SCKF (square-root cubature Kalman filter)

The Square-Root Cubature Kalman Filter (SCKF) is an extension of the Kalman filter that addresses some of its limitations. It is a nonlinear filtering algorithm used for state estimation in dynamic systems where the relationship between the state variables and the measurements is nonlinear.

To understand the SCKF, let's first review the basic concepts of the Kalman filter. The Kalman filter is an optimal recursive estimator that estimates the state of a linear system in the presence of measurement noise. It consists of two steps: the prediction step and the update step. In the prediction step, the filter predicts the current state based on the previous estimate and the system dynamics model. In the update step, it incorporates the measurement information to correct the predicted state.

However, the Kalman filter assumes that the system dynamics and measurement models are linear, which is often not the case in real-world applications. When dealing with nonlinear systems, such as those encountered in robotics, aerospace, or computer vision, the Extended Kalman Filter (EKF) is typically used. The EKF linearizes the system and measurement models using first-order Taylor expansions to apply the Kalman filter framework.

The SCKF takes a different approach by avoiding linearization. Instead, it utilizes a technique called cubature integration to approximate the nonlinear system and measurement models. Cubature integration is a method for approximating multidimensional integrals, and it provides a way to propagate the state distribution through the nonlinear models without relying on linearization.

Here are the main steps involved in the SCKF algorithm:

Initialization:

• Initialize the state estimate (mean) and covariance.
• Choose the number of sigma points (cubature points) and their weights.

Prediction Step:

• Generate sigma points by using the square-root of the state covariance matrix.
• Propagate the sigma points through the nonlinear process model to obtain predicted sigma points.
• Compute the predicted state mean and covariance by estimating the weighted mean and covariance of the predicted sigma points.

Update Step:

• Generate sigma points based on the predicted state mean and covariance.
• Propagate the sigma points through the nonlinear measurement model to obtain measurement sigma points.
• Compute the predicted measurement mean and covariance by estimating the weighted mean and covariance of the measurement sigma points.
• Compute the cross-covariance between the predicted state and predicted measurement by estimating the weighted covariance between the sigma points of the state and measurement.
• Compute the Kalman gain by multiplying the cross-covariance with the inverse of the predicted measurement covariance.
• Update the state mean and covariance using the predicted measurement mean, the Kalman gain, and the difference between the actual measurement and the predicted measurement.

Repeat the prediction and update steps for each time step in the system.

The SCKF has a few advantages over the EKF. Firstly, it avoids the errors introduced by linearization, which can lead to inaccurate state estimates. Secondly, it provides a more accurate representation of the state covariance matrix, reducing the chances of filter divergence. Thirdly, it has a square-root formulation, which ensures that the covariance matrix remains positive definite throughout the estimation process.

However, the SCKF is computationally more expensive compared to the EKF due to the increased number of sigma points required for approximations. Additionally, the choice of the sigma points and their weights can impact the filter's performance, requiring careful selection.

Overall, the SCKF is a powerful nonlinear filtering algorithm that offers improved performance and accuracy compared to the EKF, particularly in situations where linearization may result in significant estimation errors.