Clothoid Curves Research Articles

Deformation Analysis of Cosserat Rods Using Piecewise Clothoid Approximation

AbstractThis paper presents an alternative to isogeometric analysis (IGA) for static analysis of planar curved slender beams. Geometrically exact Cosserat rod model is employed to establish the governing equations. In contrast to conventional IGA, which uses NURBS as the basis function, our method incorporates clothoid curves to represent the curved geometry. We use piecewise clothoid curves to approximate the initial curvature of the undeformed beam, facilitating a seamless integration into the Cosserat rod model. A straightforward solution of the governing equations is implemented using shooting method, verifying applicability across a range of problems. Interestingly, the parameters that define the clothoid segments also appear in the governing equations of the beam. This bridges the gap between the geometric design of the beams and their static simulations. In this way, we present the proposed formulation as an alternative to conventional IGA. The notable features of the method are easy implementation, good accuracy, and convergence. Moreover, the method predicts bending stress in the beam, capturing the nonlinearity of the deformation.

Energy-based design optimization of Adept test trajectory for SCARA robots using clothoid curve

Energy-based design optimization of Adept test trajectory for SCARA robots using clothoid curve

Relaxation of Stress Concentration by Applying Clothoid Curve at the Intersection of Straight and Curved Sections in REBCO Racetrack Coil

Relaxation of Stress Concentration by Applying Clothoid Curve at the Intersection of Straight and Curved Sections in REBCO Racetrack Coil

The effect of path curvature on the stability of reversing truck–semitrailer in the presence of feedback delay

Stability analysis of the low-speed reverse motion along clothoid curve is presented for the truck–semitrailer combination. The single-track kinematic model supplemented with the steering system and the lower-level controller is applied. A higher-level controller is designed to stabilise the reverse motion of the vehicle. Linear state feedback with time delay is used with feedforward term to force the vehicle to the desired path. The linear stability analysis is done by semi-discretization and D-subdivision methods to determine the optimal control gain setup. The effects of path curvature and time delay on the stability are investigated and verified via numerical simulations. Results are validated via experiments using a small-scale test rig.

SEARCH OF OPTIMAL SPATIAL PARAMETERS FOR GEODETIC DELINEATION OF MOTORWAY INTERCHANGES WITH BRAKING TRANSITION CURVES

The most widespread grade-separated road junction — cloverleaf interchange — is a complex engineering structure, in which all the parameters and limitations of the plan and longitudinal profile must be coordinated with each other. The goals of the analysis: traffic safety, technical and economic indicators of the movement of transport units, the compactness of left-turn and right-turn exits and, as an additional, but extremely important aspect, the minimization of the area occupied by the entire transport structure, which is very important in tight urban conditions. 
 First, let's emphasize that the trajectories of cars at the intersection must correspond to their real variable speeds on left-turn exits, since this is traffic on curves of small radii. In our previous studies, it has been experimentally proven that cars entering a left-turning exit reduce their speed, and conversely, when leaving the exit, their speed increases. Therefore, it is necessary to use braking curves as transition curves of left-turn exits, and not clothoid curves, which are often used, and which are designed for constant speed of movement.
 Some braking transition curves designed for variable speed are also used in traffic junctions [4]. But, analyzing these curves, we came to the conclusion that they do not meet the classical purpose of the transition curve, namely, they do not provide the necessary radius at the end of the curve, which is equal to the radius of the circular insert for a smooth transition from a transition curve to a circular curve. This leads to dangerous moments of the car's movement at the junction of the transition and circular curves.
 Therefore, we previously derived the formulas of refined braking transition curves that meet the requirements of the radii of classical transition curves [1, 2]. In addition, it later turned out that these refined transition curves have, as an additional effect, the property of reducing the area occupied by the left-turn exit, as shown in Fig. 1.
 We offer a method for optimizing the spatial parameters of cloverleaf interchanges with refined braking transition curves.

Smooth Autonomous Patrolling for a Differential-Drive Mobile Robot in Dynamic Environments.

