## 1 Introduction  
Dead reckoning (DR) is widely adopted for navigation-assisted positioning, autonomous vehicle localization, and robot path planning. However, its performance degrades in non-cooperative scenarios where targets cannot actively provide positioning data, such as wildlife tracking (e.g., marine turtles [1] and migratory species [2]) or uncrewed aerial vehicle navigation [3]. These scenarios face three critical challenges:  
1. Sparse observations: Targets rarely emit detectable signals [4]. 2. Resource constraints: Limited energy budgets and narrow communication bandwidths. 3. Nonlinear dynamics: Nonlinear motion patterns (e.g., animal behaviors [5]). Current DR solutions fall into three categories. First is the data-driven trajectory prediction method. Deep learning methods like RNNs [6, 7], LSTMs [8], and GCNs [9] have improved sequence modeling by capturing temporal dependencies. While these models achieve higher accuracy than traditional methods, they require large training datasets and struggle with generalization to unseen scenarios [10]. Second is the multi source information fusion strategy. To overcome sparse observations, researchers combine complementary sensing modalities. In order to address the limitations of single sensor data [11, 12], scholars have proposed various multi-source information fusion schemes [13], for example, combining visual perception data with IMU data to construct robust state estimation algorithms [14; 15]. Alternatively, radar echo features can be integrated with historical trajectory data to enhance the reliability of predictions. While ref. [16] developed a novel inertial-aided Unmodulated Visible Light Positioning system (uLiDR) for resource-constrained platforms like smartphones, ref. [17] further improved accuracy by integrating IMU, magnetic field, Bluetooth, and semantic map data with a robust particle filtering method. This cross-modal information fusion [18] not only improves the reliability of prediction results, but also alleviates the problem of sparse observation data to a certain extent. Third is dynamic modeling and optimization. This is crucial to establish an accurate dynamic model for the motion characteristics of noncooperative targets. Animals' movement patterns often have strong randomness and uncertainty, as discussed in refs. [5; 19; 20]. For aircraft, UGVs, or boats, the influence of aerodynamics or fluid dynamics needs to be considered. By conducting an in-depth analysis of the motion law of the target object [21] and combining it with optimal control theory [22], the effectiveness of trajectory prediction can be significantly improved. However, in this application context, existing dead reckoning methods face three core bottlenecks:  
1. High-frequency data dependence: traditional inertial navigation systems (INS) require IMU sampling rates above 10 Hz to suppress integral drift, but in noncooperative scenarios, target observation frequencies are often lower than 5 Hz (such as satellite remote sensing revisit periods), leading to the failure of conventional Kalman filtering frameworks. 2. Dynamic model mismatch: existing methods often assume that the target follows a linear motion model, but in fact, ships often perform nonlinear avoidance maneuvers [23]. 3. Obstacles to multi-source heterogeneous data fusion include spatiotemporal asynchrony and coordinate system homogeneity in multimodal data such as radar, AIS, EO/IR in non cooperative scenarios. This study develops a novel motion-invariance-driven framework for parameter estimation and trajectory prediction that fundamentally addresses three critical challenges in navigation systems: (1) dependency on high-frequency sampling, (2) nonlinear maneuver prediction limitations, and (3) strict spatiotemporal synchronization requirements. Our primary innovations encompass three fundamental advancements. 1. Theoretical Foundation: We establish the first formal connection between motion invariance principles and dead reckoning methodologies through Lie group analysis.