Shooting Method Research Articles

Simultaneous localization, mapping, and path planning for unmanned vehicle using optimal control

Among the huge number of functionalities that are required for autonomous navigation, the most important are localization, mapping, and path planning. In this article, investigation of the path planning problem of unmanned ground vehicle is based on optimal control theory and simultaneous localization and mapping. A new approach of optimal simultaneous localization, mapping, and path planning is proposed. Our approach is mainly affected by vehicle’s kinematics and environment constraints. Simultaneous localization, mapping, and path planning algorithm requires two main stages. First, the simultaneous localization and mapping algorithm depends on the robust smooth variable structure filter estimate accurate positions of the unmanned ground vehicle. Then, an optimal path is planned using the aforementioned positions. The aim of the simultaneous localization, mapping, and path planning algorithm is to find an optimal path planning using the Shooting and Bellman methods which minimizes the final time of the unmanned ground vehicle path tracking. The simultaneous localization, mapping, and path planning algorithm has been approved in simulation, experiments, and including real data employing the mobile robot Pioneer [Formula: see text]. The obtained results using smooth variable structure filter–simultaneous localization and mapping positions and the Bellman approach show path generation improvements in terms of accuracy, smoothness, and continuity compared to extended Kalman filter–simultaneous localization and mapping positions.

Periodic orbits of diffusionless Lorenz system

The strange attractor of a chaotic system is composed of numerous periodic orbits densely covered. The periodic orbit is the simplest invariant set except for the fixed point in the nonlinear dynamic system, it not only reflects all the characteristics of the chaotic motion, but also is closely related to the amplitude generation and change of chaotic system. Therefore, it is of great significance to obtain the periodic orbits in order to analyze the dynamical behaviors of the complex system. In this paper, we study the periodic orbits of the diffusionless Lorenz equations which are derived in the limit of high Rayleigh and Prandtl numbers. A new approach to establishing one-dimensional symbolic dynamics is proposed, and the periodic orbits based on a topological structure are systematically calculated. We use the variational method to locate the cycles, which is proposed to explore the periodic orbits in high-dimensional chaotic systems. The method not only preserves the robustness characteristics of most of other methods, such as the Newton descent method and multipoint shooting method, but it also has the characteristics of fast convergence when the search process is close to the real cycle in practice. In order to apply the method, a rough loop guess must be made first based on the entire topology for the cycle to be searched, and then the variational algorithm will bring the initial loop guess to evolving toward the real periodic orbit in the system. In the calculations, the Newton descent method is used to achieve stability. Two cycles can be used as basic building blocks for initialization, searching for more complex cycles with multiple circuits around the two fixed points requires more delicate initial conditions; otherwise, it will probably lead to nonconvergence. We can initialize the loop guess for longer cycles constructed by cutting and gluing the short, known cycles. For this system, such a method yields quite a good systematic initial guess for longer cycles. Even if we deform the orbit manually into a closed loop, the variational method still shows its powerfulness for good convergence. The topological classification based on the entire orbital structure is shown to be effective. Furthermore, the deformation of periodic orbits with the change of parameters is discussed, which provides a route to the periods of cycles. The present research may provide a method of performing systematic calculation and classification of periodic orbits in other similar chaotic systems.

Maksimalna sila stiska šake u funkciji preciznosti i tačnosti gađanja iz službenog pištolja CZ 99 - generički modeli

Gun training with official weapons is one of the most important training activities in the police. The aim of this paper is to determine the relations and define the prediction model between the accuracy and precision of pistol shooting, as a specific motoric task, and the maximum muscular strength of the hand grip, as a basic indicator of general physical hand preparation. The sample covered a total of 46 subjects: students of the Faculty of Sport and Physical Education who selected the subject Shooting within the study program (N = 19, 11 male students and 8 female students), active Ministry of the Interior officers (N = 6), competitors in practical shooting, IPSC (N = 4), recreational shooters (N = 10, 8 men and 2 women), and beginners (N = 7) who have only passed one hour of basic training in firearms. For the measurement of the maximal muscle force of the hand finger flexor, a standardized test - 'grip of the hand' was used while the shooting was carried out from the CZ 99 handgun, using the precision shot method from a distance of 6 meters. Criterion variables were: accuracy and precision of shooting, while the predictive variables were the maximum forces of the left and right hand, as well as a summary indicator of the achieved maximum force level of both hands. Based on the obtained results, it can be concluded that the accuracy of shooting statistically correlates with the following variables: inversely proportional to the accuracy (r = -0.889, p = 0.000) and proportional to all variables of the maximum force of hand grip, as measures of the strength of the upper extremities ( Fmax_R, Fmax_L and Fmax_SUM at r = 0.437, p = 0.002, r = 0.439, p = 0.002 ir = 0.443, p = 0.002, respectively). In relation to the accuracy of the shooting, a statistically significant but inversely proportional correlation with all variables of the maximum force of the hand grip were determined (Fmax_R, Fmax_L and Fmax_SUM at r = -0.356, p = 0.015, r = -0.375, p = 0.010 ir = -0.369, p = 0.012, respectively). In general, based on the obtained results of this study, it can be concluded that the maximum force of the hand grip, as an indicator of arm strength, is statistically significant, and at the level of influence of 19.3% positively influences the accuracy, i.e. with 13.4% positively affects the accuracy of the shooting from the CZ 99 official handgun in a precision shooting situation from a distance of 6 meters.

