Binfeng Pan

Highly Constrained Optimal Launch Ascent Guidance

This paper is concerned with ascent guidance of multiple-stage launch vehicles subject to aerodynamic bending-moment constraint. In the first part of the paper, several new developments are made to refine and expand the capability of an optimal endoatmospheric ascent-guidance algorithm. These include new conditions for determining optimal body axes of the vehicle, handling of the aerodynamic bending-moment constraint by automated optimization of gravity-turn trajectory or by optimal closed-loop guidance, and seamless integration of the endoatmospheric guidance algorithm with exoatmospheric ascent-guidance algorithm for the upper stages of a vehicle. An extensive application ofthese developments using the vehicle and mission data ofthe Ares I Crew Launch Vehicle constitutes the second part of the paper. Extensive Monte Carlo simulations are conducted to demonstrate the algorithm and compare optimal closed-loop ascent guidance with conventional open-loop ascent guidance. The testing results help answer an age-old question on the comparison of closed-loop and open-loop ascent guidance on performance and constraint enforcement in the presence of winds and dispersions.

Double-Homotopy Method for Solving Optimal Control Problems

The homotopy method has long served as a useful tool in solving optimal control problems, particularly highly nonlinear and sensitive ones for which good initial guesses are difficult to obtain, such as some of the well-known problems in aerospace trajectory optimization. However, the traditional homotopy method often fails midway: a fact that occasional practitioners are not aware of, and a topic which is rarely investigated in aerospace engineering. This paper first reviews the main reasons why traditional homotopy fails. A new double-homotopy method is developed to address the common failures of the traditional homotopy method. In this approach, the traditional homotopy is employed until it encounters a difficulty and stops moving forward. Another homotopy originally designed for finding multiple roots of nonlinear equations takes over at this point, and it finds a different solution to allow the traditional homotopy to continue on. This process is repeated whenever necessary. The proposed method overc...

Reduced Transversality Conditions in Optimal Space Trajectories

A long-standing challenge in optimal guidance for orbital insertion of launch vehicles and orbital transfers of spacecraft is how to reduce the transversality conditions in the optimal control problem. The reduction seeks to eliminate the unknown multipliers in the transversality conditions for much improved robustness of convergence and computational speed. This paper injects a new perspective into how to approach the problem. It is shown that, for commonly seen three-dimensional optimal space flight problems, a list of six candidate-reduced transversality conditions can be precompiled (and is done so in the paper). For a given particular problem, the required reduced transversality conditions can be chosen from the list according to which of the final orbital elements are unconstrained by the specified terminal conditions. The two major achievements of this paper are to provide a “prepackaged” solution to this historically difficult problem, and to offer new insight into the interpretation of reduced tr...

Improvements to Optimal Launch Ascent Guidance

This paper presents enhancements for optimal endo- and exo-atmospheric ascent guidance. For 3-dimensional endo-atmospheric ascent, based on analysis of the necessary conditions for the optimal solution, the costate equations are formulated differently as compared to previous work. This change considerably simplifies the vector costate equations, and more importantly, helps avoid a potential numerical difficulty the previous formulation can encounter when the angle of attack is zero. For optimal exo-atmospheric ascent problems with multi-burn and -coast arcs, multiple-shooting methods have proven to be effective. But a main difficulty has been the generation of initial guesses for the burn times, coast times and the state and costate at the beginnings and ends of all the burn and coast arcs. In this paper an algorithm is developed to automate the solution process for such a problem with arbitrary given number of multi-burn and coast arcs, without the need for user-supplied initial guesses. These developments are demonstrated with numerical experiments.

Rapid Optimization of Multiburn Rocket Trajectories Revisited

This paper revisits the challenging problem of optimal rocket powered trajectory consisting of many burn and coast arcs in space. The problem is formulated as a series of N pairs of coast and finite-time burn arcs. The numerical solution to this problem is obtained by an analytical multiple shooting method. A major focus of this work is the treatment of constraints on the durations of the burn and coast arcs. These constraints arise from the requirements of non-negativeness of the flight time in each arc, upper bounds on burn times, and equality constraints on the sums of burns times across adjacent burn arcs. These constraints are treated as interior-point constraints on time. Unlike in previous work, analyses are conducted on the associated necessary conditions to determine applicable conditions without introducing additional unknowns or complexity. As a result, our algorithm has the notable ability to allow any number of arcs and conveniently handle the case when some of the durations of the arcs reduce to zero. Thus even without the knowledge of optimal number of arcs a priori in a problem, by starting with sufficiently large N ,o ur algorithm can converge to the optimal solution with correct number of arcs by nulling the unnecessary arcs. Numerical demonstrations in orbital transfer and ascent guidance are provided.

