**3.2. Lunar probe orbit compression using a B-spline**

Deep space probes that explore the Moon, Mars, and minor planets stay in the Earth's orbit after being launched by a launch vehicle. After staying in Earth's orbit for some time, they escape it by burning the engine to reach the target planet. In the case of lunar exploration satellites, they activate trans-lunar injection (TLI) in the parking orbit of the Earth to enter the lunar transfer orbit. Once a lunar exploration satellite enters the lunar transfer orbit, the satellite is tracked with ground antennae around the world, and orbit determination is performed by processing the obtained tracking data. Communication with a lunar probe corresponds to deep space communication, and a representative example is the Deep Space Network (DSN). Four antennae are in currently in operation at three locations (Goldstone, Madrid, and Canberra) [16–18]. The antennae and communication range of DSN are shown in **Figure 9**. The orbit data must be compressed as much as possible because communication is very limited in both time and range. Therefore, an algorithm for efficient orbit data compression should be developed.

#### *3.2.1. Ground control station data compression*

After orbit determination, the coordinates and velocity data of the calculated orbit are compressed as control points through the proposed procedure based on a B-spline. Compared to the conventional method, this method can stably compress even bulky data at a higher compression rate. This data is sent to the probe in the form of compressed parameters. The

**Figure 9.** Deep space network facility.

probe reconstructs the orbit based on the received parameters. Methods by which the lunar exploration satellite leaves the earth orbit and enters the lunar orbit include direct transfer, phasing loop transfer, and weak stability boundary (WSB).

The B-spline approximation method creates a three-dimensional curve with position and velocity data over time (4-D), excluding time, and then time is reconstructed by linear interpolation using the B-spline. The initial degree and control points are given, and the parameters are estimated based on the given point data. Control points are obtained from the estimated parameters and corresponding points, and the curve is reconstructed. After performing linear interpolation for the reconstructed data in line with the time, it is compared with the original data, and the control points and degrees are increased until they enter the error range.

Then, the drive command is given with the position and velocity data of the x, y, and z axes for time t. The position data can be expressed by (t, X, Y, Z) and the velocity data by (t, Vx, Vy, Vz). The next step is parameterization, where the parameter values corresponding to the given points are estimated. When a point (=1,2,…,n) is given on a secondary plane, the parameter values can be obtained as follows: si = si-1 + Δi/L, (i = 2,3,…,n), Δi = |pi-pi-1|α, and L = ∑ Δi, where s is the parameter and s1 = 0. In these equations, 1/2 or 1 can be used for α.

Next, adjustment points are calculated. The method of obtaining the curve using given points and calculated parameter values is as follows. A k-order B-spline curve with nc adjustment points is given as follows:

$$\mathbf{c}(\mathbf{s}) = \sum\_{l=1}^{n} \mathbf{b}\_{l} \, \mathbf{N}\_{l,l}(\mathbf{s}) \tag{19}$$

where b<sup>i</sup> is the adjustment point, and Ni,k is the k-order B-spline basis function. Here, if nc-3 is rt., the vector becomes T = (j = 1,…,rt). The k-order B-spline basis function is defined as follows:

$$\begin{aligned} \text{es T = (j = 1, \ldots, \text{rt). The k-order B-spin basis function is defined as} \\ N\_{i,0}(\mathbf{s}) &= \begin{cases} 1 & \text{if } u\_i \le s < u\_{i+1} \\ 0 & \text{otherwise} \end{cases} \\ N\_{i,k}(\mathbf{s}) &= \frac{s - u\_i}{u\_{i+k} - u\_i} N\_{k-1}(\mathbf{s}) + \frac{u\_{i+k+1} - s}{u\_{i+1} - u\_{i+1}} N\_{i+1,k-1}(\mathbf{s}). \end{aligned} \tag{20}$$

where *ui* is the value of the *i*th knot vector. If the given points are **p***<sup>i</sup>* (*i* = 1~n) and the parameter value corresponding to each point is *si* , the following equation must be satisfied:

$$\mathbf{p}\_{\parallel} \simeq \mathbf{c}\_{\parallel} \mathbf{s}\_{\parallel} = \sum\_{\mu=1}^{n} \mathbf{b}\_{\perp} N\_{\sqcup \lambda} \mathbf{s}\_{\parallel} \tag{21}$$

