Lie Group model and performance analysis of Triple-Frequency PPP-AR/INS Tightly Coupled Integration

Although tightly coupled Precise Point Positioning/Inertial Navigation Systems (PPP/INS) are capable of decimeter-level accuracy, conventional filtering frameworks suffer from a theoretical disconnect: attitude errors are mapped onto the special orthogonal group SO(3), while position and velocity errors are treated in Euclidean space. This mathematical heterogeneity often induces error accumulation during state propagation. To resolve this, this paper presents a multi-frequency PPP-AR/INS framework based on the Left-Invariant Lie Group. By strictly defining all state errors on the Lie group manifold, estimation consistency is significantly enhanced. Field experiments confirm the superiority of the proposed approach over traditional methods. Specifically, under open-sky conditions, the left-invariant formulation outperforms the right-invariant and conventional method by reducing 3D positioning errors by 3.3% and 9.3%, respectively. In challenging environments with partial signal blockage, the method yields improvements of 4.8% for 2D and 13.1% for 3D. Furthermore, during complete GNSS outages, the enhanced accuracy of the IMU state estimation mitigates drift, lowering 2D and 3D errors by 11.2% and 6.3%, respectively. Notably, these gains are achieved with only a marginal 2.4% increase in computational load, validating the efficiency of the method for real-time applications.