its binormal vector field with prescribed velocity.

Mixed-type curves and the lightcone frame in Minkowski 3-space

In this paper, we study mixed-type curves in Minkowski 3-space. Mixed-type curves are regular curves, and there are both non-lightlike points and lightlike points in a mixed-type curve. For non-lightlike curves and null curves in Minkowski 3-space, we can study them by a Frenet frame or a Cartan frame, respectively. But for mixed-type curves, the two frames will not work. As far as we know, no one has yet given a frame to study them in Minkowski 3-space. So, we give the lightcone frame in order to provide a tool for studying this type curves in mathematical and physical research. As an application of the lightcone frame, we define an evolute of a mixed-type curve. We also give some examples to show the evolutes.

Differential Equations of Null Quaternionic Curves

In this study, we investigate the differential equations of null quaternionic curves in Minkowski 3-space $$E_1^3$$ and Minkowski space-time $$E_1^4$$ according to components of Frenet frames. For some examples and special conditions, we get differential equations of null quarternionic curves.

The weak Frenet frame of non-smooth curves with finite total curvature and absolute torsion

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.

Invariant isogeometric formulations for three-dimensional Kirchhoff rods

In the isogeometric formulation for three-dimensional Kirchhoff rods, the invariance is required such that rigid-body modes associated with translations and infinitesimal rotations can be exactly represented. By enforcing the invariance in the isogeometric formulation of the rod, the equilibrium of forces and moments at fixed ends is preserved, and the performance is improved considerably for elements of finite sizes. We recognize that not all existing isogeometric formulations are invariant for general geometry of rods. Though an alternative formulation based on the Frenet–Serret frame and rotation angle representation was shown to be invariant, it does not work for rods with vanishing curvature, while its accuracy is affected by the low continuity of the angle of twist used. Motivated by this, two new invariant isogeometric formulations are developed, through different interpolations for the angle of twist. One uses the derivatives of the NURBS functions as the basis for the angle-of-twist interpolation, and the other uses a mixed interpolation incorporating both the Lagrange and NURBS functions. Several representative numerical examples were prepared to verify the accuracy and robustness of the two new formulations. Particularly, for rods of variable curvatures simulated by elements of finite or large sizes, the proposed formulations are demonstrated to outperform the two existing ones mentioned in predicting the displacements, end reactions and stress resultants.

High Definition Map Based Motion Plan and Control of Autonomous Vehicle on Structured Road

A high definition (HD) map based motion plan and control method of autonomous vehicle on structured road is proposed in this paper. The system is designed in a multi-layer structure: a motion planner and a motion controller, furthermore, the motion planner is consisted of a global path planner and a local trajectory planner. The inputs of the global path planner are HD map, start and goal state, Dynamic Program (DP) is used to plan a global path that connect these two states based on the lane-level structures and traffic rules extracted from HD map. The inputs of local motion planner are the planned global path, current host vehicle state and the surrounding dynamic object information, several candidate trajectories are generated in Frenet frame, and then the optimal one is chosen by a multi-term cost function calculation. Taking the vehicle dynamics under high speed into consideration, Model Predictive Control (MPC) is used in motion controller to track the planned target trajectory. Results of the conducted simulation tests show that the designed motion plan and control method works well in velocity keeping, vehicle following, stopping and going, and vehicle passing scenarios.

Evolution of quaternionic curve in the semi‐Euclidean space E24

In this paper, kinematics of semi‐real quaternionic curve in semi‐Euclidean space is obtained in terms of its curvature functions. The evolution equation of Frenet frame and curvatures of quaternionic curve are obtained. Also, examples of evolution equations of curvatures are given.

On generalized null Mannheim curves in E24

In the present paper, we introduce the notion of Cartan‐framed generalized null Mannheim curves in four‐dimensional semi‐Euclidean space with index 2, . We obtain the Cartan (or Frenet) frames and curvature functions of the generalized Mannheim mate curve by the help of curvature functions and Cartan frames of the generalized null Mannheim curve. We proved that there are no generalized null Mannheim curves with timelike principal normal N in . The related examples and their projected images are also given.

Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces

In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Euclidean and Lorentzian geometries, here we show how to build RM frames in isotropic geometries and apply them in the study of isotropic spherical curves. Indeed, through a convenient manipulation of osculating spheres described in terms of RM frames, we show that it is possible to characterize spherical curves via a linear equation involving the curvatures that dictate the RM frame motion. For the case of pseudo-isotropic space, we also discuss on the distinct choices for the absolute figure in the framework of a Cayley-Klein geometry and prove that they are all equivalent approaches through the use of Lorentz numbers (a complex-like system where the square of the imaginary unit is $+1$). Finally, we also show the possibility of obtaining an isotropic RM frame by rotation of the Frenet frame through the use of Galilean trigonometric functions and dual numbers (a complex-like system where the square of the imaginary unit vanishes).

Separation method of bending and torsion in shape sensing based on FBG sensors array.

This paper presents a theoretical method for separating bending and torsion of shape sensing sensor to improve sensing accuracy during its deformation. We design a kind of shape sensing sensor by encapsulating three fibers on the surface of a flexible rod and forming a triangular FBG sensors array. According to the configuration of FBG sensors array, we derive the relationship between bending curvature and bending strain, and set up a function about the packaging angle of FBG sensor and strain induced by torsion under different twist angles. Combined with the influence of bending and torsion on strain, we establish a nonlinear matrix equation resolving three unknown parameters including maximum strain, bending direction and wavelength shift induced by torsion and temperature. The three parameters are sufficient to separate bending and torsion, and acquire two scalar functions including curvature and torsion, which could describe 3D shape of rod according to Frenet-Serret formulas. Experimental results show that the relative average error of measurement about maximum strain, bending direction is respectively 2.65% and 0.86% when shape-sensing sensor is bent into an arc with a radius of 260 mm. The separating method also applied to 2D shape and 3D shape of reconstruction, and the absolute spatial position maximum error is respectively 3.79mm and 11.10mm when shape-sensing sensor with length 500mm is bent into arc shape with a radius 260mm and helical curve. The experiment results verify the feasibility of separating method, which would provide effective parameters for precise 3D reconstruction model of shape sensing sensor.