This becomes *n* simultaneous equations consisting of *nc* unknown numbers, which can be solved by singular value decomposition [19]. The optimal value for the error between the reconstructed value **c** and the original value **p** can be derived by adjusting the control points and degrees.

#### *3.2.2. Orbit reconstruction*

*3.1.3. Orbit reconstruction*

14 Multi-purposeful Application of Geospatial Data

The coefficients determined above are uploaded from the ground station to the satellite through communication. Using the reference orbit coefficient and residual coefficient received from the ground station, the onboard computer in the satellite reconstructs the orbit. Eqs. (12)–(18) are stored in the satellite computer, and orbit data is created if the current time is inputted. The velocity data is indirectly calculated from the position data, or can be directly created using the same method. The compressed orbit elements are converted to velocity and

Deep space probes that explore the Moon, Mars, and minor planets stay in the Earth's orbit after being launched by a launch vehicle. After staying in Earth's orbit for some time, they escape it by burning the engine to reach the target planet. In the case of lunar exploration satellites, they activate trans-lunar injection (TLI) in the parking orbit of the Earth to enter the lunar transfer orbit. Once a lunar exploration satellite enters the lunar transfer orbit, the satellite is tracked with ground antennae around the world, and orbit determination is performed by processing the obtained tracking data. Communication with a lunar probe corresponds to deep space communication, and a representative example is the Deep Space Network (DSN). Four antennae are in currently in operation at three locations (Goldstone, Madrid, and Canberra) [16–18]. The antennae and communication range of DSN are shown in **Figure 9**. The orbit data must be compressed as much as possible because communication is very limited in both time and range. Therefore, an algorithm for efficient orbit data compression should be developed.

After orbit determination, the coordinates and velocity data of the calculated orbit are compressed as control points through the proposed procedure based on a B-spline. Compared to the conventional method, this method can stably compress even bulky data at a higher compression rate. This data is sent to the probe in the form of compressed parameters. The

position data through the DCM, etc., and vary with the characteristics of the orbit.

**3.2. Lunar probe orbit compression using a B-spline**

*3.2.1. Ground control station data compression*

**Figure 9.** Deep space network facility.

The optimal solution is calculated using the B-spline method (**Figure 10**). To apply the B-spline-based approximation method to the orbit of a deep space probe, some changes

**Author details**

Address all correspondence to: yanggwa82@gmail.com

Sciences, South Korea; 2011. pp. 569-575

puter Laboratory: The United Kingdom; 2007

2006; 18-21 October 2006; Busan. IEEE; 2006. p. 1351-1354

Chonbuk National University; 2004

2016;**17**:240-252

United States; 1993

United States; 2003

United Kingdom; 2005

Studi di Napoli Parthenope; 2010

States; 2011

2003;**31**:51-59

Defense Agency for Technology and Quality, Jinju, South Korea

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**References**

**Figure 10.** Diagram of data compression via the B-spline method.

need to be made to the system currently in use. The transmission data include the number of calculated coefficients: *nc* ; the calculated coefficients: *bi* ; and the section beginning and end times: *ts* , *te* . Two system changes are required: knot vector creation and basis function evaluation.

For the former, the B-spline method requires a knot vector for calculation. This knot vector can be set in such a way that it will be automatically calculated in the probe without being transmitted from the ground to the satellite. For the latter, the B-spline method uses the basis function introduced in Section 3.2.1. Therefore, the ground control system and satellites must have a function for calculating the B-spline basis function *Ni*, *<sup>k</sup>* and calculating the B-spline curve.
