Kalman Filter Flashcards
- What is the fundamental idea behind the Kalman Filter (KF), and how does it address the limitations of traditional estimation methods?
The Kalman Filter (KF) is a recursive predictor-corrector algorithm that optimally estimates the state of a dynamic system using noisy measurements. It combines prior knowledge (prediction) with current measurements (correction) to minimize the estimated error covariance. Unlike traditional methods, the KF adapts to changing system dynamics and handles uncertainty.
- Explain the Kalman Filter prediction step and its mathematical formulation.
The prediction step involves projecting the current state estimate forward in time using the system dynamics model. Mathematically, it can be expressed as: (\hat{x}{k|k-1} = F_k \hat{x}{k-1|k-1} + B_k u_k), where (\hat{x}{k|k-1}) is the predicted state, (F_k) is the state transition matrix, (\hat{x}{k-1|k-1}) is the previous state estimate, (B_k) represents control input, and (u_k) is the control vector.
- Describe the Kalman Filter correction step and its role in improving state estimates.
The correction step updates the predicted state estimate based on the actual measurements. It combines the prediction with measurement information to obtain an improved estimate. Mathematically, it can be expressed as: (\hat{x}k = \hat{x}{k|k-1} + K_k (z_k - H_k \hat{x}_{k|k-1})), where (\hat{x}_k) is the corrected state estimate, (z_k) is the measurement, (H_k) is the measurement matrix, and (K_k) is the Kalman gain.
- What are the assumptions made in the linear Kalman Filter?
The linear Kalman Filter assumes that the system dynamics and measurement models are linear, and the noise is Gaussian and white. Additionally, it assumes that the initial state estimate and error covariance are known.
- Explain the concept of process noise and measurement noise in the Kalman Filter.
Process noise represents the uncertainty in the system dynamics (e.g., unmodeled forces, disturbances), while measurement noise accounts for inaccuracies in sensor measurements. Both are modeled as zero-mean Gaussian random variables with known covariance matrices.
- How does the Extended Kalman Filter (EKF) handle non-linear system dynamics?
The EKF linearizes the non-linear system dynamics and measurement models around the current state estimate. It computes the Jacobians of these models to approximate the Kalman gain and update the state estimate.
- What is the Unscented Kalman Filter (UKF), and how does it differ from the EKF?
The UKF avoids linearization by propagating a set of carefully chosen sigma points through the non-linear models. It captures the true mean and covariance accurately up to the third order. Unlike the EKF, it does not rely on Jacobians.
- Describe the unscented transform used in the UKF.
The unscented transform generates sigma points (representative samples) from the state distribution. These points are then propagated through the non-linear models to estimate the mean and covariance of the transformed distribution.
- What are the advantages of using the UKF over the EKF?
The UKF provides better accuracy for non-linear systems, handles non-Gaussian noise, and avoids the need for manual Jacobian computation. It is more robust and suitable for a wide range of applications.
- Give an example of a real-world application where the UKF is commonly used.
The UKF is commonly used in robotics (e.g., robot localization, sensor fusion), aerospace (e.g., spacecraft navigation), and financial modeling (e.g., stock price prediction).
- How does the Kalman Filter handle time-varying system dynamics?
The Kalman Filter adapts to time-varying system dynamics by updating the state estimate based on new measurements. It continuously adjusts the prediction and correction steps to account for changes in the underlying process.
- What is the role of the error covariance matrix in the Kalman Filter?
The error covariance matrix represents the uncertainty in the state estimate. It quantifies how much the predicted state differs from the true state. The Kalman gain adjusts the error covariance based on measurement consistency.
- Explain the concept of innovation in the Kalman Filter.
Innovation refers to the discrepancy between the predicted measurement (based on the state estimate) and the actual measurement. The Kalman Filter uses the innovation to update the state estimate and reduce the error covariance.
