Quadratic Collocation Research Articles

Learning Fuel-Optimal Trajectories for Space Applications via Pontryagin Neural Networks

This paper demonstrates the utilization of Pontryagin Neural Networks (PoNNs) to acquire control strategies for achieving fuel-optimal trajectories. PoNNs, a subtype of Physics-Informed Neural Networks (PINNs), are tailored for solving optimal control problems through indirect methods. Specifically, PoNNs learn to solve the Two-Point Boundary Value Problem derived from the application of the Pontryagin Minimum Principle to the problem’s Hamiltonian. Within PoNNs, the Extreme Theory of Functional Connections (X-TFC) is leveraged to approximate states and costates using constrained expressions (CEs). These CEs comprise a free function, modeled by a shallow neural network trained via Extreme Learning Machine, and a functional component that consistently satisfies boundary conditions analytically. Addressing discontinuous control, a smoothing technique is employed, substituting the sign function with a hyperbolic tangent function and implementing a continuation procedure on the smoothing parameter. The proposed methodology is applied to scenarios involving fuel-optimal Earth−Mars interplanetary transfers and Mars landing trajectories. Remarkably, PoNNs exhibit convergence to solutions even with randomly initialized parameters, determining the number and timing of control switches without prior information. Additionally, an analytical approximation of the solution allows for optimal control computation at unencountered points during training. Comparative analysis reveals the efficacy of the proposed approach, which rivals state-of-the-art methods such as the shooting technique and the adaptive Gaussian quadrature collocation method.

Uncertain trajectory planning integrating polynomial chaos and convex programming

The present work explores the robust trajectory optimization scheme considering both the initial state disturbance and multiple constraints. An uncertain multi-constraint optimization model has been first established. Due to the existence of the stochastic disturbance, standard numerical trajectory planning algorithms cannot be directly applied to address the considered issue. Hence, based on the intrusive polynomial chaos expansion, we present a deterministic quantification for the stochastic state and constraints, so that the transformed optimization model becomes solvable for standard numerical optimization methods. To obtain the enhanced computational performance, an hp pseudo-spectral sequential convex programming procedure combined with a penalty function and backtracking search is proposed. This is achieved by discretizing and convexifying the nonlinear dynamics/constraints using hp quadrature collocation and successive linearization, respectively, and by adjusting the confidence region manually in the iteration. The simulation of a three-dimensional interception with the specific impact angle is conducted to verify the effectiveness. The simulation results show that the initial solutions are insensitive to the convex optimization, and the control commands generated by the proposed algorithm are effective against the initial state disturbance.

End-to-End Performance Optimization for High-Speed Ascent–Entry Missions

The performance optimization for a combined ascent–entry mission subject to constraints on heating rate and heating load is studied. The ascent vehicle is modeled as a three-stage rocket that places the entry vehicle onto a suborbital exo-atmospheric trajectory after which the vehicle undergoes an unpowered entry and descent to a vertically downward terminal condition. The entry vehicle is modeled as a high-lift-to-drag-ratio vehicle that is capable of withstanding high levels of thermal and structural loads. A performance index is designed to improve control margin while attenuating phugoid oscillations during atmospheric entry. Furthermore, a mission corresponding to a prototype launch and target point is used in this study. The trajectory optimization problem is formulated as a multiple-phase optimal control problem, and the optimal control problem is solved using an adaptive Gaussian quadrature collocation method. The key features of the optimized trajectories and controls are identified, and the approach developed in this paper provides a systematic method for end-to-end ascent–entry trajectory optimization.

