Energy-Saving and Punctuality Combined Velocity Planning for the Autonomous-Rail Rapid Tram with Enhanced Pseudospectral Method

Autonomous-rail rapid transit(ART)is a new medium-capacity rapid transportation system with punctuality,comfort and convenience,but low-cost construction.Combined velocity planning is a critical approach to meet the requirements of energy-saving and punctuality.An ART velocity pre-planning and re-planning strategy based on the combination of punctuality dynamic programming(PDP)and pseudospectral(PS)method is proposed in this paper.Firstly,the longitudinal dynamics model of ART is established by a multi-particle model.Secondly,the PDP algorithm with global optimal characteristics is adopted as the pre-planning strategy.A model for determining the number of collocation points of the real-time PS method is proposed to improve the energy-saving effect while ensuring computation efficiency.Then the enhanced PS method is utilized to design the velocity re-planning strategy.Finally,simulations are conducted in the typical scenario with sloping roads,traffic lights,and intrusion of the pedestrian.The simulation results indicate that the ART with the proposed velocity trajectory optimization strategy can meet the punctuality requirement,and obtain better economy efficiency compared with the punctuality green light optimal speed advisory(PGLOSA).

Re-entry trajectory optimization using a multiple- interval Radau pseudospectral method

Aiming at increasing the calculation efficiency of the pseudospectral methods, a multiple- interval Radau pseudospectral method （RPM） is presented to generate a reusable launch vehicle （RLV） ＇s optimal re-entry trajectory. After dividing the optimal control problem into many intervals, the state and control variables are approximated using many fixed- and low-degree Lagrange polyno- mials in each interval. Convergence of the numerical discretization is then achieved by increasing the number of intervals. With the application of the proposed method, the normal nonlinear program- ming （NLP） problem transcribed from the optimal control problem can avoid being dense because of the low-degree approximation polynomials in each interval. Thus, the NLP solver can easily compute a solution. Finally, simulation results show that the optimized re-entry trajectories satisfy the path constraints and the boundary constraints successfully. Compared with the single interval RPM, the multiple-interval RPM is significantly faster and has higher calculation efficiency. The results indicate that the multiple-interval RPM can be applied for real-time trajectory generation due to its high effi- ciency and high precision.

Optimal control of attitude for coupled-rigid-body spacecraft via Chebyshev-Gauss pseudospectral method

The attitude optimal control problem （OCP） of a two-rigid-body space- craft with two rigid bodies coupled by a ball-in-socket joint is considered. Based on conservation of angular momentum of the system without the external torque, a dynamic equation of three-dimensional attitude motion of the system is formulated. The attitude motion planning problem of the coupled-rigid-body spacecraft can be converted to a dis- crete nonlinear programming （NLP） problem using the Chebyshev-Gauss pseudospectral method （CGPM）. Solutions of the NLP problem can be obtained using the sequential quadratic programming （SQP） algorithm. Since the collocation points of the CGPM are Chebyshev-Gauss （CG） points, the integration of cost function can be approximated by the Clenshaw-Curtis quadrature, and the corresponding quadrature weights can be calculated efficiently using the fast Fourier transform （FFT）. To improve computational efficiency and numerical stability, the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of state and con- trol variables. Furthermore, numerical float errors of the state differential matrix and barycentric weights can be alleviated using trigonometric identity especially when the number of CG points is large. A simple yet efficient method is used to avoid sensitivity to the initial values for the SQP algorithm using a layered optimization strategy from a feasible solution to an optimal solution. Effectiveness of the proposed algorithm is perfect for attitude motion planning of a two-rigid-body spacecraft coupled by a ball-in-socket joint through numerical simulation.

Pseudospectral method with symplectic algorithm for the solution of time-dependent SchrSdinger equations

A pseudospectral method with symplectic algorithm for the solution of time-dependent Schrodinger equations （TDSE） is introduced. The spatial part of the wavefunction is discretized into sparse grid by pseudospectral method and the time evolution is given in symplectic scheme. This method allows us to obtain a highly accurate and stable solution of TDSE. The effectiveness and efficiency of this method is demonstrated by the high-order harmonic spectra of one-dimensional atom in strong laser field as compared with previously published work. The influence of the additional static electric field is also investigated.

Study of elastic wave propagation in two phase anisotropic media by numerical modeling of pseudospectral method

When there exists anisotropy in underground media elastic parameters of the observed coordinate possibly do not coincide with that of the natural coordinate. According to the theory that the density of potential energy, dissipating energy is independent of the coordinate, the relationship of elastic parameters between two coordinates is derived for two-phase anisotropic media. Then, pseudospectral method to solve wave equations of two-phase anisotropic media is derived. At last, we use this method to simulate wave propagation in two-phase anisotropic media four types of waves are observed in the snapshots, i.e., fast P wave and slow P wave, fast S wave and slow S wave. Shear wave splitting, SV wave cusps and elastic wave reflection and transmission are also observed.

