Extended Kalman Filter - Sensor Fusion #3 - Phil's Lab #37

Summary

In this video, the extended Kalman filter (EKF) is introduced as a powerful tool for sensor fusion, building on previous videos that covered the complementary filter and basic sensor fusion concepts. The EKF combines dynamic models and sensor data to estimate system states, such as aircraft attitude. The video explores the EKF's prediction and update steps, explaining how it handles nonlinear systems and sensor noise, and how it improves state estimation over simpler methods. The practical application of the EKF in embedded systems is previewed, with a focus on real-time implementation, coding in C, and key implementation considerations.

Takeaways

• 😀 The Extended Kalman Filter (EKF) is an advanced sensor fusion technique used to estimate states of a system, like aircraft attitude (roll and pitch angles), by combining sensor data with dynamic models.
• 😀 The EKF operates in two main steps: Prediction (where the state is estimated using a dynamic model) and Update (where sensor measurements correct the state estimate).
• 😀 The Kalman gain (K) adjusts how much trust to place on the sensor data versus model predictions during the update step, optimizing the accuracy of the state estimate.
• 😀 In contrast to simpler filters, the EKF uses matrix operations to combine sensor data with a non-linear dynamic model, making it more robust and flexible.
• 😀 The prediction step uses gyroscope data to estimate the rate of change of the roll and pitch angles, while the update step uses accelerometer data to correct these estimates.
• 😀 The error covariance matrix (P) indicates the confidence in the state estimate. It increases in the prediction step and decreases during the update step as sensor measurements refine the state estimate.
• 😀 The Jacobian matrices (A and C) are used to linearize the non-linear models in the EKF, allowing for effective updates of the state and error covariance matrices.
• 😀 The use of Euler angles and trigonometric functions (like sine and cosine) can make the EKF computationally expensive, especially in embedded systems with limited resources.
• 😀 Real-time implementation of the EKF on embedded systems involves practical considerations, including computational efficiency, sensor noise filtering, and proper initialization of state estimates and matrices.
• 😀 The EKF can become unstable if the system's linearity is wrongly assumed, highlighting the need for careful tuning and the potential for system-specific optimizations.

Q & A

What is the main focus of the video regarding sensor fusion?

The video focuses on explaining the Extended Kalman Filter (EKF) as a method for combining dynamical models and sensor data to estimate the states of a system, specifically in the context of aircraft attitude estimation using accelerometers and gyroscopes.

How does the Kalman Filter differ from the complementary filter discussed in earlier videos?

The Kalman Filter is more advanced than the complementary filter, as it optimally combines sensor data with a dynamical model, adjusting the state estimate using a Kalman gain matrix. Unlike the complementary filter, which uses a fixed alpha constant, the Kalman Filter dynamically adjusts the Kalman gain at each time step to minimize the estimation error.

What role does the Kalman Gain (k) play in the Kalman Filter?

The Kalman Gain (k) determines the weight given to the sensor measurements versus the model predictions. A larger Kalman Gain means more trust is placed on the sensor data, while a smaller gain favors the model's predictions. This dynamic adjustment helps improve the accuracy of the system's state estimate.

Why is the Extended Kalman Filter (EKF) needed for sensor fusion in non-linear systems?

The EKF is used for non-linear systems because it linearizes the non-linear system and sensor models around the current state estimate. This allows the Kalman Filter's framework to handle complex, non-linear relationships, making it suitable for real-world applications like aircraft attitude estimation.

What is the difference between the prediction and update steps in the EKF?

In the prediction step, the EKF uses the system's dynamical model to predict the next state based on the current state estimate. In the update step, it incorporates sensor measurements to correct the predictions and refine the state estimate, with the Kalman Gain controlling how much weight is given to the sensor data.

What are the two primary types of noise handled by the EKF in this application?

The EKF handles two main types of noise: model noise, represented by the covariance matrix Q, and sensor noise, represented by the covariance matrix R. The model noise accounts for imperfections in the system's dynamic model, while sensor noise arises from inaccuracies in the sensor readings.

What is the significance of the error covariance matrix (P) in the EKF?

The error covariance matrix (P) indicates the uncertainty in the state estimate. It is updated at each step: after the prediction step, it increases (indicating more uncertainty), and after the update step, it decreases as the EKF incorporates sensor measurements, thus improving the estimate.

What challenges are associated with implementing the EKF in a real-time system, such as on a microcontroller?

Implementing the EKF on a microcontroller involves challenges such as the high computational cost of matrix operations (especially inversion), the need for efficient sensor data filtering, handling sensor noise, and performing trigonometric operations. Additionally, the use of Euler angles can lead to gimbal lock, which must be considered in the implementation.

How does the EKF address non-linearities in the system model?

The EKF addresses non-linearities by linearizing the system's transition function and sensor model around the current state estimate. This is achieved by calculating Jacobian matrices, which approximate the non-linear models with linear ones in the vicinity of the current state, allowing the Kalman Filter to be applied in non-linear settings.

What practical issues does the video highlight when implementing the EKF on embedded systems?

Practical issues include determining how to efficiently initialize the state estimates and covariance matrices, dealing with varying noise characteristics, filtering raw sensor data, managing the computational load of matrix operations, and addressing potential instability or incorrect assumptions about linearity in the system.