arXiv subjects

Anil V. Rao

Publications and source records attributed to Anil V. Rao.

At least 19 recordsLinked to original sources

arXiv2026

Method for Constrained Computational Guidance and Control with Application to Hypersonic Entry

A method for constrained computation guidance and control is described. The method employs guidance updates by periodically re-solving a constrained optimal control problem. Feasibility of the optimal control problem is improved by preventing constraint violations in the perturbed system. Constraint violation is prevented by augmenting the optimal control problem with a penalty term intended to increase the margin between a constraint and its limit when the perturbed system is within certain proximity of the constraint limit. The additional penalty term employs a smooth maximum approximation to increase the margin between the constraint and its limit. The method developed in this paper is demonstrated on the maximum cross-range reusable launch vehicle (RLV) entry problem, for which the reference solution contains a long-duration constrained arc. The results demonstrate that the method developed in this paper is viable for path-constrained optimal guidance and control.

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arXiv2026

An Adaptive Method for Optimal Control Problems Constrained by Parabolic Differential Equations

An adaptive direct collocation method is developed for solving optimal control problems constrained by parabolic partial differential equations. The partial differential equation is first reformulated in a variational setting, where the spatial domain is discretized using the hp-Galerkin finite element method. To address nonlinearities in the variational form, a Kirchhoff-like integral transformation is applied to linearize the dynamics. In the temporal dimension, an orthogonal collocation scheme, the hp-flipped Legendre-Gauss-Radau method, is employed to fully discretize the problem, yielding a large, sparse nonlinear programming problem. Upon solving the nonlinear programming problem, solution accuracy is assessed through an implicit residual estimation procedure. This approach evaluates the local error by solving auxiliary residual problems over selected subdomains, providing a novel means of error estimation within an orthogonal collocation framework for optimal control. Based on the computed error estimate, the mesh is adaptively refined or coarsened to meet a prescribed error tolerance. Mesh refinement is guided by the estimated regularity of the solution which is determined via the decay rate of the coefficients of a Legendre polynomial expansion. In overcollocated regions, a mesh reduction strategy is adapted from orthogonal collocation methods for application within the finite element framework. Numerical examples demonstrate that the proposed method can reduce the error by up to five orders of magnitude in both spatial and temporal dimensions.

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arXiv2026

Optimal Control of Parabolic Differential Equations Using Radau Collocation

A method is presented for the numerical solution of optimal boundary control problems governed by parabolic partial differential equations. The continuous space-time optimal control problem is transcribed into a sparse nonlinear programming problem through state and control parameterization. In particular, a multi-interval flipped Legendre-Gauss-Radau collocation method is implemented for temporal discretization alongside a Galerkin finite element spatial discretization. The finite element discretization allows for a reduction in problem size and avoids the redefinition of constraints required under a previous method. Further, a generalization of a Kirchoff transformation is performed to handle variational form nonlinearities in the context of numerical optimization. Due to the correspondence between the collocation points and the applied boundary conditions, the multi-interval flipped Legendre-Gauss-Radau collocation method is demonstrated to be preferable over the standard Legendre-Gauss-Radau collocation method for optimal control problems governed by parabolic partial differential equations. The details of the resulting transcription of the optimal control problem into a nonlinear programming problem are provided. Numerical examples demonstrate that the use of a multi-interval flipped Legendre-Gauss-Radau temporal discretization can lead to a reduction in the required number of collocation points to compute accurate values of the optimal objective in comparison to other methods. Lastly, a self-convergence analysis on each test problem illustrates that the error decays exponentially as a function of the mesh size in both the temporal and spatial dimensions.

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arXiv2025

Optimal Multi-Mode Propulsion Mission Design Using Direct Collocation

The problem of minimizing the transfer time between periodic orbits in the Earth-Moon elliptic restricted three-body problem using a multi-mode propulsion system is considered. By employing the true anomaly on the primary orbit as the independent variable and introducing normalized time as an additional state, the need to repeatedly solve Kepler's equation at arbitrary epochs is eliminated. Furthermore, a propellant constraint is imposed on the high-thrust mode to activate the multi-mode capabilities of the system and balance efficiency with maneuverability. The minimum-time optimal control problem is formulated as a three-phase trajectory consisting of a coast along the initial periodic orbit, a controlled transfer, and a coast along the terminal periodic orbit. The three-phase optimal control problem is then solved using an adaptive Gaussian quadrature direct collocation method. Case studies are presented for transfers from an L2 southern halo orbit to a near-rectilinear halo orbit, analyzing the impact of different single- and multi-mode propulsion architectures and varying propellant constraint values. Finally, the methodology developed in this paper provides a systematic framework for generating periodic orbit transfers in three-body systems using single- and multi-mode propulsion systems.

