# Structural Foundations of Nonlinear Systems with Unknown Inputs: The UID-Induced Normal Form and Minimal-Sensing Structure-from-Motion

arXiv: https://arxiv.org/abs/2610.05939 · HTML: https://mlsanity.com/paper/2610.05939/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 29 references that are in the map.
- First posted on arXiv 5 Oct 2026.

## Builds on (most cited first, 7 of 7)

- [LIC-Fusion 2.0: LiDAR-Inertial-Camera Odometry with Sliding-Window Plane-Feature Tracking](https://mlsanity.com/paper/2008.07196/) (2020)
- [Observability Analysis of Aided INS with Heterogeneous Features of Points, Lines and Planes](https://mlsanity.com/paper/1805.05876/) (2018)
- [Nonlinear Unknown Input Observability: The Analytic Solution in the case of a Single Unknown Input](https://mlsanity.com/paper/1501.02968/) (2015)
- [Observability-Aware Trajectory Optimization for Self-Calibration with Application to UAVs](https://mlsanity.com/paper/1604.07905/) (2016)
- [Nonlinear Unknown Input Observability and Unknown Input Reconstruction: The General Analytical Solution](https://mlsanity.com/paper/2201.07610/) (2022)
- [Observability, Identifiability and Sensitivity of Vision-Aided Navigation](https://mlsanity.com/paper/1311.7434/) (2013)
- [Unobservable Subspace Evolution and Alignment for Consistent Visual-Inertial Navigation](https://mlsanity.com/paper/2511.17992/) (2025)

## Abstract (arXiv)

This paper establishes the first general structural solution to the problem of state estimation for nonlinear systems driven by unknown inputs. Building upon nonlinear unknown-input observability theory, we show that every such system admits a structurally equivalent representation, referred to as the UID-induced normal form. The proposed representation decomposes the information carried by the unknown inputs into two complementary components: unknown-input directions that are structurally decoupled from the observable dynamics and observable quantities that completely represent the unknown-input information affecting the observable dynamics. As a consequence, the UID-induced normal form provides a unified structural solution to unknown-input decoupling and unknown-input reconstruction, without requiring any model or stochastic assumption on the unknown inputs. The practical significance of the proposed framework is demonstrated through a previously unexplored minimal Structure-from-Motion configuration. The proposed representation enables recursive state estimation from only three point features and a single-axis gyroscope, allowing the recovery of the three-dimensional structure and camera motion up to an unknown global scale factor. Experiments on real-world data validate the proposed framework and demonstrate the feasibility of this minimal sensing configuration.