Multiple slips effects on MHD SA-Al2O3 and SA-Cu non-Newtonian nanofluids flow over a stretching cylinder in porous medium with radiation and chemical reaction

The purpose of this communication is to examine the collective influence of velocity, and thermal slips on magnetohydrodyanmics (MHD) SA-Al2O3 and SA-Cu non-Newtonian nanofluids flow over a stretching cylinder in porous medium together with thermal radiation and chemical reaction effects. Sodium Alginate (SA-NaAlg) is taken as non-Newtonian base fluid. Two types of nanoparticles alumina or aluminum oxide (Al2O3) and copper (Cu) are suspended in sodium alginate (SA) which is taken as base fluid, an example of non-Newtonian Casson fluid. The formulated nonlinear partial differential equations with auxiliary boundary conditions are transformed into non-dimensional form by applying suitable similarity transformations. The resulting dimensionless problem is solved numerically using shooting and fourth order Runge-Kutta method. The impacts of various thermophysical parameters on local skin-friction, local Nusselt number, temperature and velocity are analyzed through graphs as well as in tabular form and discussed in detail. A comparison between SA-Al2O3 and SA-Cu nanofluids is clearly shown and in limiting sense the present results are compared with published results from literature. The results show that with magnetic parameter, skin-friction and Nusselt number both decreased, and Nusselt numbers are the highest in case of Al2O3 than Cu nanoparticles.

Closed-Form Solution of Large Deflected Cantilever Beam a gainst Follower Loading Using Complex Analysis

The literature reveals that the non-conservative deflection of an elastic cantilever beam caused by applying follower tip loading was investigated and solved by various numerical methods like: Runge Kutta, iterative shooting, finite element, finite difference, direct iterative and non-iterative numerical methods. This is due to the fact that the Euler–Bernoulli nonlinear differential equation governing the problem contains the “slope at the free end”, this slope however needs special numerical treatment. On the other hand, some of these methods fail to find numerical solutions for extremely large loading conditions. Hence, this paper is aimed to obtain a closed-form solution for solving the large deflection of a cantilever beam opposed to a concentrated point follower load at its free end. This closed-form solution when compared with other conventional numerical approaches is characterized by simplicity, stability and straightforwardness in getting the beam deflection and slopes even for extremely large loading conditions. The closed-form solution is obtained by applying complex analysis along with elliptic-integral approach. Very good results were obtained when the elastica of the beam compared with that of various numerical methods which are used in analyzing similar problem.

Shooting and Arc-Length Continuation Method for Periodic Solution and Bifurcation of Nonlinear Oscillation of Viscoelastic Dielectric Elastomers

A majority of dielectric elastomers (DE) developed so far have more or less viscoelastic properties. Understanding the dynamic behaviors of DE is crucial for devices where inertial effects cannot be neglected. Through construction of a dissipation function, we applied the Lagrange's method and theory of nonequilibrium thermodynamics of DE and formulated a physics-based approach for dynamics of viscoelastic DE. We revisited the nonlinear oscillation of DE balloons and proposed a combined shooting and arc-length continuation method to solve the highly nonlinear equations. Both stable and unstable periodic solutions, along with bifurcation and jump phenomenon, were captured successfully when the excitation frequency was tuned over a wide range of variation. The calculated frequency–amplitude curve indicates existence of both harmonic and superharmonic resonances, soft-spring behavior, and hysteresis. The underlying physics and nonlinear dynamics of viscoelastic DE would aid the design and control of DE enabled soft machines.

