INS Strapdown Inertial Navigation Solving Using Fourth-Order Runge-Kutta Method

The core challenge in Strapdown Inertial Navigation Systems (SINS) is attitude solving—how to utilize gyroscope and accelerometer measurements to update the carrier's attitude information in real-time. In SINS, attitude solving accuracy directly determines navigation performance, particularly in high-dynamic environments where traditional Euler angle or direction cosine methods may fail due to accumulated integration errors. Implementing high-order numerical integration methods like the Fourth-Order Runge-Kutta (RK4) method becomes crucial for enhancing solution accuracy.

The Fourth-Order Runge-Kutta method is a classical numerical integration algorithm that employs multi-step weighted averaging to approximate solutions to differential equations, offering superior precision and stability compared to lower-order methods like Euler integration. In SINS attitude solving, RK4 is typically applied to solve quaternion differential equations. The implementation workflow consists of four key stages:

Initial Condition Setup: Generate initial quaternions based on the carrier's starting attitude (using Euler angles or direction cosine matrices) as the computation starting point. Gyroscope Data Preprocessing: Compensate for gyroscope output errors (including bias calibration and scale factor correction) and convert angular velocity data into angular increment format required for quaternion differential equations. RK4 Step Computation: Divide the angular increment within the current time step into four stages (k1 to k4), compute intermediate quaternion states at each stage, and obtain the next time step's quaternion through weighted averaging. Normalization Processing: Since quaternions must maintain unity norm (magnitude of 1), perform normalization correction after each update to prevent numerical drift.

Simulation data validates algorithm effectiveness—for example, simulating 3D rotational motions (pitch, roll, yaw) and comparing RK4 results against theoretical trajectories to evaluate precision and real-time performance. Practical implementations must address real-world factors like sensor noise and computational latency, where Kalman filtering algorithms can further enhance robustness through multi-sensor data fusion.

Extended Considerations: While RK4 outperforms lower-order methods in accuracy, its computational demands require precision-real-time tradeoffs in resource-constrained embedded systems. For ultra-high-speed carriers (e.g., missiles), adaptive step-size Runge-Kutta variants (like RKF45) can dynamically adjust integration steps to balance efficiency and stability.