## 2 The Theoretical Basis of Tightly-Coupled INS/RTK Integrations
### Conventional Tightly Coupled RTK/INS Integrations with AR in the Ambiguity Domain  
In this subsection, we introduce two conventional RTK/INS integrated techniques, both of which complete the AR in the ambiguity domain. As we mentioned above, the typical tightly coupled RTK/INS algorithms generally include two stages: INS-aided AR, and the correction process. The total accuracy is guaranteed by the ability to successfully find a fixed solution. According to the different way that INS participates in the AR process, the existing tightly coupled algorithm is divided into two categories [10; 11], as mentioned in the introduction Section 1. The first implementation is depicted in Figure 1. It uses INS measurements to help estimate float ambiguities through an EKF, and then searches for integer solutions in the ambiguity domain by LAMBDA. Compared with the ambiguity covariance matrix obtained only from the double-differenced carrier phase and pseudo-range observations, this tightly coupled algorithm takes into account a priori motion information, so the covariance matrix obtained by EKF has a more efficient search space. The filter states include position, velocity, attitude, biases of inertial sensors, and float ambiguities. Therefore, if no change in the satellite tracking status is detected, the integer solutions will be used as the initial values of filter states at the next moment, as shown by the red arrow in Figure 1. However, if there is an undetected cycle slip, the incorrect initial value can cause the subsequent filtering process to fail. At the same time, the accuracy of this tightly-coupled algorithm is affected by the quality of pseudorange measurements, which can cause a bias in the ambiguity estimation of a continuous tracking satellite.  
The second kind of implementation is illustrated in Figure 2. Float ambiguities are obtained by least squares estimation. The next step is to search for integer solutions in the ambiguity domain  
Figure 1: The first implementation of tightly-coupled Inertial Navigation System (INS)/real-time kinematic (RTK) integration using an extended Kalman filter (EKF).  
and correct the state variables using the fixed carrier-phase observations. If no valid integer solution is found, pseudorange observations will be used to correct the state variables by an EKF instead. For the float ambiguity estimation process, there are not only double-differenced carrier-phase and pseudorange observations as constraints, but also the positioning result of the INS solution as additional observables, as shown by the green arrow in Figure 2. The inertial navigation assistance makes it practicable to complete the AR process when there are fewer than four satellites in a single constellation system. In contrast to the first kind of implementation, this algorithm only utilizes the current epoch observations in the AR process, so it is not affected by frequent changes in satellite visibility. From the second flowchart, it also can be clearly concluded that the quality of the pseudorange observation directly affects the success rate when searching for a fixed solution in the ambiguity domain. To avoid this undesirable influence, our new ARPD-RTK/INS algorithm searches for integer ambiguities in the position domain.