Six-DOF trajectory optimization for reusable launch vehicles via Gauss pseudospectral method

To be close to the practical flight process and increase the precision of optimal trajectory, a six-degree-offreedom（6-DOF） trajectory is optimized for the reusable launch vehicle（RLV） using the Gauss pseudospectral method（GPM）. Different from the traditional trajectory optimization problem which generally considers the RLV as a point mass, the coupling between translational dynamics and rotational dynamics is taken into account. An optimization problem is formulated to minimize a performance index subject to 6-DOF equations of motion, including translational and rotational dynamics. A two-step optimal strategy is then introduced to reduce the large calculations caused by multiple variables and convergence confinement in 6-DOF trajectory optimization. The simulation results demonstrate that the 6-DOF trajectory optimal strategy for RLV is feasible.

The Lanczos-Chebyshev Pseudospectral Method for Solution of Differential Equations

In this paper, we propose to replace the Chebyshev series used in pseudospectral methods with the equivalent Chebyshev economized power series that can be evaluated more rapidly. We keep the rest of the implementation the same as the spectral method so that there is no new mathematical principle involved. We show by numerical examples that the new approach works well and there is indeed no significant loss of solution accuracy. The advantages of using power series also include simplicity in its formulation and implementation such that it could be used for complex systems. We investigate the important issue of collocation point selection. Our numerical results indicate that there is a clear accuracy advantage of using collocation points corresponding to roots of the Chebyshev polynomial.

Modified Laguerre spectral and pseudospectral methods for nonlinear partial differential equations in multiple dimensions

The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial differential equations. Some results on the modified Laguerre orthogonal approximation and interpolation are established, which play important roles in the related numerical methods for unbounded domains. As an example, the modified Laguerre spectral and pseudospectral methods are proposed for two-dimensional Logistic equation. The stability and convergence of the suggested schemes are proved. Numerical results demonstrate the high accuracy of these approaches.

Direct trajectory optimization based on a mapped Chebyshev pseudospectral method

In view of generating optimal trajectories of Bolza problems, standard Chebyshev pseudospectral （PS） method makes the points＇ accumulation near the extremities and rarefaction of nodes close to the center of interval, which causes an ill-condition of differentiation matrix and an oscillation of the optimal solution. For improvement upon the difficulties, a mapped Chebyshev pseudospectral method is proposed. A conformal map is applied to Chebyshev points to move the points closer to equidistant nodes. Condition number and spectral radius of differentiation matrices from both methods are presented to show the improvement. Furthermore, the modification keeps the Chebyshev pseudospectral method＇s advantage, the spectral convergence rate. Based on three numerical examples, a comparison of the execution time, convergence and accuracy is presented among the standard Chebyshev pseudospectral method, other collocation methods and the proposed one. In one example, the error of results from mapped Chebyshev pseudospectral method is reduced to 5% of that from standard Chebyshev pseudospectral method.

SH-wavefield simulation for a laterally heterogeneous whole-Earth model using the pseudospectral method

We present a scheme to simulate SH-wave propagation in a whole-Earth model with arbitrary lateral heterogeneities employing the Fourier pseudospectral method. Wave equations are defined in two-dimensional cylindrical coordinates and the model is taken through a great circle of the Earth. Spatial derivatives in the wave equations are calculated in the wavenumber domain by multiplication, and the transformation between spatial and wavenumber domains is performed via fast Fourier transformation. Because of the high accuracy and high speed of the Fourier pseudospectral method, the scheme enables us to calculate a short-wavelength global SH-wavefield with accurate waveforms and arrival times for models with heterogeneities that can be approximated as azimuthally symmetric. Comparing with two-dimensional simulation methods based on an axisymmetric model, implementing the seismic source in the present scheme is more convenient. We calculated the global SH-wavefield for the preliminary reference Earth model to identify the generation, reflection and refraction of various seismic phases propagating in the Earth. Applications to a heterogeneous global model with low-velocity perturbation above the core-mantle boundary were conducted to analyze the effect of lateral heterogeneity on global SH-wave propagation.

MULTISYMPLECTIC FOURIER PSEUDOSPECTRAL METHOD FOR THE NONLINEAR SCHR■DINGER EQUATIONS WITH WAVE OPERATOR

In this paper, the multisymplectic Fourier pseudospectral scheme for initial-boundary value problems of nonlinear SchrSdinger equations with wave operator is considered. We investigate the local and global conservation properties of the multisymplectic discretization based on Fourier pseudospectral approximations. The local and global spatial conservation of energy is proved. The error estimates of local energy conservation law are also derived. Numerical experiments are presented to verify the theoretical predications.

ON NON-ISOTROPIC JACOBI PSEUDOSPECTRAL METHOD

In this paper, a non-isotropic Jacobi pseudospectral method is proposed and its appli- cations are considered. Some results on the multi-dimensional Jacobi-Gauss type interpolation and the related Bernstein-Jackson type inequalities are established, which play an important role in pseudospectral method. The pseudospectral method is applied to a twodimensional singular problem and a problem on axisymmetric domain. The convergence of proposed schemes is established. Numerical results demonstrate the efficiency of the proposed method.