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arXiv2025

Integral Form of Legendre-Gauss-Lobatto Collocation for Optimal Control

A new method is described for solving optimal control problems using direct collocation at Legendre-Gauss-Lobatto points. The approach of this paper employs a polynomial approximation of the right-hand side vector field of the differential equations and leads to the following important outcomes. First, the first-order optimality conditions of the LGL integral form are derived, which lead to a full-rank transformed adjoint system and novel costate estimate. Next, a derivative-like form of the LGL collocation method is obtained by multiplying the system by the inverse of an appropriate full-rank block of the integration matrix. The first-order optimality conditions of the LGL derivative-like form are then derived, leading to an equivalent full-rank transformed adjoint system and secondary novel costate estimate which is related to the costate estimate of the integral form via a linear transformation. Then, it is shown that a second integral form can be constructed by including an additional noncollocated support point, but such a point is superfluous and has no impact on the solution to the nonlinear programming problem. Finally, the method is demonstrated on two benchmark problems: a one-dimensional initial value optimal control problem with an analytic solution and a time-variant orbit raising optimal control problem.

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arXiv2024

Adaptive Mesh Refinement and Error Estimation Method for Optimal Control Using Direct Collocation

An adaptive mesh refinement and error estimation method for numerically solving optimal control problems is developed using Legendre-Gauss-Radau direct collocation. In regions of the solution where the desired accuracy tolerance has not been met, the mesh is refined by either increasing the degree of the approximating polynomial in a mesh interval or dividing a mesh interval into subintervals. In regions of the solution where the desired accuracy tolerance has been met, the mesh size may be reduced by either merging adjacent mesh intervals or decreasing the degree of the approximating polynomial in a mesh interval. Coupled with the mesh refinement method described in this paper is a newly developed relative error estimate that is based on the differences between solutions obtained from the collocation method and those obtained by solving initial-value and terminal-value problems in each mesh interval using an interpolated control obtained from the collocation method. Because the error estimate is based on explicit simulation, the solution obtained via collocation is in close agreement with the solution obtained via explicit simulation using the control on the final mesh, which ensures that the control is an accurate approximation of the true optimal control. The method is demonstrated on three examples from the open literature, and the results obtained show an improvement in final mesh size when compared against previously developed mesh refinement methods.

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arXiv2024

Desensitized Optimal Guidance Using Adaptive Radau Collocation

An optimal guidance method is developed that reduces sensitivity to parameters in the dynamic model. The method combines a previously developed method for guidance and control using adaptive Legendre-Gauss-Radau (LGR) collocation and a previously developed approach for desensitized optimal control. Guidance updates are performed such that the desensitized optimal control problem is re-solved on the remaining horizon at the start of each guidance cycle. The effectiveness of the method is demonstrated on a simple example using Monte Carlo simulation. It is found that the method reduces variations in the terminal state as compared to either desensitized optimal control without guidance updates or a previously developed method for optimal guidance and control.

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arXiv2024

Modified Legendre-Gauss Collocation Method for Solving Optimal Control Problems with Nonsmooth Solutions

A modified form of Legendre-Gauss orthogonal direct collocation is developed for solving optimal control problems whose solutions are nonsmooth due to control discontinuities. This new method adds switch-time variables, control variables, and collocation conditions at both endpoints of a mesh interval, whereas these new variables and collocation conditions are not included in standard Legendre-Gauss orthogonal collocation. The modified Legendre-Gauss collocation method alters the search space of the resulting nonlinear programming problem and enables determining accurately the location of the nonsmoothness in the optimal control. The transformed adjoint system of the modified Legendre-Gauss collocation method is then derived and shown to satisfy a discrete form of the continuous variational necessary conditions for optimality. The method is motivated via a control-constrained triple-integrator minimum-time optimal control problem where the solution possesses a two-switch bang-bang optimal control structure. In addition, the method developed in this paper is compared with existing Gaussian quadrature collocation methods. The method developed in this paper is shown to be capable of accurately solving optimal control problems with a discontinuous optimal control.