Today, mobile robots have a wide range of real-world applications where they can replace or assist humans in many tasks, such as search and rescue, surveillance, patrolling, inspection, environmental monitoring, etc. These tasks usually require a robot to navigate through a dynamic environment with smooth, efficient, and safe motion. In this paper, we propose an online smooth-motion-planning method that generates a smooth, collision-free patrolling trajectory based on clothoid curves. Moreover, the proposed method combines global and local planning methods, which are suitable for changing large environments and enabling efficient path replanning with an arbitrary robot orientation. We propose a method for planning a smoothed path based on the golden ratio wherein a robot's orientation is aligned with a new path that avoids unknown obstacles. The simulation results show that the proposed algorithm reduces the patrolling execution time, path length, and deviation of the tracked trajectory from the patrolling route compared to the original patrolling method without smoothing. Furthermore, the proposed algorithm is suitable for real-time operation due to its computational simplicity, and its performance was validated through the results of an experiment employing a differential-drive mobile robot.

Physical Consistent Path Planning for Unmanned Surface Vehicles under Complex Marine Environment

The increasing demand for safe and efficient maritime transportation has underscored the necessity of developing effective path-planning algorithms for Unmanned Surface Vehicles (USVs). However, the inherent complexities of the ocean environment and the non-holonomic properties of the physical system have posed significant challenges to designing feasible paths for USVs. To address these issues, a novel path planning framework is elaborately designed, which consists of an optimization model, a meta-heuristic solver, and a Clothoid-based path connector. First, by encapsulating the intricate nature of the ocean environment and ship dynamics, a multi-objective path planning problem is designed, providing a comprehensive and in-depth portrayal of the underlying mechanism. By integrating the principles of the candidate set random testing initialization and adaptive probability set, an enhanced genetic algorithm is devised to fully exploit the underlying optimization problem in constrained space, contributing to the global searching ability. Accounting for the non-holonomic constraints, the fast-discrete Clothoid curve is capable of maintaining and improving the continuity of the path curve, thereby promoting strong coordination between the planning and control modules. A thorough series of simulations and comparisons conducted in diverse ocean scenarios has conclusively demonstrated the effectiveness and superiority of the proposed path planning framework.

Clothoid-Based Path Planning for a Formation of Fixed-Wing UAVs

Unmanned aerial vehicles (UAVs) are playing an increasingly crucial role in many applications such as search and rescue, delivery services, and military operations. However, one of the significant challenges in this area is to plan efficient and safe trajectories for UAV formations. This paper presents an optimization procedure for trajectory planning for fixed-wing UAV formations using graph theory and clothoid curves. The proposed planning strategy consists of two main steps. Firstly, the geometric optimization of paths is carried out using graphs for each UAV, providing piece-wise linear paths whose smooth connections are made with clothoids. Secondly, the geometric paths are transformed into time-dependent trajectories, optimizing the assigned aircraft speeds to avoid collisions by solving a mixed-integer optimal control problem for each UAV of the flight formation. The proposed method is effective in achieving suboptimal paths while ensuring collision avoidance between aircraft. A sensitivity analysis of the main parameters of the algorithm was conducted in ideal conditions, highlighting the possibility of decreasing the length of the optimal path by about 4.19%, increasing the number of points used in the discretization and showing a maximum path length reduction of about 10% compared with the average solution obtained with a similar algorithm using a graph based on random directions. Furthermore, the use of clothoids, whose parameters depend on the UAV performance constraints, provides smoother connections, giving a significant improvement over traditional straight-line or circular trajectories in terms of flight dynamics compliance and trajectory tracking capabilities. The method can be applied to various UAV formation scenarios, making it a versatile and practical tool for mission planning.

