GMKF: Generalized Moment Kalman Filter for Polynomial Systems with Arbitrary Noise (2403.04712v2)

Abstract: This paper develops a new filtering approach for state estimation in polynomial systems corrupted by arbitrary noise, which commonly arise in robotics. We first consider a batch setup where we perform state estimation using all data collected from the initial to the current time. We formulate the batch state estimation problem as a Polynomial Optimization Problem (POP) and relax the assumption of Gaussian noise by specifying a finite number of moments of the noise. We solve the resulting POP using a moment relaxation and prove that under suitable conditions on the rank of the relaxation, (i) we can extract a provably optimal estimate from the moment relaxation, and (ii) we can obtain a belief representation from the dual (sum-of-squares) relaxation. We then turn our attention to the filtering setup and apply similar insights to develop a GMKF for recursive state estimation in polynomial systems with arbitrary noise. The GMKF formulates the prediction and update steps as POPs and solves them using moment relaxations, carrying over a possibly non-Gaussian belief. In the linear-Gaussian case, GMKF reduces to the standard Kalman Filter. We demonstrate that GMKF performs well under highly non-Gaussian noise and outperforms common alternatives, including the Extended and Unscented Kalman Filter, and their variants on matrix Lie group.

• The paper introduces the GMKF that extends Kalman Filtering to polynomial systems by managing arbitrary noise via higher-order moment relaxations.
• It formulates state estimation as a polynomial optimization problem, enabling optimal estimation under non-Gaussian noise conditions.
• Empirical results highlight the method's superior performance and robustness compared to traditional filters, especially for real-world robotics applications.

An Insightful Overview of the Generalized Moment Kalman Filter for Polynomial Systems with Arbitrary Noise

Introduction to \GMKFlong (\GMKF)

State estimation remains a pivotal problem in robotics, often approached through the lens of Kalman Filtering (KF) when dealing with linear systems and Gaussian noise. However, the extension of KF to nonlinear systems or systems perturbed by non-Gaussian noise necessitates sophisticated techniques that stray from the foundational principles of KF, losing its optimality guarantees. Addressing these limitations, recent advances introduce the \GMKFlong (\GMKF), a novel filtering approach tailored for polynomial systems subjected to arbitrary noise. This paper dissects the formulation and implications of \GMKF, comparing it against traditional Kalman Filtering and its variants on matrix Lie groups.

Batch Estimation with Arbitrarily Distributed Noise

The cornerstone of \GMKF is its capacity to handle arbitrarily distributed noise in polynomial systems by considering higher-order moments up to $r$, drastically extending the efficacy of state estimation beyond the Gaussian assumption. By casting the batch state estimation as a Polynomial Optimization Problem (POP) and leveraging moment relaxations, the \BPUE (Best Polynomial Unbiased Estimator) guarantees provably optimal estimates under certain conditions. This development not only aligns with the Generalized Method of Moments (GMM) in econometrics for dealing with data of unknown distributions but also enriches the theoretical framework of polynomial optimization in state estimation.

Recursive Estimation and the Essence of \GMKF

Transitioning from batch to a recursive setup, \GMKF innovatively encapsulates the prediction and update steps within polynomial systems, resembling the traditional Kalman Filter structure but enriched with the robustness against non-Gaussian noise. It formulates these steps as POPs, solved through moment relaxations, enabling the propagation of non-Gaussian beliefs across time. Notably, in the linear-Gaussian scenario, \GMKF collapses to the standard Kalman Filter, underscoring its generality and versatility.

Theoretical Underpinnings and Practical Implications

The theoretical foundation of \GMKF reveals that the dual solutions of the moment relaxations facilitate the computation of an SOS belief, interpreted as an approximated covariance in noiseless cases. This feature not only highlights the method's fidelity in capturing the true distribution of the system's state but also signifies the potential for refined uncertainty quantification in polynomial systems. Moreover, the empirical validation of \GMKF against non-Gaussian noise showcases superior performance compared to conventional filters, suggesting its effectiveness in real-world robotics applications.

Future Perspectives on \GMKF

Despite its promising attributes, \GMKF necessitates further exploration, particularly in theoretical conditions ensuring the moment relaxation's rank-1 solutions and the observability properties of the polynomial system. Furthermore, establishing a fully probabilistic interpretation of the SOS belief and its connection to the estimation error remains an enticing avenue for research. Lastly, refining the choice of the moment order $r$ for balancing computational complexity and estimation accuracy could significantly enhance the practicality of \GMKF.

Conclusion

The introduction of the \GMKFlong marks a significant step forward in the domain of state estimation for polynomial systems, particularly in the presence of arbitrary noise. By meticulously integrating insights from polynomial optimization and moment theory, \GMKF not only extends the Kalman Filtering paradigm to accommodate non-Gaussian uncertainties but also opens up new horizons for robust state estimation in robotics and beyond.