Coast Arcs in Optimal Multiburn Orbital Transfers

for the correct switching time and suffer from potential failures, this paper shows that the correct zero of the switching function, with each of the gravity modelings, can either be obtained in closed form or determined accurately and rapidly in a fail-safe way by solving a polynomial of, at most, degree 5. Numerical examples provide clear comparison of the results and show how, when other methods fail, the proposed approach still ensures the correct solution.

Automatic load relief numerical predictor-corrector guidance for low L/D vehicles return from low Earth orbit

The poor maneuverability of low lift-to-drag ratio (L/D) vehicles results in low precision of reference-trajectory guidance and difficulty in meeting load constraints. This paper presents an automatic load relief numerical predictor-corrector method for the guidance of low L/D vehicles return from low Erath orbit. By elaborately designing a bank-angle profile in each guidance circle and selecting appropriate iteration parameters, the goal of automatic load relief is achieved, which greatly reduces the maximum peak load. With the use of coupled guidance and landing error feedback algorithms, the guidance precision is improved. Aerodynamic coefficients of the vehicle and landing errors are filtered to further increase the robustness of the algorithm. Extensive Monte Carlo simulations are conducted to evaluate and verify the design features of the algorithm. The test results show that the algorithm consistently offers very satisfactory performance even in highly dispersed cases. Such an algorithm holds distinct potential for onboard applications.

A quadratic homotopy method for fuel-optimal low-thrust trajectory design

Solving fuel-optimal low-thrust trajectory problems is a long-standing challenging topic, mainly due to the existence of discontinuous bang–bang controls and small convergence domain. Homotopy methods, the principle of which is to embed a given problem into a family of problems parameterized by a homotopic parameter, have been widely applied to address this difficulty. Linear homotopy methods, the homotopy functions of which are linear functions of the homotopic parameter, serve as useful tools to provide continuous optimal controls during the homotopic procedure with an energy-optimal low-thrust trajectory optimization problem as the starting point. However, solving energy-optimal problem is still not an easy task, particularly for the low-thrust orbital transfers with many revolutions or asteroids flyby, which is typically solved by other advanced numerical optimization algorithms or other homotopy methods. In this paper, a novel quadratic homotopy method, the homotopy function of which is a quadratic function of the homotopic parameter, is presented to circumvent this possible difficulty of solving the initial problem in the existing linear homotopy methods. A fixed-time full-thrust problem is constructed as the starting point of this proposed quadratic homotopy, the analytical solution of which can be easily obtained under a modified linear gravity approximation formulation. The criterion of energy-optimal problem is still involved in the homotopic procedure to provide continuous optimal controls until the original fuel-optimal problem is solved. Numerical demonstrations in an Earth to Venus rendezvous problem, a geostationary transfer orbit (GTO) to geosynchronous orbit (GEO) orbital transfer problem with many revolutions, and an Earth to Mars rendezvous problem with an asteroid flyby are presented to illustrate the applications of this proposed homotopy method.

Rapid Prototyping of a Guidance and Control System for Missiles

This paper describes rapid prototyping of a guidance and control system for missiles to improve its overall design process eectiveness. A rapid prototyping design process is developed based on MATLAB/Simulink/RTW and Skyfly. An air-to-surface missile is used as a design example. A high fidelity 6-DOF missile simulation model is built for rapid prototyping test and a trimmed simulation model is presented as validation and linear equations of motion are provided for guidance and control analysis. Simulation study results indicate that the proposed rapid prototyping process is practically feasible for eectively developing a guidance and control system of an air-to-surface missile.

Robust Trajectory Tracking Guidance for Low L/D Lunar Return Vehicles Using Command Filtered Backstepping Approach

AbstractA reentry guidance method for low lift-to-drag (L/D) ratio vehicles in a lunar return mission is presented. Different from the reference drag tracking scheme, method in this paper tracks a ...