Trajectory generation of heat load test based on gauss pseudospectral method

A new trajectory generation for heat load test is proposed based on gauss pseudospectral method within limit range. Firstly,with multiple path constraints and flight task requirements taken into consideration, heat load parameters are introduced into the dynamics equations. In order to solve the problem of generating such a trajectory within limit range rapidly, the dynamics equations have been normalized by Earth related parameters. Secondly, since the gauss pseudospectral method is just employed to solve the discrete nonlinear programming problem, transformations are developed, which can relate the Lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem. In addtion, another approach of trajectory generation by tracking the given heat rate is also presented. Finally, simulation results with common aero vehicle(CAV-H) show that the trajectories obtained by both methods can well perform the heat load test with high stagnation heating rate and the large total aeroheating amount; meanwhile, gauss pseudospectral method is better than the compared one in the given range. Furthermore, the 3-D trajectory states and control variables, angle of attack and bank, which are generated by gauss pseudospectral method, can change smoothly.

High-fidelity trajectory optimization for aeroassisted vehicles using variable order pseudospectral method

In this study,the problem of time-optimal reconnaissance trajectory design for the aeroassisted vehicle is considered.Different from most works reported previously,we explore the feasibility of applying a high-order aeroassisted vehicle dynamic model to plan the optimal flight trajectory such that the gap between the simulated model and the real system can be narrowed.A highly-constrained optimal control model containing six-degree-of-freedom vehicle dynamics is established.To solve the formulated high-order trajectory planning model,a pipelined optimization strategy is illustrated.This approach is based on the variable order Radau pseudospectral method,indicating that the mesh grid used for discretizing the continuous system experiences several adaption iterations.Utilization of such a strategy can potentially smooth the flight trajectory and improve the algorithm convergence ability.Numerical simulations are reported to demonstrate some key features of the optimized flight trajectory.A number of comparative studies are also provided to verify the effectiveness of the applied method as well as the high-order trajectory planning model.

MIXED LEGENDRE-HERMITE PSEUDOSPECTRAL METHOD FOR HEAT TRANSFER IN AN INFINITE PLATE

A new mixed Legendre-Hermite interpolation is introduced. Some approximation results are established. Mixed Legendre-Hermite pseudospectral method is proposed for non-isotropic heat transfer in an infinite plate. Its convergence is proved. Numerical results show the efficiency of this approach.

A novel trajectory planning strategy for aircraft emergency landing using Gauss pseudospectral method

To improve the survivability during an emergency situation, an algorithm for aircraft forced landing trajectory planning is proposed. The method integrates damaged aircraft modelling and trajectory planning into an optimal control framework, in order to deal with the complex aircraft flight dynamics, a solving strategy based on Gauss pseudospetral method （GPM） is presented. A 3-DOF nonlinear mass-point model taking into account the wind is developed to approximate the aircraft flight dynamics after loss of thrust. The solution minimizes the forced landing duration, with respect to the constraints that translate the changed dynamics, flight envelope limitation and operational safety requirements. The GPM is used to convert the trajectory planning problem to a nonlinear programming problem （NLP）, which is solved by sequential quadratic programming algorithm. Simulation results show that the proposed algorithm can generate the minimum-time forced landing trajectory in event of engine-out with high efficiency and precision.

A LEGENDRE PSEUDOSPECTRAL METHOD FOR SOLVINGNONLINEAR KLEIN-GORDON EQUATION

A Legendre pseudospectral scheme is proposed for solving initial-boundary value problem of nonlinear Klein-Gordon equation. The numerical solution keeps the discrete conservation. Its stability and convergence are investigated. Numerical results are also presented, which show the high accuracy. The technique in the theoretical analysis provides a framework for Legendre pseudospectral approximation of nonlinear multi-dimensional problems.

Multi-Symplectic Fourier Pseudospectral Method for the Kawahara Equation

In this paper,we derive a multi-symplectic Fourier pseudospectral scheme for the Kawahara equation with special attention to the relationship between the spectral differentiation matrix and discrete Fourier transform.The relationship is crucial for implementing the scheme efficiently.By using the relationship,we can apply the Fast Fourier transform to solve the Kawahara equation.The effectiveness of the proposed methods will be demonstrated by a number of numerical examples.The numerical results also confirm that the global energy and momentum are well preserved.

JACOBI PSEUDOSPECTRAL METHOD FOR FOURTH ORDER PROBLEMS

In this paper, we investigate Jacobi pseudospectral method for fourth order problems. We establish some basic results on the Jacobi-Gauss-type interpolations in non-uniformly weighted Sobolev spaces, which serve as important tools in analysis of numerical quadratures, and numerical methods of differential and integral equations. Then we propose Jacobi pseudospectral schemes for several singular problems and multiple-dimensional problems of fourth order. Numerical results demonstrate the spectral accuracy of these schemes, and coincide well with theoretical analysis.

ANALYSIS OF CHEBYSHEV PSEUDOSPECTRAL METHOD FOR MULTI-DIMENSIONAL GENERALIZED SRLW EQUATIONS

The Chebyshev pseudospectral approximation of the homogeneous initial boundary value problem for a class of multi-dimensional generalized symmetric regularized long wave (SRLW) equations is considered. The fully discrete Chebyshev pseudospectral scheme is constructed. The convergence of the approximation solution and the optimum error of approximation solution are obtained.