Non-uniform Observability for Fast Moving Horizon Estimation with application to the SLAM problem

The paper proves that, for the second-order 2D SLAM model with h0(z)=z and relative measurements (bearing, range, optical flow, Doppler), any input that drives the robot to track a circular trajectory yields weak regular persistence (and hence weak regular observability) under an L1-closeness condition to the ideal circle; the argument proceeds by lower-bounding an observability Gramian block for the landmark coordinates along the circle and transferring this bound to the actual trajectory using a Lipschitz/L1 stability estimate, together with the h0(z)=z contribution to the Gramian in the sensor-centric setting (Theorem 33 with (47)–(52) and Proposition 28) . The candidate solution establishes the same conclusion via the same core ideas: (i) explicit Jacobians H(i) for the four sensors, (ii) positive definiteness of the circular-reference landmark Gramian sA(i) (Proposition 32) with the rc≠‖χc−ℓ‖ restriction only needed for optical flow (i=3), and (iii) robustness from the reference to the actual trajectory via an L1 bound, then (iv) combination with the direct-state output h0(z)=z to control the robot-state component. These steps mirror the paper’s structure (Assumption 27, Proposition 28, and Theorem 33) and differ mainly in presentation (a “good set” decomposition and a direct lower bound on the moving-horizon cost versus the paper’s Hessian/Schur-complement route). One small conservatism in the candidate solution is imposing rc≠‖χc−ℓ‖ for all sensors, while the paper only requires it for optical flow (i=3) (Proposition 32) . Overall, the reasoning aligns and the result matches the paper’s claim.