###### Abstract  
Joint localization and synchronization (JLAS) is a technology that simultaneously determines the spatial locations of user nodes and synchronizes the clocks between user nodes (UNs) and anchor nodes (ANs). This technology is crucial for various applications in wireless sensor networks. Existing solutions for JLAS are either computationally demanding or not resilient to noise. This paper addresses the challenge of localizing and synchronizing a mobile user node in broadcast-based JLAS systems using sequential one-way time-of-arrival (TOA) measurements. The AN position uncertainty is considered along with clock offset and skew. Two redundant variables that couple the unknowns are introduced to pseudo-linearize the measurement equation. In projecting the equation to the nullspace spanned by the coefficients of the redundant variables, the affection of them can be eliminated. While the closed-form projection solution provides an initial point for iteration, it is suboptimal and may not achieve the Cramer-Rao lower bound (CRLB) when noise or AN position error is relatively large. To improve performance, we propose a novel robust iterative solution (RIS) formulated through factor graphs and developed via message passing. The RIS outperforms the common Gauss-Newton iteration, especially in high-noise scenarios. It exhibits a lower root mean-square error (RMSE) and a higher probability of converging to the optimal solution, while maintaining manageable computational complexity. Both analytical results and numerical simulations validate the superiority of the proposed solution in terms of performance, resilience, and computational load. joint localization and synchronization; sequential one-way time-of-arrival; anchor position error; nullspace projection; robust iteration +
Footnote †: journal: Medical  
through the geometric relationship of AN and UN positions. However, obtaining the TOF requires perfect synchronization between ANs and UNs, which incurs additional communication and computation costs [7; 8; 9; 10; 11]. In contrast, measuring the time-of-arrival (TOA) does not necessitate synchronization between ANs and UNs. If the emitted signals contain a time label, the TOA can be determined by comparing the emission time to the local time. However, the lack of synchronization means that the TOA includes the clock offset between ANs and UNs, introducing additional errors when converting the TOA to a range, which can result in significant errors in UN position estimation. Thanks to the emergence of joint localization and synchronization (JLAS) [9; 10; 11], the issues of finding the position of the UN and achieving synchronization can be addressed simultaneously. JLAS is a technique in wireless communication and sensor networks that simultaneously determines node positions and synchronizes their clocks [12; 13; 14; 15; 16]. By exchanging signals and processing measurements like TOA and signal strength, JLAS reduces errors and enhances efficiency compared to separate localization and synchronization methods. It is essential for applications like environmental monitoring, the Internet of Things (IoT), and autonomous systems. Despite challenges such as non-line-of-sight conditions and clock drift, JLAS provides robust performance in diverse environments. JLAS operates by transmitting signals from ANs to UNs, which capture and parse these signals to obtain measurements such as the TOA [10] and TOF. Without synchronization between ANs and UN, the TOA is a more popular and widely used measurement type in JLAS since it can provide high precision and simple implementation. In TOA schemes, three methods commonly to identify the signals from ANs to UN are frequency division (FD), code division (CD), and time division (TD). FD distinguishes ANs in the frequency domain, allowing TOAs between ANs and the UN to be obtained within one time slot.