# Summary of Lunar Constellation Navigation and Orbit Determination Technology
## 3 Overview of Single-Satellite Navigation and Orbit Determination Methods
### Radio Measurement Orbit Determination Technology
The transmitter frequency \(f_{T}\) at a tracking station on Earth can be constant or ramped. If it is a constant frequency, it is obtained from the record of the OD file for the data point. If the transmitter frequency is ramped, it is specified as a series of contiguous ramps. Each ramp has a start time, an end time, the frequency \(f\) at the start time and the constant derivative \(\dot{f}\) of \(f\) (the ramp rate) which applies between the start time and the end time for the ramp. When the spacecraft is the transmitter, the frequency of the transmitted signal (for all one-way data types except GPS/TOPEX observables) is:  
\[f_{T}=C_{2}f_{S/C} \tag{2}\]  
where \(f_{S/C}\) is the S-band value of the spacecraft transmitter frequency and \(C_{2}\) converts it to the transmitted frequency for the downlink band for the data point (obtained from the data record for the data point on the OD file). 2. Radio ranging  
In ranging, continuous wave ranging signals can be divided into harmonic signals (such as sidetone signals) and non-harmonic signals (such as pseudo-code signals). Sidetone ranging uses'sidetone' (a single-frequency sine wave) to modulate the sidetone with a fixed frequency as a baseband signal onto a carrier wave and transmits it to the target. The target forwards the signal to the receiving end to demodulate. After processing, by comparing the phase difference between the received signal and the transmitted signal, the total distance between the transmitting point, the target, and the receiving point can be obtained. Increasing the sidetone frequency can achieve high-precision ranging, and this method has a narrow bandwidth and high acquisition speed. The basic principle of pseudo-code ranging is the same as that of sidetone ranging. The distance is calculated based on the relationship between the propagation delay of the ranging signal and the speed of light. The difference is that sidetone ranging uses phase difference to calculate distance, while pseudo-code ranging uses autocorrelation functions to measure delay. Pseudo-code ranging uses the long pseudo-code period as the de-ambiguation signal and uses random chips as high-frequency sidetones. Ranging can be completed with only one signal. Code-tone hybrid ranging is a hybrid system using pseudo-code and side-tone, which combines their respective advantages. It uses high-frequency sidetones to improve measurement accuracy and pseudo-code to decode distance blur. The computed values of three-way ramped Sequential Ranging Assembly (SRA) and Planetary Ranging Assembly (PRA) range observables are calculated as follows [19]:  
\[\rho_{3}(ramped)=\left[\int_{T_{B}}^{t_{3}(ST)_{R}}F(t_{3})dt_{3}-\int_{T_{A}}^{ t_{1}(ST)_{T}}F(t_{1})dt_{1}\right],\text{modulo}M\text{ range units} \tag{3}\]  
The reception time \(t_{3}(ST)_{R}\) in station time \(ST\) at the receiving electronics at the receiving station on Earth is equal to the data time tag \(TT\). The corresponding transmission time \(t_{1}(ST)_{T}\) in station time ST at the transmitting electronics at the transmitting station on Earth is calculated as follows:  
\[t_{1}(ST)_{T}=t_{3}(ST)_{R}-\rho\text{ \ \ s} \tag{4}\]  
\(\rho\) is the precision round-trip light time. The quantities \(T_{B}\) and \(T_{A}\) are zero-phase times at the receiving and transmitting stations, respectively. At \(T_{B}\), the phase of the transmitter ranging code at the receiving station is zero. The phase of the transmitted ranging code at the transmitting station is zero at \(T_{A}\). \(F(t_{3})\) and \(F(t_{1})\) are conversion factors at the receiving station and transmitting station, respectively, depending upon the uplink band and exciter type at the station. The first integral in Equation (3) is the phase of the transmitter ranging code at the reception time \(t_{3}(ST)_{R}\) at the receiving electronics at the receiving station on Earth.