Existence, uniqueness, and approximate solutions for the general nonlinear distributed-order fractional differential equations in a Banach space

The purpose of this paper is to provide sufficient conditions for the local and global existence of solutions for the general nonlinear distributed-order fractional differential equations in the time domain. Also, we provide sufficient conditions for the uniqueness of the solutions. Furthermore, we use operational matrices for the fractional integral operator of the second kind Chebyshev wavelets and shifted fractional-order Jacobi polynomials via Gauss–Legendre quadrature formula and collocation methods to reduce the proposed equations into systems of nonlinear equations. Also, error bounds and convergence of the presented methods are investigated. In addition, the presented methods are implemented for two test problems and some famous distributed-order models, such as the model that describes the motion of the oscillator, the distributed-order fractional relaxation equation, and the Bagley–Torvik equation, to demonstrate the desired efficiency and accuracy of the proposed approaches. Comparisons between the methods proposed in this paper and the existing methods are given, which show that our numerical schemes exhibit better performances than the existing ones.

Minimum-Time Earth-to-Mars Interplanetary Orbit Transfer Using Adaptive Gaussian Quadrature Collocation

The problem of minimum-time, low-thrust, Earth-to-Mars interplanetary orbital trajectory optimization is considered. The minimum-time orbital transfer problem is modeled as a four-phase optimal control problem where the four phases correspond to planetary alignment, Earth escape, heliocentric transfer, and Mars capture. The four-phase optimal control problem is then solved using a direct collocation adaptive Gaussian quadrature collocation method. The following three models are used in the study: 1) circular planetary motion, 2) elliptic planetary motion, and 3) elliptic planetary motion with gravity perturbations, where the transfer begins in a geostationary orbit and terminates in a Mars-stationary orbit. Results for all three cases are provided, and one particular case is studied in detail to show the key features of the optimal solutions. Using the particular value thrust specific force of , it was found that the minimum times for cases 1, 2, and 3 are, respectively, 215, 196, and 198 d with departure dates, respectively, of 1 July 2020, 30 June 2020, and 28 June 2020. Finally, the problem formulation developed in this study is compared against prior work on an Earth-to-Mars interplanetary orbit transfer where it is found that the results of this research show significant improvement in transfer time relative to the prior work.

A high-order semi-Lagrangian method for the consistent Monte-Carlo solution of stochastic Lagrangian drift–diffusion models coupled with Eulerian discontinuous spectral element method

The explicit semi-Lagrangian method for solution of Lagrangian transport equations as developed in Natarajan and Jacobs (2020) is adopted for the solution of stochastic differential equations that is consistent with Discontinuous Spectral Element Method (DSEM) approximations of Eulerian conservation laws. The method extends the favorable properties of DSEM that include its high-order accuracy, its local and boundary fitted properties and its high performance on parallel platforms for the concurrent Monte-Carlo, semi-Lagrangian and Eulerian solution of a class of time-dependent problems that can be described by coupled Eulerian–Lagrangian formulations. Such formulations include the probabilistic models used for the simulation of chemically reacting turbulent flows or particle-laden flows. Consistent with an explicit, DSEM discretization, the semi-Lagrangian method seeds particles at Gauss quadrature collocation nodes within a spectral element. The particles are integrated explicitly in time according to a drift velocity and a Wiener increment forcing and form the nodal basis for an advected interpolant. This interpolant is mapped back in a semi-Lagrangian fashion to the Gauss quadrature points through a least squares fit using constraints for element boundary values. Stochastic Monte-Carlo samples are averaged element-wise on the quadrature nodes. The stable explicit time step Wiener increment is sufficiently small to prevent particles from leaving the element’s bounds. The semi-Lagrangian method is hence local and parallel and does not have the grid complexity, and parallelization challenges of the commonly used Lagrangian particle solvers in particle-mesh methods for solution of Eulerian–Lagrangian formulations. Formal proof is presented that the semi-Lagrangian algorithm evolves the solution according to the Eulerian Fokker–Planck equation. Numerical tests in one and two dimensions for drift–diffusion problems show that the method converges exponentially for constant and non-constant advection and diffusion velocities.

