**3. Auxiliary navigation system using data compression technique**

The analysis of space orbital motion generally means the integration of the nonlinear orbital motion equation. For this analysis, special perturbation [12], which is a numerical method, or general perturbation, which is an analytical method, is used. Special perturbation has a small error due to numerical integration during orbit propagation, but requires a high-performance onboard computer. General perturbation has a small computational load during orbit propagation, but the numerical integration error increases substantially with time. To improve this shortcoming, low-orbit satellites generate a residual, which is the difference between the reference orbit and the true orbit. The residuals, which exhibit periodic characteristics, are approximated using the coefficients of trigonometric and Fourier functions, before being transmitted to the satellites along with the reference orbital elements. The satellite computer can improve the precision of orbit propagation and greatly reduce the computational load by generating a reconstruction orbit [13, 14] using coefficients and reference orbital elements received from the ground station.

However, the approximation method using residuals cannot be applied to deep space probes for the Moon and Mars because they do not rotate around the Earth repeatedly, like satellites. As a solution, an auxiliary navigation system using B-spline data compression has been proposed [3]. The data compression rate must be increased because deep space communication is expensive and limited in communication time. For Earth and lunar orbits where takeoff and landing occur, intensive control and communication are essential due to the danger and unpredictability. However, in the transition segment, which is stable and accounts for the majority of navigation, communication time can be saved, and stable navigation data can be received by sending compressed orbit data calculated on the ground to the probes.

#### **3.1. Satellite orbit compression using the Fourier technique**

Satellite orbit compression using the Fourier technique, and development of an auxiliary navigation system using this technique, have mainly been studied using a low-orbit satellite model [1, 2, 12, 15]. The overall operation concept of an onboard orbit propagator is

*i*(*t*) = *i*

an argument of latitude of the orbit.

regression is used for this purpose.

**Figure 8.** Definition of various orbits.

*3.1.2. Residual reproduction*

<sup>0</sup> + *i* <sup>1</sup> *t* + *i*

Ω(*t*) = Ω<sup>0</sup> + Ω<sup>1</sup> *t* + Ω<sup>2</sup> *t*<sup>2</sup> (15)

*ω*(*t*) = *ω*<sup>0</sup> + *ω*<sup>1</sup> *t* + *ω*<sup>2</sup> *t*<sup>2</sup> (16)

*M*(*t*) = *M*<sup>0</sup> + *M*<sup>1</sup> *t* + *M*<sup>2</sup> *t*<sup>2</sup> (17)

where *n* is the mean motion, *e* is the eccentricity, *i* is the inclination angle, Ω is the right ascen-

Here, the coefficients are determined using the least squares curve fit or a similar technique based on the precise orbit prediction data of actual orbits created by numerical integration. However, in the case of a near-circular orbit, the argument of perigee cannot be defined or the curve fitting may be inaccurate. Therefore, instead of *ω* and *M*, *u* = *ω* + *f* can be used, which is

The residual means the difference between the actual orbit data and the designed reference orbit (**Figure 8**). The Fourier series coefficient must be determined, and the least squares

*r* − *r*<sup>∗</sup> = *r* = *b*<sup>0</sup> + ∑*bk* sin(*ku*) + ∑*bk* cos(*ku*) (18)

sion of ascending node, *ω* is the argument of perigee, and *M* is the mean anomaly.

<sup>2</sup> *t*<sup>2</sup> (14)

Introduction to Navigation Systems http://dx.doi.org/10.5772/intechopen.71047 13

**Figure 7.** Diagram of onboard orbit propagator operation.

shown in **Figure 7**. First, an actual orbit is created through accurate modeling and numerical integration of orbital motions, which is available on the ground. Then, a reference orbit is created that is sufficiently close and has a known solution. After defining the residual, which is the difference between the reference orbit and the actual orbit, a few approximate functions are obtained by reflecting the characteristics of orbital motion. The corresponding coefficients are sent to the satellite. The satellite determines the position and velocity of the satellite by substituting the pre-embedded numbers in the onboard computer (Orbit Reconstruction).

The orbit precision predicted through such orbit reconstruction is closely related to the selection of the reference orbit and residual reproduction function. However, the selection criteria for the reference orbit and residual reproduction function must be based on the calculation power of the satellite onboard computer, the data transmission protocol between the ground station and the satellite, and the required precision of the orbit propagation result.

#### *3.1.1. Creation of reference orbit*

Essentially, orbit data is created by numerical integration using the initial time, position, and velocity data on the ECI coordinate system, as well as precise orbit modeling. The data determined in this way is converted to orbit elements, and the reference orbit is established. The general reference orbit can be expressed in first or second order polynomials, as shown below, and the coefficients are interpolated using the least squares method.

$$n(t) = n\_0 + n\_1 t + n\_2 t^2 \tag{12}$$

$$e(t) = e\_0 + e\_1 t + e\_2 t^2 \tag{13}$$

#### Introduction to Navigation Systems http://dx.doi.org/10.5772/intechopen.71047 13

$$\dot{i}(t) = \dot{i}\_0 + \dot{i}\_1 t + \dot{i}\_2 t^2 \tag{14}$$

Ω(*t*) = Ω<sup>0</sup> + Ω<sup>1</sup> *t* + Ω<sup>2</sup> *t*<sup>2</sup> (15)

$$
\omega(t) = \omega\_0 + \omega\_1 t + \omega\_2 t^2 \tag{16}
$$

$$M(t) = M\_0 + M\_1 t + M\_2 t^2 \tag{17}$$

where *n* is the mean motion, *e* is the eccentricity, *i* is the inclination angle, Ω is the right ascension of ascending node, *ω* is the argument of perigee, and *M* is the mean anomaly.

