## 5 Positioning and Orbit Determination of Space Targets
### A Model for Determining Orbital Parameters of Space Debris Based on Space-Based Observations  
Accurate evaluation of orbital precision is very difficult because, first, there is no way to know the true criterion for precision assessment, the true value of the orbit, and secondly, it is often difficult to obtain higher precision orbits that can be used as a criterion. Therefore, the evaluation of orbital precision can only be an approximately statistical result under certain assumptions. The orbital precision can be evaluated by comparing the instantaneous orbital roots, the average orbital roots, or the spatial debris position and velocity components of the time series [27]. In this paper, referring to the Gooding initial orbit determination method of space-based optical measurement, three sets of angular measurement data and the position information of the measurement platform are used to estimate the position and velocity of the space target. In addition, the space coordinate position is converted into six orbital elements using the star tracker coordinates. The orbital elements of elliptical motion are commonly expressed in terms of six Kepler orbital elements (as shown in Figure 12), namely: orbital half-length diameter \(a\), orbital eccentricity \(e\), argument of perigee \(\omega\), orbital inclination \(i\), longitude of ascending node \(\Omega\) and mean anomaly \(M\). \(M\) is an orbital element describing the position of the satellite in orbit:  
\[M\ =\ n(t-\tau) \tag{35}\]  
where \(n\ =\ \sqrt{\frac{\mu}{a^{3}}}\), \(n\) is often called the translation and \(\tau\) is the moment of perigee of the debris. The angle between the directional diameter \(\overset{\rightharpoonup}{r}\) and the perigee is called the true anomaly \(f\). The orbital equation and energy integral are then converted into:  
\[ \begin{array}{rcl}r&=&\frac{a\left(1-\tau^{2}\right)}{\Gamma+\varepsilon \cos f}\\ v^{2}&=&\mu\left(\frac{2}{\tau}-\frac{1}{a}\right)\end{array}  \tag{36}\]  
Figure 12: Schematic diagram of the six orbital elements. The spatial coordinate position is transformed into the six orbital elements. The coordinates \(\stackrel{{\rightharpoonup}}{{r}}\) and \(\stackrel{{\rightharpoonup}}{{v}}\) at a given moment are known, and the six orbital elements are calculated as follows:  
1. Calculate \[a=\frac{\mu r}{2\mu-rv^{2}}\] (37)
2. Calculate \(e\sin E,e\cos E,e,E\) \[ \begin{array}{c}e\sin E=\frac{\stackrel{{\rightharpoonup}}{{r}} \cdot\stackrel{{\rightharpoonup}}{{\Box}}}{\mu\stackrel{{ \rightharpoonup}}{{\Box}}^{2}}\\ e\cos E=1-\frac{r}{a}\end{array} \] (38)
3. Calculate \(i,\Omega\) \[ \begin{array}{c}\stackrel{{\rightharpoonup}}{{h}}=\stackrel{{ \rightharpoonup}}{{r}}\times\stackrel{{\rightharpoonup}}{{v}}= \sqrt{\mu p}\left[\begin{array}{c}\sin\Omega\sin i\\ -\cos\Omega\sin i\\ \cos i\end{array}\right]\\ \cos i=\frac{h_{3}}{\sqrt{\mu p}}\\ \tan\Omega=-\frac{h_{1}}{h_{2}}\end{array} \] (39)
4. Calculate \(\omega\) The true anomaly angle \(f\) can be calculated according to the following formula. \[f=E+2\tan^{-1}\left(\frac{e\sin E}{1+\sqrt{1-e^{2}}-e\cos E}\right) \tag{40}\]  
The eccentric anomaly \(u\) can be calculated according to the following formula:  
\[\left[ \begin{array}{c}r\cos u\\ r\sin u\\ 0\end{array} \right]=R_{1}(i)R_{3}(\Omega)\stackrel{{\rightharpoonup}}{{r}} \tag{41}\]  
\[\omega=u-f \tag{42}\]
5. Calculate mean anomaly \(M\) \[M=E-e\sin E\] (43)  
By calculating the positioning algorithm in the previous section, a higher accuracy position determination of the target can be accomplished. If the spatial positions of the target at two moments can be obtained, the initial orbital parameters of the space debris can be calculated by Equations (37)-(43). If there are multiple measurements of space debris in one orbital arc, the initial orbital elements can be further obtained with higher accuracy by calculating the average orbital elements. The location information of each orbital arc segment of space debris can be solved through the positioning algorithm in the previous section.