Ascent Trajectory Optimization for Air-Breathing Hypersonic Vehicles Based on IGS-MPSP

This paper proposes an improved Generalized Quasi-Spectral Model Predictive Static Programming (GS-MPSP) algorithm for the ascent trajectory optimization for hypersonic vehicles in a complex flight environment. The proposed method guarantees the satisfaction of constraints related to the state and control vector while retaining its high computational efficiency. The spectral representation technique is used to describe the control variables, which reduces the number of decision variables and makes the control input smooth enough. Through Taylor expansion, the constraints are transformed into an inequality containing only decision variables, such that it can be added into GS-MPSP framework. By Gauss quadrature collocation method, only a few collocation points are needed to solve the sensitivity matrix, which greatly accelerates the calculation. Subsequently, the analytical expression is obtained by combining the static optimization with the penalty function method. Finally, the simulation results demonstrate that the proposed improved GS-MPSP algorithm can achieve both high computational efficiency and high terminal precision under the constraints.

Optimal Guidance and Control of a Low-Altitude Skid-to-Turn Vehicle

The problem of optimal guidance of a low-altitude skid-to-turn vehicle is considered. In particular, the skid-to-turn vehicle is both guided and controlled using a recently developed computational optimal guidance and control method that employs Gaussian quadrature collocation. First, this recently developed method is used to re-solve the optimal control problem on a shrinking horizon at constant guidance cycles. The performance of this computational guidance and control method is then analyzed and compared against a classical LQ follower. It is found that, although the computational efficiencies of both methods are comparable, the computational method for optimal guidance and control is significantly more accurate than that of the LQ follower. The results of this study show that the recently developed computational method for optimal guidance and control has the potential to improve performance over traditional methods when solving an optimal control problem in real time.

Generalized quasi-spectral model predictive static programming method using Gaussian quadrature collocation

A new generalized quasi-spectral model predictive static programming (GS-MPSP) method is proposed to efficiently solve a class of terminal-constrained optimal control problems with specified or free terminal time. A spectral representation method is used to model the profile of the control vector, then an infinite-dimensional optimization problem in a continuous-time framework is transformed into a small-dimensional static programming problem minimizing a certain performance index. Using the Gauss quadrature collocation method, the computation of the sensitivity matrix can be converted to the solution of a group of linear equations and algebraic summation at a few collocation nodes. Subsequently, the spectral coefficients and terminal time are efficiently obtained to eliminate terminal output deviations by solving the static programming problem. A simulation case with a scenario of intercepting a high-speed target with a specified impact angle in the midcourse phase was conducted. The results indicate that the proposed GS-MPSP approach has increased computational efficiency compared to traditional methods.

Frictional receding contact problem of a functionally graded layer resting on a homogeneous coated half-plane

This paper investigates the frictional receding contact problem between a functionally graded (FG) layer resting on a homogeneous coated half-plane when the system indented by a rigid cylindrical punch. The shear modulus of the upper graded layer is assumed to vary exponentially in the depth direction. Upon loading, the advancing contact and receding contact occur between the FG layer and rigid punch and between the FG layer and coating, respectively. Under the assumptions of sliding contact, the shear and normal contact stresses are related through Coulomb’s law of friction. The contact problem is converted analytically using Fourier integral transforms and the proper boundary conditions into a system of two singular integral equations. The unknowns of these integral equations consist of the contact stresses under the punch and between the FG layer and coating as well as the dimensions of the contact zones. The Gauss–Jacobi quadrature collocation method is then employed to transform the singular integral equations into a system of nonlinear equations which are solved using an appropriate iterative algorithm to compute the contact stresses and the dimensions of the contact areas. The main purpose of this paper is to examine the influence of several parameters on the upper and lower contact stresses, which include the material inhomogeneity parameter, friction coefficient, punch radius, applied load, thickness of homogeneous layer, shear modulus of homogeneous layer and shear modulus of homogeneous half-plane. One of the most important conclusions from this study is that the contact stresses and contact areas can be controlled with the addition of the homogeneous layer.