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arXiv2024

Numerical Optimization Study of a Constrained Hypersonic Reentry Vehicle

The trajectory optimization of the atmospheric entry of a reusable launch vehicle is studied. The objective is to maximize the crossrange of the vehicle subject to two control-inequality path constraints, two state-inequality path constraints, and one mixed state-and-control inequality path constraint. In order to determine the complex switching structure in the activity of the path constraints, a recently developed method for solving state-path constrained optimal control problems is used. This recently developed method is designed to algorithmically locate the points of activation and deactivation in the path constraints and partition the domain of the independent variable into subdomains based on these activation and deactivation points. Additionally, in a domain where a state-inequality path constraint is found to be active, the method algorithmically determines and enforces the additional necessary conditions that apply on the constrained arc. A multiple-domain formulation of Legendre-Gauss-Radau direct collocation is then employed to transcribe the optimal control problem into a large sparse nonlinear programming problem. Two studies are performed which analyze a variety of problem formulations of the hypersonic reusable launch vehicle. Key features of the constrained trajectories are presented, and the method used is shown to obtain highly accurate solutions with minimal user intervention.

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arXiv2024

Method for Solving State-Path Constrained Optimal Control Problems Using Adaptive Radau Collocation

A new method is developed for accurately approximating the solution to state-variable inequality path constrained optimal control problems using a multiple-domain adaptive Legendre-Gauss-Radau collocation method. The method consists of the following parts. First, a structure detection method is developed to estimate switch times in the activation and deactivation of state-variable inequality path constraints. Second, using the detected structure, the domain is partitioned into multiple-domains where each domain corresponds to either a constrained or an unconstrained segment. Furthermore, additional decision variables are introduced in the multiple-domain formulation, where these additional decision variables represent the switch times of the detected active state-variable inequality path constraints. Within a constrained domain, the path constraint is differentiated with respect to the independent variable until the control appears explicitly, and this derivative is set to zero along the constrained arc while all preceding derivatives are set to zero at the start of the constrained arc. The time derivatives of the active state-variable inequality path constraints are computed using automatic differentiation and the properties of the chain rule. The method is demonstrated on two problems, the first being a benchmark optimal control problem which has a known analytical solution and the second being a challenging problem from the field of aerospace engineering in which there is no known analytical solution. When compared against previously developed adaptive Legendre-Gauss-Radau methods, the results show that the method developed in this paper is capable of computing accurate solutions to problems whose solution contain active state-variable inequality path constraints.

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arXiv2022

Minimum-Fuel Earth-Based Orbit Transfers Using Multiple-Domain Adaptive Radau Collocation

A numerical optimization study of minimum-fuel Earth-based orbital transfers from low-Earth orbit (LEO) to either medium-Earth orbit (MEO), high-Earth orbit (HEO), or geostationary orbit (GEO), is performed. Various values of maximum allowable thrust acceleration are considered for each type of transfer (LEO-to-MEO, LEO-to-HEO, or LEO-to-GEO). A key aspect of the study performed in this paper is that the optimal thrusting structure is not assumed to be known a priori, but is determined as part of the solution process. In order to determine the optimal thrusting structure, a recently developed bang-bang and singular optimal control (BBSOC) method is employed together with multiple-domain Legendre-Gauss-Radau quadrature collocation. Key results obtained in this study include not only the number of switches in the optimized thrust, but also the total impulse. Furthermore, it is found that, as the maximum allowable thrust acceleration decreases, the total impulse is less than the total impulse obtained from a previous study where a burn-coast-burn thrusting structure was assumed a priori. For each type of transfer a particular value of maximum allowable thrust acceleration is chosen to highlight in more detail the key features of the optimal solutions. This study provides improved results over previous studies and provides improved insight into the optimal thrusting structure required in order to accomplish each type of orbital transfer using the least amount of fuel.

