## 1 Introduction  
Nowadays, the topic of CL has caught much attention [1; 2; 3]. CL is a technique that enables a group of agents to estimate their positions and orientations relative to each other and their surroundings. This method has many practical applications, including in robotics, where it is essential for enabling robots to navigate and operate in complex environments. One specific use case of CL is in the development of firefighter rescue systems. The precise localization of firefighters is critical in this context to ensure their safety and improve their response effectiveness during emergencies. A group of firefighters can commonly be separated into the commander and soldiers in firefighting rescue scenarios, depending on the responsibilities they undertake. As illustrated in Figures 1 and 2, the commander usually stands in the open sky far away from the fires, observing the fire situation and judging the possibility of danger. However, soldiers are frequently close to the front lines of firefighting in global navigation satellite system (GNSS) denied environments, such as forests and indoors. Because sensor nodes should be arranged in advance, it is infeasible to use the existing infrastructure to locate them in such a case [4; 5]. Therefore, inertial navigation as an autonomous navigation scheme is preferred [6; 7; 8]. only maintain the localization results for some time without divergence. It is essential to introduce other sensors to assist with inertial navigation. The geomagnetic field can be measured by the magnetometer to obtain the heading [15; 16; 17], the barometer measures pressure to calculate altitude [18; 19; 20], and GNSS can be used to obtain a position directly [21; 22; 23]. The above methods are commonly used in individual localization, but the relationship of firefighters can also be utilized in CL. UWB can provide a relative range of measurements among firefighters so that it can be used in CL. UWB technology stands out among many radio technologies because of its high temporal resolution, high-ranging precision, and resistance to multipath effects [24; 25; 26]. Many studies have been conducted on the combination of an inertial navigation system (INS) and UWB technology in CL. Most of them required fixed anchors and were only suitable for individual localization. In [27], the error divergence of the INS was constrained by a loosely coupled system of INS and UWB that used the UWB position estimation from the trilateration method. In [28], a tightly coupled method was applied to the foot-mounted INS and UWB system, and from this, precise position results could be obtained. In addition, UWB range measurements may suffer from NLOS errors due to environmental effects, and the direct use of inaccurate range measurements can lead to poor localization results. CL can be divided into two main structures: centralized cooperative localization (CCL) and the DCL. In the CCL structure, there is a need for a fusion center that analyzes information from the entire team. Although this method can achieve accurate estimates of position, it has a high communication cost and lacks robustness when some members are disconnected [29; 30]. In the DCL structure, every member could serve as a local fusion center and exchange information with each other. However, this method suffers from a double-counting problem in calculating the cross-correlations between the state estimates of members. In [31], a covariance intersection method that inflated the measurement noise variance was proposed. However, it was conservative, and the localization accuracy was not good. In [32], a cross-correlations compensation method was introduced to the DCL but suffered from a high computational cost. In [33], a recursive DCLEKF algorithm was proposed that was able to obtain satisfactory CL accuracy at a small computation cost. However, this does not take into account the effects of the inaccurate measurement noise covariance matrix and how the CL accuracy degrades if inaccurate measurement noise is used.