Extended Kalman Filter Implementation for Three-Attitude-Angle Data Fusion

The Extended Kalman Filter (EKF) is a widely used state estimation method for nonlinear systems, particularly suitable for attitude angle fusion problems based on multi-sensor data. In attitude estimation, it's essential to fuse data from triaxial angular rate sensors (gyroscopes) and triaxial accelerometers to improve the accuracy of attitude angle estimation.

### Core Methodology State Definition: Typically selects attitude angles (roll, pitch, yaw) and their angular velocities as state variables. In code implementation, the state vector would be initialized as [roll, pitch, yaw, roll_rate, pitch_rate, yaw_rate]. Prediction Phase: Uses gyroscope outputs to establish a motion model and predict attitude angle changes for the next time step. Since gyroscopes measure angular velocity, numerical integration (e.g., using Euler integration method) is applied to obtain predicted attitude angles. The implementation would involve calculating the state transition matrix and propagating the covariance matrix. Observation Update Phase: Gravity acceleration components from accelerometers are used to correct predicted attitude angles. Given the known gravity direction, accelerometers can compute preliminary pitch and roll angles, but they are significantly affected by dynamic acceleration interference and shouldn't be used alone. The implementation requires constructing an observation matrix that relates accelerometer measurements to attitude angles. Nonlinear Model Handling: EKF adapts to the Kalman filter framework by linearizing nonlinear models (such as attitude rotation matrices) using Jacobian matrices, ensuring stable state estimation. The implementation involves calculating partial derivatives of the system model at each time step.

### Implementation Considerations Data Preprocessing: Gyroscope data typically contains bias noise requiring dynamic compensation or calibration. Code should include bias estimation algorithms and real-time offset subtraction. Observation Noise Optimization: During high dynamic conditions (like rapid movements), accelerometer reliability decreases. The implementation should adapt the observation noise matrix to reduce correction weight and prevent erroneous updates using conditional logic based on acceleration magnitude. Covariance Tuning: The covariance matrices of state noise and observation noise significantly impact filtering performance. Parameter tuning should consider actual sensor error characteristics, often implemented through empirical testing and optimization algorithms.

### Advanced Extensions Integrating magnetometers can further correct yaw angles and reduce heading drift. Implementation would involve adding magnetic field measurements to the observation model. For high-dynamic environments, consider using adaptive EKF or complementary filtering for auxiliary optimization. Adaptive EKF implementation would require real-time adjustment of noise covariance matrices based on innovation sequences.

Through proper modeling and parameter tuning, EKF can effectively fuse gyroscope and accelerometer data to provide stable attitude angle estimation. The complete implementation typically involves initialization, prediction-update loops, and quaternion or Euler angle conversion functions for practical applications.