An explicit semi-Lagrangian, spectral method for solution of Lagrangian transport equations in Eulerian-Lagrangian formulations

An explicit high-order semi-Lagrangian method is developed for the solution of Lagrangian transport equations in Eulerian-Lagrangian formulations. The method is consistent with an explicit, discontinuous spectral element method (DSEM) discretization of the Eulerian formulation. The semi-Lagrangian method seeds particles at Gauss quadrature collocation nodes within a spectral element. The particles are integrated explicitly in time and form the nodal basis for an advected interpolant. This interpolant is mapped back in a semi-Lagrangian fashion to the Gauss quadrature points through a least squares fit using constraints for element boundary values and optional constraints for mass and energy preservation. The stable explicit time step of the DSEM solver is sufficiently small to prevent particles seeded at the Gauss quadrature points from leaving the element’s bounds. The semi-Lagrangian method is hence local and parallel and does not have the grid complexity, and parallelization challenges of the commonly used Lagrangian particle solvers in particle-mesh methods for solution of Eulerian-Lagrangian formulations. Numerical tests in one and two dimensions for linear and non-linear advection show that the method converges exponentially. The use of mass and energy constraints can improve accuracy depending on the order of accuracy of the time integrator.

Local meshless differential quadrature collocation method for time-fractional PDEs

This paper is concerned with the numerical solution of time- fractional partial differential equations (PDEs) via local meshless differential quadrature collocation method (LMM) using radial basis functions (RBFs). For the sake of comparison, global version of the meshless method is also considered. The meshless methods do not need mesh and approximate solution on scattered and uniform nodes in the domain. The local and global meshless procedures are used for spatial discretization. Caputo derivative is used in the temporal direction for both the values of $ \alpha \in (0,1) $ and $ \alpha\in(1,2) $. To circumvent spurious oscillation casued by convection, an upwind technique is coupled with the LMM. Numerical analysis is given to asses accuracy of the proposed meshless method for one- and two-dimensional problems on rectangular and non-rectangular domains.

Convergence and error analysis of a spectral collocation method for solving system of nonlinear Fredholm integral equations of second kind

This paper presents a new numerical approximation method to solve a system of nonlinear Fredholm integral equations of second kind. Spectral collocation method and their properties are applied to determine the general solution procedure for nonlinear Fredholm integral equations (FIEs). The convergence and error analysis of spectral collocation method are incorporated for the given nonlinear model. Legendre–Gauss–Lobatto (LGL) points are used as collocation points with various Legendre–Gauss quadrature with weight functions. The use of Legendre polynomials, together with the Gauss quadrature collocation points is well known for the accurate approximations that converge exponentially. Finally, we validate our theoretical results with a number of numerical examples, which further enhance the efficiency of our proposed scheme.

Computational Method for Optimal Guidance and Control Using Adaptive Gaussian Quadrature Collocation

A method is described for computational optimal guidance and control using adaptive Gaussian quadrature collocation and sparse nonlinear programming. The method employs adaptive Legendre–Gauss–Radau (LGR) quadrature collocation using a mesh truncation and remapping procedure at the start of each guidance cycle, thereby retaining only the mesh points associated with the unexpired horizon. Using this approach, the starting mesh for a guidance cycle is smaller than the starting mesh used on the previous guidance cycle. The nonlinear programming problem is then solved on the reduced mesh corresponding to the unexpired horizon to generate the control that is used on the current guidance cycle. It is noted that the mesh points on the unexpired horizon are well placed for rapidly solving the LGR quadrature collocation approximation of the optimal control problem. The method developed in this paper is demonstrated on two challenging aerospace optimal control problems. For both examples, the dynamics are simulated on each guidance cycle using a perturbed dynamic model in the absence and presence of a computation time delay. The results of this study demonstrate that the method developed in this paper is viable as a computational method for optimal guidance and control.

Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method

In this paper, a local meshless differential quadrature collocation method based on radial basis functions is proposed for the numerical simulation of one-dimensional Klein–Gordon, two-dimensional coupled Burgers’, and regularized long wave equations. Both local and global meshless collocation procedures are used for spatial discretization, which convert the mentioned partial differential equations into a system of ordinary differential equations. The obtained system has been solved by the forward Euler difference formula. An upwind technique is utilized in the case of the convection-dominated coupled Burgers’ model equation. Having no need for the mesh in the problem domain and being less sensitive to the variation of the shape parameter as compared to global meshless methods are the salient features of the local meshless method. Both rectangular and non-rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy, and the ease of implementation of the proposed method are shown via test problems.

