Unbiased Minimum-Variance Filter for Flexible Landing with Mixed State Constraints

Flexible landing on small celestial bodies involves mixed state constraints arising from bounded structural deformation and unit-quaternion attitude representation. Incorporating these constraints into unbiased minimum-variance (UMV) filtering is challenging because the corresponding gain-design problem is high-dimensional and nonconvex. This article develops a UMV-constrained filter for flexible landing with mixed state constraints. It is shown that, by introducing an increment vector and a tailored objective function, the original gain-design problem can be reformulated, without loss of optimality, as a bilevel optimization problem. The bilevel problem comprises an inner problem with a closed-form solution and an outer problem that optimizes over the low-dimensional increment vector, thereby enabling dimensionality reduction in the optimization variables. By analyzing the structure of its solution space, a spatial decomposition strategy is developed to split the outer problem into three subproblems. Two subproblems admit closed-form solutions derived via the Lagrange multipliers method, whereas the remaining one is solved numerically. In the considered flexible-landing scenario, the proposed bilevel reformulation and spatial decomposition strategy enable efficient computation of the UMV-constrained estimate, with an average runtime of 1.23×10^-2s. Simulation results further demonstrate that the proposed filter explicitly satisfies the mixed constraints and provides more accurate position and attitude estimates than the unconstrained filter and the estimated projection method.