Fuel-Optimal Ascent Trajectory Problem for Launch Vehicle via Theory of Functional Connections

Shiyao Li,Xinguo Li,Yushen Yan,Kun Zhang,Erdal Kayacan

doi:10.1155/2021/2734230

Abstract

In this study, we develop a method based on the Theory of Functional Connections (TFC) to solve the fuel-optimal problem in the ascending phase of the launch vehicle. The problem is first transformed into a nonlinear two-point boundary value problem (TPBVP) using the indirect method. Then, using the function interpolation technique called the TFC, the problem’s constraints are analytically embedded into a functional, and the TPBVP is transformed into an unconstrained optimization problem that includes orthogonal polynomials with unknown coefficients. This process effectively reduces the search space of the solution because the original constrained problem transformed into an unconstrained problem, and thus, the unknown coefficients of the unconstrained expression can be solved using simple numerical methods. Finally, the proposed algorithm is validated by comparing to a general nonlinear optimal control software GPOPS-II and the traditional indirect numerical method. The results demonstrated that the proposed algorithm is robust to poor initial values, and solutions can be solved in less than 300 ms within the MATLAB implementation. Consequently, the proposed method has the potential to generate optimal trajectories on-board in real time.

Highlights

With the recent development in space exploration, launch vehicles are very important as they are the only means for humans to explore space from the earth
A launch vehicle mission has been planned over a long period, and the trajectory was designed in advance, and it cannot be updated during flight, which means it is not robust or flexible
We proposed a new approach to solve the fueloptimal problem in the ascending phase of the launching vehicle using Theory of Functional Connections (TFC)

Summary

Introduction

With the recent development in space exploration, launch vehicles are very important as they are the only means for humans to explore space from the earth. The primary aim of the trajectory planning algorithm is to solve the optimal control problem that is generally based on nonlinear dynamics, which achieves specific performance indicators under the constraints of state and control variables. The solution of such problems is mainly achieved using the indirect method [1,2,3] and the direct method [4,5,6]. The sequential convex optimization algorithm is a convexification method based on linearization, which increases the dependence on the reference trajectory.

Theory of Functional Connections

Fuel-Optimal Problem in the Ascending Phase of the Launch Vehicle

Solution via TFC

Simulations

Conclusion