State Transition Tensors Research Articles

Semi-Analytic Probability Density Function for System Uncertainty

In general, the behavior of science and engineering is predicted based on nonlinear math models. Imprecise knowledge of the model parameters alters the system response from the assumed nominal model data. One proposes an algorithm for generating insights into the range of variability that can be expected due to model uncertainty. An automatic differentiation tool builds the exact partial derivative models required to develop a state transition tensor series (STTS)-based solution for nonlinearly mapping initial uncertainty models into instantaneous uncertainty models. The fully nonlinear statistical system properties are recovered via series approximations. The governing nonlinear probability distribution function is approximated by developing an inverse mapping algorithm for the forward series model. Numerical examples are presented, which demonstrate the effectiveness of the proposed methodology.

Modified Picard Integrator for Spaceflight Mechanics

This paper investigates the applicability of the Parker–Sochacki Picard iteration method for spaceflight mechanics. Notably, solving initial value problems, parallel integration of multiple trajectories, and high-order state transition tensor computations are addressed. This paper presents a systematic approach to transforming vector fields to polynomial form, which is required for the method investigated. The application of this approach is presented on a few vector fields, including spherical harmonics gravity field. The initial cost of transforming the vector field can be justified by the performance of the initial value problem solver. Additionally, this polynomial form of the vector field is used in the calculation of state transition tensors with differentiation of polynomials only. This reduces the computations required to obtain state transition tensors for some problems and is discussed in this paper. An efficient representation of vector fields in polynomial form benefits the implementation of the method for the parallel architecture of graphic processing units. The performance of the parallel integration method is discussed along with the execution of the algorithm on a graphic processing unit and a central processing unit.

Analytical Nonlinear Propagation of Uncertainty in the Two-Body Problem

One topic of recent interest in the field of space situational awareness is the accurate and consistent representation of an observed object’s uncertainty under nonlinear dynamics. This paper presents amethod of analytical nonlinear propagation of uncertainty under two-bodydynamics. In particular, the probability density function over state space and itsmean and covariancematrix are expressed analytically for all time via a special solution of the Fokker-Planck equations for deterministic Hamiltonian systems. The state transition tensor concept is used to express the solution flow of the dynamics. Some numerical examples, where a second-order state transition tensor is found to sufficiently capture the nonlinear effects, are also discussed.

A high order method for estimation of dynamic systems

An analytical approach is presented for developing an estimation framework (called the Jth Moment Extended Kalman Filter (JMEKF)). This forms an important component of a class of architectures under investigation to study the interplay of major issues in nonlinear estimation such as model nonlinearity, measurement sparsity and initial condition uncertainty in the presence of low process noise. Utilizing an automated nonlinear expansion of the model about the current best estimated trajectory, a Jth order approximate solution for the departure motion dynamics about a nominal trajectory is derived in the form of state transition tensors. This solution is utilized in evaluating the evolution of statistics of the departure motion as a function of the statistics of initial conditions. The statistics thus obtained are used in the determination of a state estimate assuming a Kalman update structure. Central to the state transition tensor integration about a nominal trajectory, is the high order sensitivity calculations of the nonlinear models (dynamics and measurement), being automated by OCEA (Object Oriented Coordinate Embedding Method), a computational tool generating the required various order partials of the system differential equations without user intervention. Working in tandem with an OCEA automation of the derivation of the state transition tensor differential equations is a vector matrix representation structure of tensors of arbitrary rank, facilitating faster and more accurate computations. High order moment update equations are derived to incorporate the statistical effects of the innovations process more rigorously, improving the effectiveness of the estimation scheme. Numerical simulations on an orbit estimation example investigate the gain obtained in using the proposed methodology in situations where the classical extended Kalman filter’s domain of convergence is smaller. The orbit estimation example presented examines a situation that requires us to determine the position and velocity state of the orbiter from range, azimuth and elevation measurements being made available sparsely.

Nonlinear Semi-Analytic Methods for Trajectory Estimation

Nonlinear semi-analytic filtering methods to sequentially estimate spacecraft states and their associated uncertainties are presented. We first discuss the state transition tensors that characterize the localized nonlinear behavior of the trajectory statistics and illustrate the importance of higher-order effects on orbit uncertainty propagation. We then present a semi-analytic filtering method by implementing the state transition tensors to sequentially updatethe filter informationwith contributionsfromeachmeasurement, whichrequiresnointegration once the tensors are computed. A sun–Earth halo orbit about the L1 point is considered as an example with realistic orbit uncertainties, and the results are compared with the extended Kalman filter and unscented Kalman filter.