Norm-Constrained Kalman Filtering

Abstract

The problem of estimating the state vector of a dynamical system from vector measurements, when it is known that the state vector satisfies norm equality constraints is considered. The case of a linear dynamical system with linear measurements subject to a norm equality constraint is discussed with a review of existing solutions. The norm constraint introduces a nonlinearity in the system for which a new estimator structure is derived by minimizing a constrained cost function. It is shown that the constrained estimate is equivalent to the brute force normalization of the unconstrained estimate. The obtained solution is extended to nonlinear measurement models and applied to the spacecraft attitude filtering problem.

IMU and magnetometer modeling for smartphone-based PDR

“…Specifically, it has to be ensured that the unity of q I k is preserved. Thus, a norm-constrained Kalman filter where part of the state vector is subject to norm constraints is required [17], [18]. The algorithm is listed in Algorithm 1 where 1) Setx 0|0 = µ 0 and P 0|0 = Σ 0 , k ← 1.…”

“…Note that the only difference between the regular unscented Kalman filter and its norm-constrained version is step 4 where the normalization is performed: The constrained statesx c k|k can simply normalized after a regular filter update (as it has been shown that this is equivalent to calculating a constrained filter gain, see [17], [18]). …”

Sequential Pose Estimation Using Linearized Rotation Matrices

“…However, using quaternions within an EKF requires either a normalization step or a norm constraint [24] within the filter to maintain a valid rotation. We avoid these complexities and the issues of representation by using the group structure of rotations in our EKF formulation.…”

Linear Covariance Analysis and Epoch State Estimators

“…The state estimation errors propagate to any other time according to equation (12). Therefore, the various components of the error covariances at such other times are given by…”

“…11 For this application, two filters are run concurrently, one constrained by the condition that the distance of closest approach is greater than some effective hard body radius, and the other constrained so that the closest approach distance is less than the hard body radius. The constraint is implemented using the technique of Zanetti, et al 12 An epoch state filter with epoch time in the future is required because the time of closest approach, where the constrained update is enforced, is in the future of all the measurements.…”