A Proposal of 3D Feature Based on Occupancy of Point Cloud in Multiscale Shell Region

SUMMARYIn this paper, we propose a novel keypoint detection and feature description method called “SHORT” (Shell Histograms and Occupancy from Radial Transform) for fast 3D object recognition. Conventional keypoint detection and feature description methods such as the SHOT method have been necessary to calculate many normal vectors or other statistical values from the point cloud data in local regions, and therefore its computational costs are expensive. By contrast, the SHORT method consists of a fast keypoint detector that does not calculate statistics and a fast feature descriptor that uses a small number of points in the restricted local regions. The keypoint detector uses the occupancy measure which can be estimated by only counting the number of points in multiple spherical shell regions. Also, the feature descriptor uses a small number of points included in distinctive shell regions of multiple scales. Experimental results in 3D object recognition using real dataset show that the processing speed of the proposed method is approximately nine times faster than that of comparative methods.

A fast, spectrally accurate solver for the Falkner--Skan equation

We present a new numerical technique, the Gegenbauer homotopy analysis method, which allows for the construction of iterative solutions to nonlinear differential equations. This technique is a numerical extension of the semi-analytic homotopy analysis method that exhibits spectral convergence while performing sparse matrix operations in Gegenbauer space. This technique is used to present solutions to the Falkner--Skan equation, a well known problem in boundary layer fluid dynamics. These solutions are compared to previously published works, and the convergence properties exhibited by this new technique are considered. References N. S. Asaithambi. A numerical method for the solution of the Falkner–Skan equation. Applied Mathematics and Computation , 81 :259–264, 1997. doi:10.1016/S0096-3003(95)00325-8 H. Blasius. Grenzschichten in flussigkeiten mit kleiner reibung. Zeitschrift fur Angewandte Mathematik und Physik , 56 :1–37, 1908. T. Cebeci and H. B. Keller. Shooting and parallel shooting methods for solving the Falkner–Skan boundary-layer equation. Journal of Computational Physics , 7 :289–300, 1971. doi:10.1016/0021-9991(71)90090-8 V. M. Falkner and S. W. Skan. Some approximate solutions of the boundary layer equations. Philosophical Magazine , 12 :865–896, 1930. R. Fazio. Blasius problem and Falkner–Skan model: Topfer's algorithm and its extension. Computers and Fluid , 75 :202–209, 2013. doi:10.1016/j.compfluid.2012.12.012 S. J. Liao. The proposed homotopy analysis technique for the solution of nonlinear problems . PhD thesis, Shanghai Jiao Tong University, 1992. S. S. Motsa, G. T. Marewo, P. Sibanda and S. Shateyi. An improved spectral homotopy analysis method for solving boundary layer problems. Boundary Value Problems , 2011(1) :3, 2011. doi:10.1186/1687-2770-2011-3 S. Olver and A. Townsend. A fast and well-conditioned spectral method. SIAM Review , 55(3) :462–489, 2013. doi:10.1137/120865458 S. M. Rassoulinejad-Mousavi and S. Abbasbandy. Analysis of forced convection in a circular tube filled with a Darcy–Brinkman–Forcheimer porous medium using spectral homotopy analysis method. Journal of Fluids Engineering , 133(10) :101207–101207–9, 2011. doi:10.1115/1.4004998 H. Saberi Nik, S. Effati, S. S. Motsa, and M. Shirazian. Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems. Numerical Algorithms , 65(1) :171–194, 2014. doi:10.1007/s11075-013-9700-4

Entropy generation and heat transfer in boundary layer flow over a thin needle moving in a parallel stream in the presence of nonlinear Rosseland radiation

Analysis of entropy generation and heat transfer in the boundary layer flow over a thin needle moving in a parallel stream is performed in this work. Energy dissipation and nonlinear radiation terms are incorporated in the energy equation. It is assumed that the free stream velocity u∞ is in the direction of positive x−axis (axial direction) and the thin needle moves in the direction of free stream velocity. The problem is self-similar in the presence of viscous dissipation and non-linear Rosseland thermal radiation. The reduced self-similar governing equations are solved numerically using shooting and fourth order Runge-Kutta method. The expressions for dimensionless volumetric entropy generation rate and Bejan number are also obtained by selecting suitable similarity variables. The effects of the Eckert number, heating parameter, radiation parameter, Prandtl number, velocity ratio parameter and dimensionless size of a thin needle are described graphically in detail. The analysis reveals that the entropy generation decreases by decreasing the size of the thin needle. Entropy generation number increases with the increasing values of the Eckert number, Prandtl number and the temperature parameter. Moreover, it is observed that the Bejan number decreases by increasing the thermal radiation parameter. Validation of present analysis is performed by comparing the obtained results with those available in the existing literature and found a very good agreement.