Here, the coefficients are determined using the least squares curve fit or a similar technique based on the precise orbit prediction data of actual orbits created by numerical integration. However, in the case of a near-circular orbit, the argument of perigee cannot be defined or the curve fitting may be inaccurate. Therefore, instead of *ω* and *M*, *u* = *ω* + *f* can be used, which is an argument of latitude of the orbit.

#### *3.1.2. Residual reproduction*

shown in **Figure 7**. First, an actual orbit is created through accurate modeling and numerical integration of orbital motions, which is available on the ground. Then, a reference orbit is created that is sufficiently close and has a known solution. After defining the residual, which is the difference between the reference orbit and the actual orbit, a few approximate functions are obtained by reflecting the characteristics of orbital motion. The corresponding coefficients are sent to the satellite. The satellite determines the position and velocity of the satellite by substituting the pre-embedded numbers in the onboard computer (Orbit

The orbit precision predicted through such orbit reconstruction is closely related to the selection of the reference orbit and residual reproduction function. However, the selection criteria for the reference orbit and residual reproduction function must be based on the calculation power of the satellite onboard computer, the data transmission protocol between the ground

Essentially, orbit data is created by numerical integration using the initial time, position, and velocity data on the ECI coordinate system, as well as precise orbit modeling. The data determined in this way is converted to orbit elements, and the reference orbit is established. The general reference orbit can be expressed in first or second order polynomials, as shown below,

*n*(*t*) = *n*<sup>0</sup> + *n*<sup>1</sup> *t* + *n*<sup>2</sup> *t*<sup>2</sup> (12)

*e*(*t*) = *e*<sup>0</sup> + *e*<sup>1</sup> *t* + *e*<sup>2</sup> *t*<sup>2</sup> (13)

station and the satellite, and the required precision of the orbit propagation result.

and the coefficients are interpolated using the least squares method.

Reconstruction).

*3.1.1. Creation of reference orbit*

**Figure 7.** Diagram of onboard orbit propagator operation.

12 Multi-purposeful Application of Geospatial Data

The residual means the difference between the actual orbit data and the designed reference orbit (**Figure 8**). The Fourier series coefficient must be determined, and the least squares regression is used for this purpose.

$$r - r^\* = \delta r = b\_o + \sum b\_i \sin(ku) + \sum b\_i \cos(ku) \tag{18}$$

**Figure 8.** Definition of various orbits.

#### *3.1.3. Orbit reconstruction*

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

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,

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,

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

> *i*=1 *nc* **b***<sup>i</sup> Ni*,*<sup>k</sup>*

is rt., the vector becomes T = (j = 1,…,rt). The k-order B-spline basis function is defined as

1 *if ui* ≤ *s* < *ui*+1 0 *otherwise* ,

> ) = ∑ *t*=1 *nc*

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

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

is the adjustment point, and Ni,k is the k-order B-spline basis function. Here, if nc-3

*ui*+*k*+1 <sup>−</sup> *<sup>s</sup>* \_\_\_\_\_\_\_\_ *ui*+*p*+1 − *ui*+1

**b***<sup>i</sup> Ni*,*<sup>k</sup>*(*si*

*Ni*+1,*k*−<sup>1</sup> (*s*).

, the following equation must be satisfied:

(*s*) (19)

Introduction to Navigation Systems http://dx.doi.org/10.5772/intechopen.71047 15

(*i* = 1~n) and the parameter

) (21)

unknown numbers, which can be

(20)

where s is the parameter and s1 = 0. In these equations, 1/2 or 1 can be used for α.

phasing loop transfer, and weak stability boundary (WSB).

points is given as follows:

where b<sup>i</sup>

follows:

where *ui*

and degrees.

*3.2.2. Orbit reconstruction*

**c**(*s*) = ∑

*Ni*,0

This becomes *n* simultaneous equations consisting of *nc*

*Ni*,*<sup>k</sup>*

value corresponding to each point is *si*

**p***<sup>i</sup>* ≃ **c**(*si*

(*s*) <sup>=</sup> {

is the value of the *i*th knot vector. If the given points are **p***<sup>i</sup>*

*Nk*−<sup>1</sup> (*s*) +

(*s*) <sup>=</sup> *<sup>s</sup>* <sup>−</sup> *<sup>u</sup>* \_\_\_\_\_*<sup>i</sup> ui*+*<sup>k</sup>* − *ui*
