# Remarks on stochastic cloning and delayed-state filtering

## TL;DR

This paper demonstrates that a properly derived delayed-state Kalman filter can handle correlated delayed measurements as effectively as stochastic cloning, without the need for state augmentation, offering computational advantages.

## Contribution

It revisits the delayed-state Kalman filter, showing it matches stochastic cloning in performance and clarifies misconceptions about handling delayed measurements without augmentation.

## Key findings

- Delayed-state Kalman filter yields the same results as stochastic cloning.
- Two equivalent formulations of the delayed-state Kalman filter are presented.
- One formulation can reduce computational and storage costs for certain problems.

## Abstract

Many estimation problems in aerospace navigation and robotics involve measurements that depend on prior states. A prominent example is odometry, which measures the relative change between states over time. Accurately handling these delayed-state measurements requires capturing their correlations with prior state estimates, and a widely used approach is stochastic cloning (SC), which augments the state vector to account for these correlations. This work revisits a long-established but often overlooked alternative--the delayed-state Kalman filter--and demonstrates that a properly derived filter yields exactly the same state and covariance update as SC, without requiring state augmentation. Moreover, two equivalent formulations of the delayed-state Kalman filter (DSKF) are presented, providing complementary perspectives on how the prior-state measurement correlations can be handled within the generalized Kalman filter. These formulations are shown to be comparable to SC in asymptotic computational and memory complexity, while one DSKF formulation can offer reduced arithmetic and storage costs for certain problem dimensions. Our findings clarify a common misconception that Kalman filter variants are inherently unable to handle correlated delayed-state measurements, demonstrating that an alternative formulation achieves the same results without state augmentation.

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Source: https://tomesphere.com/paper/2508.21260