Higher Order Algorithm for Solving Lambert’s Problem

Abstract

This work presents a high-order perturbation expansion method for solving Lambert’s problem. The necessary condition for the problem is defined by a fourth-order Taylor expansion of the terminal error vector. The Taylor expansion partial derivative models are generated by Computational Differentiation (CD) tools. A novel derivative enhanced numerical integration algorithm is presented for computing nonlinear state transition tensors, where only the equation of motion is coded. A high-order successive approximation algorithm is presented for inverting the problems nonlinear necessary condition. Closed-form expressions are obtained for the first, second,third, and fourth order perturbation expansion coefficients. Numerical results are presented that compare the convergence rate and accuracy of first-through fourth-order expansions. The initial p-iteration starting guess is used as the Lambert’s algorithm initial condition. Numerical experiments demonstrate that accelerated convergence is achieved for the second-, third-, and fourth-order expansions, when compared to a classical first-order Newton method.

Appendix A: Index Equation Based Model

State transition tensor model naturally consists of objects that represent multidimensional arrays. Unfortunately, conventional numerical integrators are designed to work with systems defined by a vector-based array model. The works for the state part of the integration model which is an (n × 1) vector system. To make use of the vector-based integration algorithms, the multidimensional array tensor objects are mapped to a vector data structure. This step required that the multidimensional array objects are packed and unpacked several times during the current integration time step. The packing and unpacking operations are valid but very inefficient computationally. The packing/unpacking operations are eliminated by introducing a user defined data structure. The data structure consists of the following generalized scalar function (1 × 1):

α := {t, x, φ₁, φ₂, φ₃, φ₄} where the internal data structure consists of five named sub-components, including: t which denotes time (double precision) in the simulation, x denotes the (n × 1) state vector (double precision derivative enhanced variables), and φ₁, φ₂, φ₃, and φ₄ denote double precision first (n × n) through fourth (n × n × n × n × n) order state transition tensors. The unpacking and packing operations are eliminated by overloading generalized scalar structure for scalar multiplication, addition, subtraction, and division operations to allow derivative approximations to be computed. This approach allows all numerical calculations are performed in the native tensor dimension state. The derivative enhanced state vector enables the calculation of state vector gradient operations ∇f , ∇²f, ∇³f, and ∇⁴f required for numerically evaluating the state transition tensor ordinary differential equations

$$\begin{array}{@{}rcl@{}}\dot{x}_{i,j} &=& \nabla_{i,r} x_{r,j}\dot{x}_{i,jk} = \nabla_{i,rs} x_{r,j}x_{s,k} + \nabla_{i,r}x_{r,jk}\\ \dot{x}_{i,jkl} &=& \nabla_{i,rsq} x_{r,j}x_{s,k} x_{q,k} + \nabla_{i,rs} x_{r,j,l}x_{s,k} + \nabla_{i,rs}x_{r,j}x_{s,k,l} + \nabla_{i,rn} x_{r,jk}x_{n,l} + \nabla_{i,r} x_{r,jkl}\\ \dot{x}_{i,jklm} &=& \nabla_{i,rspq} x_{r,j} x_{s,k}x_{p,l}x_{q,m} + \nabla_{i,rsp}(x_{r,jm}x_{s,k}x_{p,l} + x_{r,j}x_{s,km}x_{s,l} + x_{r,j}x_{s,k}x_{p,lm}) \\&&+ \nabla_{i,rsp} (x_{r,jl}x_{s,k} + x_{r,j}x_{s,kl} + x_{r,jk}x_{s,l}) x_{p,m} \\&&+ \nabla_{i,rs}(x_{r,jlm}x_{s,k} + x_{r,jl}x_{s,km} x_{r,jm}x_{s,kl} + x_{r,j}x_{s,klm} + x_{r,jkm}x_{s,l} \\&&+ x_{r,jk}x_{s,lm} + x_{r,jkl}x_{s,m}) + \nabla_{i,r}x_{r,jklm} \end{array} $$

where i, r, j, k, l, m are indices from 1, 2, .. 6. The time derivative of equation α := {t, x, φ₁, φ₂, φ₃, φ₄} is given by $\dot {\alpha }$:= {1, f, $\dot {\varphi _{1}},\dot {\varphi _{2}},\dot {\varphi _{3}},\dot {\varphi _{4}}$} subject to the following initial condition structure

$$\alpha:= \{0,x_{0},I_{6\times6},0_{6\times6\times6},0_{6\times6\times6\times6},0_{6\times6\times6\times6\times6}\}$$

where I_6×6 denotes a (6 × 6) identity matrix. As a result, only a scalar variable appears in all integration and derivative calls. This approach has been successfully implemented in Fortran 2003, and cross validated with other formulations.