Analytic characterization of measurement uncertainty and initial orbit determination on orbital element representations

Abstract

This paper presents an approach to characterize the uncertainty associated with the state vector obtained from the Herrick-Gibbs orbit determination approach using transformation of variables. The approach is applied to estimate the state vector and its probability density function for objects in low Earth orbit using sparse observations. The state vector and associated uncertainty estimates are computed in Cartesian coordinates and Keplerian elements. The approach is then extended to accommodate the $J_2$ perturbation where the state vector is written in terms of mean orbital elements. The results obtained from the analytical approach presented in this paper are validated using Monte Carlo simulations and compared with the often utilized similarity transformation for Kepler, mean, and nonsingular elements. The measurement uncertainty characterization obtained is used to initialize conventional nonlinear filters as well as operate a Bayesian approach for orbit determination and object tracking.

Appendix 1

This appendix demonstrates the TOV technique for instances where the desired domain has the same number of base functions as the initial domain and when the desired domain has less base functions than the initial domain.

1.1 Transformation of variables for $\mathfrak {R}^n\rightarrow \mathfrak {R}^m \, , \, m = n$

Consider the two-dimensional mapping of Cartesian coordinates, $\mathbf{x} \doteq \left( x, y\right)$, from a range and angle pair set, $\mathbf{z}\doteq \left( \rho ,\alpha \right)$, Eq. (26). Let the spherical domain be the initial domain with a given continuous joint PDF, $p\left( \mathbf{z}\right)$, the Cartesian domain PDF, $p\left( \mathbf{x}\right)$, is computed by applying Eq. (1) and the result shown in Eq. (27).

$$\begin{aligned}&\begin{array}{ccl|ccl} \rho &{}=&{} \sqrt{x^2+y^2}&{}x&{}=&{}\rho \cos (\alpha )\\ \alpha &{}=&{} \arctan \left[ \displaystyle \frac{x}{y}\right] &{}y&{}=&{}\rho \sin (\alpha )\\ \end{array} \end{aligned}$$

(26)

$$\begin{aligned}&p\left( \mathbf{x}\right) =p\bigg (\mathbf{z}=\psi \left( \mathbf{x}\right) \bigg )\bigg \vert \rho \bigg \vert _{\mathbf{z}=\psi \left( \mathbf{x}\right) }^{-1} \end{aligned}$$

(27)

1.2 Transformation of variables for $\mathfrak {R}^n\rightarrow \mathfrak {R}^m\, ,\, m < n$ (auxiliary variable method)

When concerned with a subset of variables related to the initial domain, the method of auxiliary variables (Papoulis 1991) can be applied to populate a square Jacobian. Let the auxiliary variable be the angle, $\alpha$, this results in a square Jacobian, Eq. (28), and joint and marginal PDFs shown by Eq. (29). With $\rho$ as the auxiliary variable, Eq. (30) would result. Equations (29) and (30) are derived differently based upon elimination of auxiliary variables but are equivalent due to the invariance of total probability.

$$\begin{aligned}&\displaystyle \frac{\partial \left( x, \alpha \right) }{\partial \left( \rho ,\ \alpha \right) }=\underbrace{ \left[ \begin{array}{cc} \cos (\alpha )&{}-\rho \sin (\alpha )\\ 0&{}1 \end{array}\right] }_{J} \left[ \begin{array}{c} \rho \\ \alpha \end{array}\right] \end{aligned}$$

(28)

$$\begin{aligned}&\displaystyle p\left( x,\ \alpha \right) = p\bigg (\rho = \frac{x}{\cos (\alpha )},\ \alpha = \alpha \bigg )\bigg \vert \cos (\alpha )\bigg \vert ^{-1} \nonumber \\&\displaystyle p\left( x\right) = \displaystyle \int \limits _{\alpha _0}^{\alpha _f}p\left( x,\ \alpha \right) d\alpha \end{aligned}$$

(29)

$$\begin{aligned}&\displaystyle p\left( x,\ \rho \right) =p\bigg (\rho =\rho ,\ \alpha = \arccos (x/\rho )\bigg )\bigg \vert \ \rho \sin (\alpha )\bigg \vert _{\alpha =\arccos (x/\rho )}^{-1} \nonumber \\&\displaystyle p\left( x\right) = \displaystyle \int \limits _{\rho _0}^{\rho _f}p\left( x,\ \rho \right) d\rho \end{aligned}$$

(30)

1.3 Transformation of variables for $\mathfrak {R}^n\rightarrow \mathfrak {R}^m\, ,\, m < n$ (Dirac delta method)

The reduced state variable PDF representation for continuous or discrete random variables can be computed using the Dirac delta method (Au and Tam 1999). The method applies the Dirac generalized function in order to transform only the needed variables to the state(s) of interest, the continuous random variables the application is given by Theorem 1.