Road Shape Classification-Based Matching Between Lane Detection and HD Map for Robust Localization of Autonomous Cars

Many map matching-based localization algorithms estimate the autonomous car's pose using a registration between lanes measured by a camera and the High-Definition (HD) map. However, the registration methods based on numerical optimization, such as the Iterative Closest Point (ICP) and Normal Distributions Transform (NDT), can cause underdetermined problems due to no unique solution when matching under-constrained lane shapes such as straight lines and circular arcs. This paper proposes a robust localization algorithm with centimeter-level accuracy by applying different matching techniques depending on the road shape. We proposed a road shape classification-based map matching algorithm to overcome the under-constrained problems, which have no unique solution due to insufficient constraints. The proposed algorithm classifies lane segments into line, arc, and clothoid curves considering their curvature characteristics. After that, we find correction information through a map matching and covariance estimation method using lane pairs with the same type of their shape. For the under-constrained shapes, the geometry-based map-matching algorithm and covariance estimation method are applied to avoid underdetermined results. Finally, the measurement calculated from the correction information and predicted pose of the egovehicle is exploited for the measurement update of the Extended Kalman Filter (EKF). The proposed method was quantitatively evaluated in the simulation environment, which contains various road shapes, and qualitatively validated for experimental data from an autonomous driving platform. The proposed algorithm shows more robust matching capability, efficient computation time, and higher accuracy than the localization system based on ICP-based lane matching.

Clothoid-Based Lane-Level High-Definition Maps: Unifying Sensing and Control Models

Autonomous vehicles rely on lane-level high-definition (HD) maps for self-localization and trajectory planning. Current mapping, however, relies on simple line models, while clothoid curves have unexplored potential. Clothoids, well known in road design, are often chosen to model the vehicle trajectory in planning and control systems as they describe the road with higher fidelity. For this reason, we propose two vision-based pipelines for generating lane-level HD maps using clothoid models. The first pipeline performs mapping with known poses, requiring precise real-time kinematics GPS (RTK GPS) measurements; the second copes with noisy localizations, solving the simultaneous localization and mapping (SLAM) problem. Both pipelines rely on a line detection algorithm to identify each line marking and perform a graph-based optimization to estimate the map.

Underwater Submarine Path Planning Based on Artificial Potential Field Ant Colony Algorithm and Velocity Obstacle Method.

Navigating safely in complex marine environments is a challenge for submarines because proper path planning underwater is difficult. This paper decomposes the submarine path planning problem into global path planning and local dynamic obstacle avoidance. Firstly, an artificial potential field ant colony algorithm (APF-ACO) based on an improved artificial potential field algorithm and improved ant colony algorithm is proposed to solve the problem of submarine underwater global path planning. Compared with the Optimized ACO algorithm proposed based on a similar background, the APF-ACO algorithm has a faster convergence speed and better path planning results. Using an inflection point optimization algorithm greatly reduces the number and length of inflection points in the path. Using the Clothoid curve fitting algorithm to optimize the path results, a smoother and more stable path result is obtained. In addition, this paper uses a three-dimensional dynamic obstacle avoidance algorithm based on the velocity obstacle method. The experimental results show that the algorithm can help submarines to identify threatening dynamic obstacles and avoid collisions effectively. Finally, we experimented with the algorithm in the submarine underwater semi-physical simulation system, and the experimental results verified the effectiveness of the algorithm.

Potential field‐based path planning for emergency collision avoidance with a clothoid curve in waypoint tracking

AbstractA novel potential field‐based model curve fitting method (PF‐MCF) is presented in this paper to handle emergency collision avoidance in waypoint tracking (following waypoints from a leading vehicle). It is reported that the PF has high performance on real‐time obstacle avoidance in robotic path planning. However, some inherent limitations exist in the PF, such as no passage between closely spaced obstacles and oscillations in the presence of obstacles, which is rarely discussed in the research field of PF‐based autonomous vehicles (AVs), especially in waypoint tracking. To solve these problems, we propose the PF‐MCF method to transform the clothoid curve into a quadratic programming form and solve the optimization under reasonable constraints to consider the different waypoints from the PF and the leading vehicle. The proposed PF‐MCF method is validated via MATLAB/Simulink and CarSim to avoid multiple obstacle vehicles in waypoint tracking compared with the two latest methods (PF‐based model predictive controller and PF‐based curve fitting method). The simulation results confirm the efficiency of finding a passage in closely spaced obstacle vehicles and eliminating oscillations (under 0.5 [deg] in tire steering angle) in the presence of obstacles compared to other latest methods.