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

SummaryThis paper presents a fast numerical method, based on the indirect shooting method and Proper Orthogonal Decomposition (POD) technique, for solving distributed optimal control of the wave equation. To solve this problem, we consider the first‐order optimality conditions and then by using finite element spatial discretization and shooting strategy, the solution of the optimality conditions is reduced to the solution of a series of initial value problems (IVPs). Generally, these IVPs are high‐order and thus their solution is time‐consuming. To overcome this drawback, we present a POD indirect shooting method, which uses the POD technique to approximate IVPs with smaller ones and faster run times. Moreover, in the presence of the nonlinear term, to reduce the order of the nonlinear calculations, a discrete empirical interpolation method (DEIM) is applied and a POD/DEIM indirect shooting method is developed. We investigate the performance and accuracy of the proposed methods by means of 4 numerical experiments. We show that the presented POD and POD/DEIM indirect shooting methods dramatically reduce the CPU time compared to the full indirect shooting method, whereas there is no significant difference between the accuracy of the reduced and full indirect shooting methods.

Multi-class analysis for simultaneous determination of pesticides, mycotoxins, process-induced toxicants and packaging contaminants in tea

This study attempts at uniting the analysis of four different classes of contaminants for both liquid and solid tea samples. A total of 32 compounds, classified as pesticides, mycotoxins, process-induced toxicants or packaging contaminants, were carefully chosen for their diversity of structures and physicochemical properties. The proposed method combines a sample treatment strategy coming from metabolomics with liquid chromatography analysis using a silica bonded C18-pentafluorophenyl column coupled to high resolution mass spectrometry. For tea brew, dilute and shoot method provides good quantification (70–120% recoveries and <20% RSD) for more than 80% of compounds. For tea leaves, strong matrix effects are observed, thus, matrix-matched calibration is required to reach good performances, i.e. 63% of compounds quantified and 81% detected at 10µg/kg. Finally, method performances were evaluated against existing regulations, and it appears that 69% of contaminants are quantified and 91% detected at levels lower than their respective European regulation limits.

Combination of optimal control approaches for aircraft conflict avoidance via velocity regulation

SummaryThis paper deals with optimal control applied to one of the most crucial and challenging problems in Air Traffic Management that of aircraft conflict avoidance. We propose an optimal control model where aircraft separation is achieved by changing the speeds of aircraft, and the integral over a time window of their squared accelerations is minimized. Pairwise aircraft separation constraints constitute the main difficulties to be handled. We propose an original decomposition of the problem into 3 zones, in such a way that in two of them, no conflict occurs. Then, using the Pontryagin maximum principle, 2 new formulations of the original optimal control problem are proposed and solved via direct shooting methods. Thanks to our decomposition, these numerical methods are applied on subproblems having reduced size with respect to the original one, thus improving the efficiency of the solution process. Thirty problem instances are numerically solved, showing the effectiveness of the proposed approaches.

Traveling wave solutions for diffusive predator–prey type systems with nonlinear density dependence

Traveling wave solution for a class of diffusive predator–prey system with nonlinear density dependence is considered. Using methods of topological shooting, we show the existence of a non-negative traveling wave solution connecting a boundary equilibrium to the co-existence steady state with the help of a Wazewski-like set together with Lyapunov function constructed elaborately. This means that the traveling wave solution established by Huang (2012) can be preserved in the presence of the nonlinear density dependence for the predator and the results of Huang (2012) are generalized.

Geometric optimal control and applications to aerospace

This article deals with applications of optimal control to aerospace problems with a focus on modern geometric optimal control tools and numerical continuation techniques. Geometric optimal control is a theory combining optimal control with various concepts of differential geometry. The ultimate objective is to derive optimal synthesis results for general classes of control systems. Continuation or homotopy methods consist in solving a series of parameterized problems, starting from a simple one to end up by continuous deformation with the initial problem. They help overcoming the difficult initialization issues of the shooting method. The combination of geometric control and homotopy methods improves the traditional techniques of optimal control theory.A nonacademic example of optimal attitude-trajectory control of (classical and airborne) launch vehicles, treated in details, illustrates how geometric optimal control can be used to analyze finely the structure of the extremals. This theoretical analysis helps building an efficient numerical solution procedure combining shooting methods and numerical continuation. Chattering is also analyzed and it is shown how to deal with this issue in practice.

