Constrained Unscented Kalman Filtering with Improved Reliability for Small Celestial Body Relative Navigation

Constrained filtering problems are ubiquitous in deep space exploration missions. Through integrating physical constraints into the estimation framework, the performance of the filtering algorithm may be improved. For constrained filtering problems, the estimated state should always satisfy the constraints. To this end, one useful technique is to construct a constrained least square problem to restrain the estimated state within the constraint region. Moreover, for constrained UKF, which is applicable to many nonlinear spacecraft systems, the sigma points are also required to stay within the constraint region. In previous studies, this is done by introducing a scaling parameter in unscented transformation to project the sigma points to the constraint region while preserving their statistical properties. Unfortunately, the sigma points will degrade after scaling if the constrained estimation from the last step is on the boundary of the constraints. The degradation will significantly increase the error in prediction covariance, which will further lead to unreliable state estimations or even failure of the UKF with scaled unscented transformation. To address this problem, a new sigma point projection method for constrained UKF is developed in this paper. Instead of directly projecting the unconstrained sigma points to the real constraint region, a redundant term is first introduced to construct a relaxed constraint boundary. Then, a non-zero scaling parameter is optimized taking into account both the new constraint boundary and limits on the scaling parameter. The optimal scaling parameter is used in the projection of the unconstrained sigma points so that degradation of the sigma points can be avoided. The proposed sigma point projection method is further incorporated into a constrained UKF to generate precise state estimations. Finally, the effectiveness of the proposed filtering algorithm is verified through numerical simulations in a small celestial body relative navigation scenario with state constraints. The performance of the method is evaluated, providing insights for future development of an optimal projection method for UKF under complicated constraints.