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The Tiger Optimization Software - A Pseudospectral Optimal Control Package
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The fields of trajectory optimization and optimal control are closely connected. Practical trajectory optimization techniques are built on the indirect and direct methods for solving optimal control problems. Indirect methods seek to satisfy the first-order necessary conditions of optimality, while direct methods discretize the problem and transcribe it into a large-scale nonlinear programming (NLP) problem. This study provides a basic overview of the direct and indirect methods. The indirect necessary conditions and Pontryagin's Minimum Principle are reviewed, and the approaches taken by direct methods are presented. The focus of this study then shifts to pseudospectral direct methods, which belong to the class of collocation methods. The theoretical groundwork of pseudospectral methods is laid, leading to a pseudospec-tral formulation and software for solving general optimal control problems. Several types of pseudospectral methods are presented, including the Legendre-Gauss and Chebyshev-Gauss methods. Special attention is given to the Legendre-Gauss-Radau (LGR) method, which is the primary transcription employed by the MATLAB-based Tiger Optimization Software (TOPS). TOPS is a general-purpose pseudospectral optimal control software developed by the author as part of their research in the Aero-Astro Computational and Experimental (ACE) Lab. A multi-interval pseudospectral method is presented, and the discrete form of the optimality conditions are derived. The study shifts focus from theory to application, and the practical aspects of pseudospectral optimal control methods are discussed. Another objective of this research is to compile the author's knowledge with regard to implementing pseudospectral techniques, thus enabling the reader to easily implement their own pseudospectral method. Several mesh refinement methods are compared and their merits are compared. In addition, several techniques that improve the efficiency of an NLP solver are presented, including efficient and exact derivative formulas and several scaling techniques. Finally, several example optimal control problems are solved using TOPS. The solutions obtained from TOPS are compared to the solutions obtained using indirect techniques to verify their accuracy and demonstrate the capabilities of TOPS.
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Rapid and accurate rendezvous and proximity operations for spacecraft are crucial to the success of most space missions. In this paper, a sequential convex programming method, combined with the first-order and second-order Birkhoff pseudospectral methods, is proposed for the autonomous rendezvous and proximity operations of spacecraft. The original nonlinear and nonconvex close-range rendezvous problem with thrust constraints and no-fly zone constraints is converted into its convex version by using the sequential convexification techniques; then, the Birkhoff pseu-dospectral method is used to transcribe the dynamic constraints into a series of linear algebraic equality constraints, in other words, a convex second-order conic programming problem with a relatively small condition number. Thus, the resulting problem can be accurately and efficiently solved by a convex solver. The simulation results indicate that the proposed methods, especially the second-order Birkhoff pseudospectral method, have obvious advantages over other methods in computational efficiency and sensitivity.
This study investigates a complicated reentry trajectory optimization problem for a reusable launch vehicle (RLV) in the presence of two control constraints and three state path constraints. Upon using traditional indirect methods, the considered RLV reentry trajectory optimization problem is converted into a complicated 12-point boundary-value problem that requires a priori knowledge about the structure of control and state constraints. Instead, a recently developed indirect method, the uniform trigonometrization method (UTM) is used, which leads to a two-point boundary-value problem, making the solution procedure notably easier. New features are introduced into the UTM framework in order to incorporate mixed state-control path constraints. Two scenarios are considered in which the control and state path constraints are 1) inactive and 2) active, respectively. A numerical continuation strategy involving backward-in-time propagation of the dynamics from the final point to the initial point with sequential implementation of the path constraints is proposed. The results are validated with a direct pseudospectral method. This study is the first instance of applying the UTM to a three-degree-of-freedom RLV reentry trajectory optimization problem in the presence of multiple linear and polynomial-form control inputs and mixed state-control path constraints.
A method is developed for solving bang-bang and singular optimal control problems using adaptive Legendre–Gauss–Radau 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.
Closed-loop feedback-driven control laws can be used to solve low-thrust many-revolution trajectory design and guidance problems with minimal computational cost. Lyapunov-based control laws offer the benefits of increased stability whilst their optimality can be increased by tuning their parameters. In this paper, a reinforcement learning framework is used to make the parameters of the Lyapunov-based Q-law state-dependent, increasing its optimality. The Jacobian of these state-dependent parameters is available analytically and, unlike in other optimisation approaches, can be used to enforce stability throughout the transfer. The results focus on GTO-GEO and LEO-GEO transfers in Keplerian dynamics, including the effects of eclipses. The impact of the network architecture on the behaviour is investigated for both time- and mass-optimal transfers. Robustness to navigation errors and thruster misalignment is demonstrated using Monte Carlo analyses. The resulting approach offers potential for on-board autonomous transfers and orbit reconfiguration.
Indirect formalism of optimal control theory is used to generate minimum-time and minimum-fuel trajectories for formation of two spacecraft (deputies) relative to a chief satellite. For minimum-fuel problems, a hyperbolic tangent smoothing method is used to facilitate numerical solution of the resulting boundary-value problems by constructing a one-parameter family of smooth control profiles that asymptotically approach the theoretically optimal, but non-smooth bang-bang thrust profile. Impact of the continuation parameter on the solution of minimum-fuel trajectories is analyzed. The fidelity of the dynamical model is improved beyond the two-body dynamics by including the perturbation due to the Earth’s second zonal harmonic, J 2 . In addition, a particular formation is investigated, where the deputies are constrained to lie diametrically opposite on a three-dimensional sphere centered at the chief.
Variational approach to optimal control theory converts trajectory optimization problems into two- or multiple-point boundary-value problems, which consist of costates (i.e., Lagrange multipliers associated with the states). Estimating missing values of the non-intuitive costates is an important step in solving the resulting boundary-value problems. By leveraging costate vector mapping theorem, we extend the method of Adjoint Control Transformation (ACT), called Mapped ACT (MACT), to alternative sets of coordinates/elements for solving low-thrust trajectory optimization problems. In particular, extension of the ACT method to the set of modified equinoctial elements and an orbital element set based on the specific angular momentum and eccentricity vectors (h-e) is demonstrated. The computational and robustness efficiency of the MACT method is compared against the traditionally used random initialization of costates by solving 1) interplanetary rendezvous maneuvers and 2) an Earth-centered, orbit-raising problem with and without the inclusion of J2 perturbation. For the considered problems, numerical results indicate two to three times improvement in the percent of convergence of the resulting boundary-value problems when the MACT method is used compared to the random initialization method. Results also indicate that the h-e set is also a contender and suitable choice for solving low-thrust trajectory optimization problems.
Space mission design places a premium on cost and operational efficiency. The search for new science and life beyond Earth calls for spacecraft that can deliver scientific payloads to geologically rich yet hazardous landing sites. At the same time, the last four decades of optimization research have put a suite of powerful optimization tools at the fingertips of the controls engineer. As we enter the new decade, optimization theory, algorithms, and software tooling have reached a critical mass to start seeing serious application in space vehicle guidance and control systems. This survey paper provides a detailed overview of recent advances, successes, and promising directions for optimization-based space vehicle control. The considered applications include planetary landing, rendezvous and proximity operations, small body landing, constrained attitude reorientation, endo-atmospheric flight including ascent and reentry, and orbit transfer and injection. The primary focus is on the last ten years of progress, which have seen a veritable rise in the number of applications using three core technologies: lossless convexification, sequential convex programming, and model predictive control. The reader will come away with a well-rounded understanding of the state-of-the-art in each space vehicle control application, and will be well positioned to tackle important current open problems using convex optimization as a core technology.
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 9.8×10−4 m⋅s−2, 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.