Non-Abelian Higgs models: Paving the way for asymptotic freedom

Asymptotically free renormalization group trajectories can be constructed in nonabelian Higgs models with the aid of generalized boundary conditions imposed on the renormalized action. We detail this construction within the languages of simple low-order perturbation theory, effective field theory, as well as modern functional renormalization group equations. We construct a family of explicit scaling solutions using a controlled weak-coupling expansion in the ultraviolet, and obtain a standard Wilsonian RG relevance classification of perturbations about scaling solutions. We obtain global information about the quasi-fixed function for the scalar potential by means of analytic asymptotic expansions and numerical shooting methods. Further analytical evidence for such asymptotically free theories is provided in the large-N limit. We estimate the long-range properties of these theories, and identify initial/boundary conditions giving rise to a conventional Higgs phase.

Time-optimal aircraft trajectories in climbing phase and singular perturbations

This article deals with the singularly perturbed aircraft minimum time-to-climb problem. First, we introduce a reduced-order problem with affine dynamics with respect to the control and analyze it with the tools from geometric control: maximum principle combined with second order optimality conditions. Then, we compute candidates as minimizers using multiple shooting and homotopy methods for both problems, that we briefly compare numerically. Particular attention is paid to the singular extremals and we discuss about their local optimality.

Rapid Indirect Trajectory Optimization on Highly Parallel Computing Architectures

A graphics-processing-units-accelerated indirect trajectory optimization methodology that uses the multiple shooting method and continuation is developed using the CUDA platform. The algorithm is designed to exploit the parallelism inherent in the indirect shooting method while maximizing computational efficiency. The resulting rapid optimal control framework enables the construction of high quality optimal trajectories that satisfy constraints and the necessary conditions of optimality. The performance of the framework is highlighted by construction of maximum terminal velocity trajectories for a hypothetical long-range weapon system. Various hypothetical mission scenarios that enforce different combinations of initial, terminal, interior point, and path constraints that demonstrate the rapid construction of complex trajectories are used to compare performance of the graphics-processing-units-accelerated solver to MATLAB®’s bvp4c. Trajectory problems of this kind were previously considered impractical to solve using indirect methods. The graphics-processing-units-accelerated solver is found to be two to four times faster than MATLAB®’s bvp4c for a small-dimensional system, even while running on graphics processing units hardware that is five years behind the state of the art.

Verification of Radiation Isocenter on Linac Beam 6 MV using Computed Radiography

Radiation isocenter is more important part of quality assurance for the linear accelerator (Linac) due to radiation isocenter is a main location in irradiation radiotherapy, isocenter can shift when the gantry and collimator rotation. In general, the radiation isocenter verification using a special film. This research was conducted radiation isocenter verification using computed radiography with digital image processing techniques. Image acquisition was done using the modalities of Linac 6 MV with star shot method is star-shaped beam due to rotation of the collimator, gantry and couch. Then do the delineation on each beam to determine the centroid and beam diameter. By the results of verification of radiation isocenter performed on collimator and the couch, it shows that the size diameter for rotational collimator is 0.632 mm and 0.458 mm for the couch. Based on AAPM report 40 about the size of the Linac radiation isocenter diameter used in this study is still in good condition and worth to be operated because the value of the radiation isocenter diameter is below 2 mm.

A Python Program for Solving Schrödinger’s Equation in Undergraduate Physical Chemistry

In undergraduate physical chemistry, Schrodinger’s equation is solved for a variety of cases. In doing so, the energies and wave functions of the system can be interpreted to provide connections with the physical system being studied. Solving this equation by hand for a one-dimensional system is a manageable task, but it becomes time-consuming once students aim to make various changes and investigate the impact of those changes on the results. To address this challenge, numerical methods, such as the shooting and linear finite-difference methods, have been utilized to quickly solve Schrodinger’s equation. In this technology report, we use the Python programming environment and the three-point finite-difference numerical method to find the solutions and plot the results (wave functions or probability densities) for a particle in an infinite, finite, double finite, harmonic, Morse, or Kronig–Penney finite potential energy well. We believe that this technology report will educate undergraduates on the basic ...

A Hybrid Direct–Indirect Approach for Solving the Singular Optimal Control Problems of Finite and Infinite Order

This paper presents a hybrid approach to solve singular optimal control problems. It combines the direct Euler method with a modified indirect shooting method. The presented method circumvents the main difficulties and drawbacks of both the direct and indirect methods, when applied to the singular optimal control problems. This method does not require a priori knowledge of the switching structure of the solution and it can be applied to finite or infinite order singular optimal control problems. It provides not only an approximate optimal solution for the problem but, remarkably, it also produces the switching times. We illustrate the features of this new approach treating numerically through two optimal control problems, one of finite order and the other with infinite order.