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arXiv2022

Minimum-Time Reorientation of Axisymmetric Rigid Spacecraft Using Three Controls

A minimum-time reorientation of an axisymmetric rigid spacecraft controlled by three torques is studied. The orientation of the body is modeled such that the attitude kinematics are representative of a spin-stabilized spacecraft. The optimal control problem considered is shown to have a switching control structure. Moreover, under certain assumptions, the solutions contain segments that lie on a singular arc. A numerical optimization study is performed using a recently developed method that is designed to accurately solve bang-bang and singular optimal control problems. The optimality conditions for the resulting optimal control problem are derived and analyzed for a variety of cases. Also, the results obtained in this study are compared to a previous method existing in the literature. The key features of the optimized trajectories and controls are identified, and the aforementioned method for solving bang-bang and singular optimal control problems is shown to efficiently and accurately solve the problem under consideration.

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arXiv2022

Method for Solving Bang-Bang and Singular Optimal Control Problems using Adaptive Radau Collocation

A method is developed for solving bang-bang and singular optimal control problems using adaptive Legendre-Gauss-Radau (LGR) collocation. The method is divided into several parts. First, a structure detection method is developed that identifies switch times in the control and analyzes the corresponding switching function for segments where the solution is either bang-bang or singular. Second, after the structure has been detected, the domain is decomposed into multiple domains such that the multiple-domain formulation includes additional decision variables that represent the switch times in the optimal control. In domains classified as bang-bang, the control is set to either its upper or lower limit. In domains identified as singular, the objective function is augmented with a regularization term to avoid the singular arc. An iterative procedure is then developed for singular domains to obtain a control that lies in close proximity to the singular control. The method is demonstrated on four examples, three of which have either a bang-bang and/or singular optimal control while the fourth has a smooth and nonsingular optimal control. The results demonstrate that the method of this paper provides accurate solutions to problems whose solutions are either bang-bang or singular when compared against previously developed mesh refinement methods that are not tailored for solving nonsmooth and/or singular optimal control problems, and produces results that are equivalent to those obtained using previously developed mesh refinement methods for optimal control problems whose solutions are smooth.

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arXiv2021

End-to-End Ascent-Entry Mission Performance Optimization Using Gaussian Quadrature Collocation

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 vehicle onto a suborbital exo-atmopheric 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. A key aspect of the optimized trajectories is that, for particular ranges of maximum allowable heating rate and heating load during entry, relatively small adjustments made during ascent can potentially decrease the control effort required during atmospheric entry. Outside of these ranges for maximum allowable heating rate and heating load, however, it is found that the required control effort increases and eventually saturates the commanded angle of attack upon initial descent. 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.

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arXiv2021

Nonsingular Euler Parameterizations for Motion of a Point Mass in Atmospheric Flight

Three parameterizations are developed for modeling translational motion of a point mass in atmosphere flight over a central rotating body. Unlike well-known parameterizations such as spherical coordinate parameterizations, where position and velocity are parameterized using a magnitude an an Euler angle rotation sequence, the method presented in this research employs Euler parameters. Consequently, singularities and trigonometric functions are eliminated from the differential equations of motion. As a result, the new parameterizations presented in this paper offer computational advantages over standard parameterizations that employ Euler angle sequences. Finally, an example is studied where an atmospheric vehicle moves while in vertical flight, demonstrating the nonsingular nature of the formulations developed in this paper.

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arXiv2021

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 $0.00098\times 10^{-4}~\textrm{m}\cdot\textrm{s}^{-2}$, it was found that the minimum times for cases (1), (2), and (3) are, respectively, 215 d, 196 d, 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.

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arXiv2020

Nonsingular Parameterization for Modeling Translational Motion Using Euler Parameters

A parameterization is described for quantifying translational motion of a point in three-dimensional Euclidean space. The parameterization is similar to well-known parameterizations such as spherical coordinates in that both position and velocity are decoupled into magnitude and orientation components. Unlike these standard parameterizations, where principal rotation sequences are employed, the method presented in this research employs Euler parameters. By using Euler parameters instead of Euler angles, singularities and trigonometric functions are removed from the equations of motion. The parameterization is demonstrated on two examples, where it is found that the new parameterization offers both mathematical and computational advantages over other commonly used parameterizations.

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arXiv2020

Modified Legendre-Gauss-Radau Collocation Method for Solving Optimal Control Problems with Nonsmooth Solutions

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre-Gauss-Radau orthogonal direct collocation method. This modified Legendre-Gauss-Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre-Gauss-Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh interval. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre-Gauss-Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass-Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.

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