Theorem 1

Suppose that $z_i,\,i=\left[ 1,n\right]$, are continuous random variables with joint probability distribution $p\left( z_1,\ z_2, \dots , \ z_n\right)$. Let ${\fancyscript{D}}$ be the $n$-dimensional set of every possible outcome of the $z_i$’s. Then the continuous random variable

$$\begin{aligned} x = \psi ^{-1}\left( z_1,\ z_2, \dots , \ z_n\right) \end{aligned}$$

has the probability distribution given by use of the Kronecker delta $\delta _{a,b}$ in the form

$$\begin{aligned} p\left( x\right) =\displaystyle \int \limits _{{\fancyscript{D}}^\mathbf{z}}p\left( z_1,\ z_2, \dots , \ z_n\right) \delta \left[ \psi ^{-1}\left( z_1,\ z_2, \dots , \ z_n\right) -x\right] dz_1dz_2\dots dz_n. \end{aligned}$$

(31)

To compute the result of Theorem 1, properties of the Kronecker delta are required and given by Eq. (32). The composition property in Eq. (32) is computed using the roots, $y_n$, of the function $f$, hence $f(y_n)=0$. For the translation property the limits of integration can be over any domain surrounding the critical points where the Kronecker delta is not zero. The Kronecker delta scaling, sifting, and translation properties allow for the theorem to be carried out.

$$\begin{aligned} \begin{array}{lcl} Scaling&{} &{}\\ &{}\delta (ay)&{}=\left| \displaystyle \frac{\partial (ay)}{\partial a}\right| ^{-1}\delta (y)=\displaystyle \frac{1}{\vert a\vert }\delta (y)\\ Translation(sifting)&{} &{}\\ &{}f(a)&{}=\displaystyle \int \limits _{-\infty }^\infty f(y)\delta (y-a)dy \\ Composition&{} &{}\\ &{}\delta \bigg (f(y)\bigg )&{}=\displaystyle \sum _n \delta (y-y_n)\left| \frac{\partial f(y)}{\partial y}\right| _{y_n}^{-1}\\ &{}\mathrm{Where:}&{}\quad f(y_n)=0\quad \mathrm{and }\quad \displaystyle \frac{\partial f(y)}{\partial y}\ne 0 \end{array} \end{aligned}$$

(32)

Application of Theorem 1 allows for Eq. (29) to be computed without the need for evaluating the $2\times 2$ Jacobian, shown in Eq. (33). The roots of $x-\rho \cos (\alpha )=0$ are $\rho =x/\cos (\alpha )$ and $\alpha = \arccos (x/\rho )$, using the first root to replace the range random variable results in the scaling factor to be $\frac{\partial }{\partial x}\left( x/\cos (\alpha )\right) = \sec (\alpha )$. Once the distribution is properly scaled, it can then be sifted to alleviate dependence on the range random variable, leaving only the angle random variable to be integrated over to produce the desired marginal PDF.

$$\begin{aligned} p\left( x\right)&= \displaystyle \int \limits _{{\fancyscript{D}}^\rho } \int \limits _{{\fancyscript{D}}^{\alpha }}p\left( \rho ,\alpha \right) \delta \left[ \rho \cos (\alpha )-x\right] d\rho d\alpha \nonumber \\&= \displaystyle \int \limits _{{\fancyscript{D}}^\rho } \int \limits _{{\fancyscript{D}}^{\alpha }}\frac{p\left( \rho ,\alpha \right) }{\left| \cos (\alpha )\right| }\delta \left[ \rho -\frac{x}{\cos (\alpha )}\right] d\rho d\alpha \nonumber \\&= \displaystyle \int \limits _{\alpha _0}^{\alpha _f}\frac{1}{\cos (\alpha )}p \left( \rho =\frac{x}{\cos (\alpha )},\alpha \right) d\alpha \end{aligned}$$

(33)

Equation (30) can be reproduced in a similar manner by utilizing the angle root, $\alpha = \arccos (x/\rho )$, and using its derivative with respect to the state of interest, $x$, for the scaling factor as shown by Eq. (34). Further examples of the application of the method can be found in (Khuri 2004; Shamilov et al. 2006; Hari and Venugopalakrishna 2008).

$$\begin{aligned} p\left( x\right)&= \displaystyle \int \limits _{{\fancyscript{D}}^\rho } \int \limits _{{\fancyscript{D}}^{\alpha }}p\left( \rho ,\alpha \right) \delta \left[ \rho \cos (\alpha )-x\right] d\rho d\alpha \nonumber \\&= \displaystyle \int \limits _{{\fancyscript{D}}^\rho } \int \limits _{{\fancyscript{D}}^{\alpha }}\frac{p\left( \rho ,\alpha \right) }{\rho \sin (\alpha )}\delta \left[ \alpha -\arccos \left( \displaystyle \frac{x}{\rho }\right) \right] d\rho d\alpha \nonumber \\&= \displaystyle \int \limits _{\rho _0}^{\rho _f}\left| \frac{1}{\rho \sin (\alpha )}\right| _{\alpha = \arccos \left( \displaystyle \frac{x}{\rho }\right) }p \left( \rho =\rho ,\alpha =\arccos \left( \displaystyle \frac{x}{\rho } \right) \right) d\rho \end{aligned}$$

(34)