Out-of-Plane Free Vibration Analysis of Continuous Curved Girders with Combined Linetypes Using Differential Quadrature Element Method

In this study, the out-of-plane free vibration equations of circular curved and clothoid transition curved girders with various boundary conditions were derived. The continuous curved girder with combined linetypes was discretized into separate curved girder elements. Based on the differential quadrature element method (DQEM), the Chebyshev–Gauss–Labatto unequal mesh division was applied to discretize the vibration equations of the continuous curved girder into those of the curved girder elements, along with the boundary conditions, internal geometric compatibility conditions, and force balance conditions. The out-of-plane natural frequency equations were derived considering the boundary conditions by the substitution method. The free vibration was solved for continuous curved girders with combined linetypes of circular and clothoid transition curves. The effectiveness of the DQEM was verified, and the effect of the number of grid points on the accuracy of the solution was evaluated. In addition, the effects of the bending/torsional stiffness ratio, warping coefficient, and boundary conditions on the natural frequencies of continuous curved girders with combined linetypes were investigated. The results showed that the DQEM can be efficiently used to solve the free vibration of multispan continuous curved girders with various linetype combinations. The frequencies of the continuous curved girder with combined linetypes decrease with the weakening of boundary constraints, but increase with increasing warping coefficient and bending/torsional stiffness ratio.

Closed form parametrisation of 3D clothoids by arclength with both linear varying curvature and torsion

The extension from the planar case to three dimensions of the clothoid curve (Euler spiral) is herein presented, that is, a curve parametrised by arc length, whose curvature and torsion are linear (affine) functions of the arc length. The problem is modelled as a linear time variant system and its stability is studied with Lyapunov techniques. Closed form solutions in terms of the standard Fresnel integrals are for the first time presented and they are valid for a wide family of clothoids; for the remaining cases, numerical methods of order four, based on Lie algebra, Magnus Expansions and Commutator-Free Expansions are provided. These geometric integrators have been optimised to be easily implemented with few lines of code and require only matrix-vector products (avoiding explicit matrix exponentials at each iteration). The achieved computational efficiency has a dramatic impact for several real-time applications, especially for designing trajectories for flying vehicles, as previous approaches were fully numeric and could not take advantage of the present closed form solutions, which render the computation of the 3D clothoid as computationally expensive as its planar companion.

Mathematical Modelling of Vertical Looped Water Slide Using Matlab

Abstract: A conceptual mathematical model of a water slide with vertical loops is developed. The principle used is the conservation of energy. The thrill experienced by a rider on a water slide is mainly due to the variation of G-force acting on the rider through the course of the ride. The geometry of the slide is developed by plotting G-force variation with the arc length of the loop. The G-force exposure limits should meet with the standards set by the F24 committee on amusement parks and rides. The coordinates of the slide geometry are determined by using Euler’s method of discretized equations. Keywords: G-Force, Centripetal acceleration, Clothoid curve, Weightlessness, Potential Energy, Kinetic Energy

Clothoid curve optimization for parallel parking path planning and tracking control

In order to solve the problem of abrupt curvature change at the connection between arcs and straight lines in circle-line-circle (C-L-C) combined parallel parking paths, curvature optimization was carried out by using a cycloid curve, and the trajectory curvature and heading Angle were made to meet the pose requirements of parallel parking. By establishing the kinematics model state equation and taking the optimized C-L-C trajectory as the reference trajectory, the error model was obtained. The model predictive controller based on the error model was designed, and the effectiveness of the path planning and model predictive controller was verified on Carsim and Matlab/Simulink co-simulation platform.