Minimum-Fuel Low-Earth-Orbit Aeroglide and Aerothrust Aeroassisted Orbital Transfer Subject to Heating Constraints

A numerical optimization study of minimum-fuel low-Earth-orbit aeroglide and aerothrust aeroassisted orbital transfer of a small spacecraft subject to constraints on heating rate and heating load with a required inclination change is considered. The aeroassisted orbital transfer is formulated as a three-phase optimal control problem consisting of an exoatmospheric deorbit phase, an atmospheric flight phase that may or may not include thrust, and a second exoatmospheric flight phase. The two different variations of the three-phase optimal control problem that arise from the trajectory design are solved using a high-accuracy adaptive Gaussian quadrature collocation method. The fuel consumption of both the aeroglide and aerothrust aeroassisted maneuvers are assessed as functions of the maximum allowable heating rate, the maximum allowable integrated heat load, the maximum lift-to-drag ratio of the vehicle, and the initial mass of the vehicle. Finally, the key features of the optimal aeroglide and aerothrust maneuvers are identified.

A Numerical Solution for a Frictional Contact Problem between an Orthotropic Strip and Punch System

In this study, a numerical solution of elasticity problem is examined. This problem is a plane contact problem. The frictional contact problem for an elastic strip under a rigid punch system is considered. The frictional contact problem is related to infinite length elastic strip in contact with N punches under the influence of horizontal and vertical forces. The lower boundary of the strip is hinged. The solution of contact problems is often reduced to the solution of an integral equation. This integral equation system can be derived from contact problem by using the basic equations of elasticity theory and the given boundary conditions. The singular integral equation system is solved with the help of Gauss Jacobi Quadrature Collocation Method. The frictional contact problem for a homogenous and orthotropic elastic layer are investigated numerically the pressure distribution under the punch system due to the geometrical and mechanical properties of elastic layer are examined and the results are shown in the graphics and tabular form.

Local RBF method for multi-dimensional partial differential equations

In this paper, a local meshless differential quadrature collocation method is utilized to solve multi-dimensional reaction–convection–diffusion PDEs numerically. In some cases, global version of the meshless method is considered as well. The meshless methods approximate solution on scattered and uniform nodes in both local and global sense. In the case of convection-dominated PDEs, the local meshless method is coupled with an upwind technique to avoid spurious oscillations. For this purpose, a physically motivated local domain is utilized in the flow direction. Both regular and irregular geometries are taken into consideration. Numerical experiments are performed to demonstrate effective applications and accuracy of the meshless method on regular and irregular domains.

Robust a posteriori stress analysis for quadrature collocation approximations of nonlocal models via nonlocal gradients

As alternatives to partial differential equations (PDEs), nonlocal continuum models given in integral forms avoid the explicit use of conventional spatial derivatives and allow solutions to exhibit desired singular behavior. It is of practical interest to develop robust numerical schemes not only for the numerical solution of nonlocal models but also for the evaluation of suitably defined derivatives of solutions. The latter motivates the development of a nonlocal analog of gradient recovery for numerical solution of PDEs. For structure mechanical models, this leads to a posteriori nonlocal stress analysis. We illustrate that when smooth solutions are found in nonlocal models, one may compute local gradients of nonlocal solutions using conventional techniques like that for PDEs. More generically however, we present a framework for stress analysis of nonlocal solutions based on nonlocal gradient operators and their asymptotically compatible discretization. We demonstrate that the nonlocal gradient recovery is consistent in the local limit and is more advantageous than using local gradients of nonlocal solutions. Superconvergence properties of some special nonlocal gradient operators are identified for nonlocal continuum models. Moreover, methods are presented to preserve such features in the numerical discretization. Both computational observations and theoretical insights are provided to substantiate our findings.

SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10 different input distributions or histograms.