Optimal Smooth Paths Based on Clothoids for Car-like Vehicles in the Presence of Obstacles

Automated Guided Vehicles are increasingly used for material transfer in factory and warehouse environments amongst humans and human operated vehicles. Safe and efficient operation is challenging when there is a mix of human and automated traffic as fixed guide paths can become blocked more frequently. In this work we aim to show smooth and efficient paths based on clothoid curves can be used to automatically plan diversions which can be traversed at high speed by automated fork-lift vehicles, which are car-like in the sense they have a limited turning radius and angular acceleration. The approach, based on numerical optimisation within convex region constraints is described in detail, and numerical results for curvature and sharpness are compared to a cubic spline on a small number of simulated environments. The clothoid spline is less affected, in terms of its objective function, by a shift in the obstacle boundaries than a cubic spline, for obstacle shifts below 0.5m. The clothoid spline takes longer to converge for but the output path has attractive qualities like lower peak sharpness, enabling high speed operation. The method is therefore most useful for applications where path quality is important and updates are required less frequently. Changing the objective function by increasing weighting parameter b allowed the path shape to be tuned to reduce the peak sharpness, at the cost of increasing the total length. With b > 100, convergence was poor because parts of the spline were pushed outside the assigned region, an artefact arising from the constraints only being enforced at the start and end of each segment. The analytical Jacobian of the constraints was effective at reducing the number of function evaluations to reach convergence.

Evaluation of Safety in Horizontal Curves of Roads Using a Multi-Body Dynamic Simulation Process.

Road transportation poses one of the significant public health risks. Several contributors and factors strongly link public health and road safety. The design and advancement of higher-quality roads can significantly contribute to safer roads and save lives. In this article, the safety aspect of the roads’ horizontal curves under the standard of the American Association of State Highway Transportation Officials (AASHTO) is evaluated. Several factors, including vehicle weight, vehicle dimensions, longitudinal grades, and vehicle speed in the geometric design of the horizontal curves, are investigated through a multi-body dynamic simulation process. According to the AASHTO, a combination of simple circular and clothoid transition curves with various longitudinal upgrades and downgrades was designed. Three vehicles were used in this simulation, including a sedan, a bus, and a 3-axle truck. The analysis was based on the lateral friction between the tire and the pavement and also the safety margin parameter. The results showed that designers must differentiate between light and heavy vehicles, especially in curves with a high radius. Evaluation of longitudinal grade impacts indicated that the safety margin decreases when the vehicle is entering the curve. Safety margin reduction on the clothoid curve takes place with a lower grade toward the simple circular curve. By increasing the speed, the difference between lateral friction demand obtained from simulation and lateral friction demand proposed by AASHTO grows. The proposed novel methodology can be used for evaluating road safety.

Exploring Benefits of Using Blending Splines as Transition Curves

Track geometry is a fundamental subject in railway construction. With the demand for increased capacity in terms of load and speed, the need for suitable transitions between consecutive track sections is highly relevant. Properly constructed transition curves lead to improved travel comfort, increased safety, and reduced wear. The well known clothoid curve is widely used as a transition curve; however, the linear curvature is not sufficiently smooth to meet the requirements for railways carrying high speed trains or heavy hauls. Blending spline curves are flexible spline constructions possessing favourable smoothness properties at the end points, which makes them considerable for use as transition curves. This paper demonstrates some selected blending splines applied as transition curves between two existing circular arc segments selected from the Ofotbanen railway. The main results in this paper are related to the smoothness at the end points and the behaviour of the curvature of the curves, where the new transition curves were shown to be smoother than the original clothoid. Another new result is the observation that the proposed method allows for the improvement of existing railways without forcing extensive changes to the original track. Some representative examples are included to highlight the flexibility of this first instance of blending splines as transition curves.

Greyhound racing ideal trajectory path generation for straight to bend based on jerk rate minimization

This paper presents methods for modelling and designing an ideal path trajectory between straight and bend track path segments for racing greyhounds. To do this, we numerically generate clothoid and algebraic curve segments for racing quadrupeds using a sequential vector transformation method as well as using a helper equation for approaching ideal clothoid segments that would respect greyhound kinematic parameters and boundary conditions of the track. Further, we look into the limitations of using a clothoid curve for racing dog track path design and propose a smooth composite curve for track transition design which roughly maintains G3 curvature continuity for smooth jerk to overcome limitations of a clothoid transition. Finally, we show results from race data modelling and past injury data, which provide a strong indication of clothoid curve segments improving the dynamics and safety of racing greyhounds while reducing injuries.