Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Existence of Linear Observed Systems on Manifolds with Connection

Changwu Liu, Yuan Shen

TL;DR

This work generalizes the concept of linear observed systems from Euclidean spaces and Lie groups to dynamics on arbitrary smooth manifolds endowed with an affine connection ${\nabla}$. It formalizes two complementary properties—exact linearization with state-independent Jacobians and preintegration—and proves their equivalence under the manifold-connection framework; curvature constraints, such as ${\mathcal{R}}=0$ or ${\nabla}{\mathcal{R}}=0$, become necessary conditions for existence. A key result shows that the flat Cartan connection recovers the Lie-group characterization of linear observed systems, thus linking the new theory to known group-affine observer results. An explicit non-Lie-group example on $S^2$ demonstrates the applicability of the framework to curved state spaces, while discussions clarify metric compatibility and the broader implications for invariant filtering in navigation and robotics. Overall, the paper provides a rigorous geometric foundation for designing linear-like observers on curved manifolds with practical implications for GNSS/IMU-based estimation and related navigation problems.”

Abstract

Linear observed systems on manifolds are a special class of nonlinear systems whose state spaces are smooth manifolds but possess properties similar to linear systems. Such properties can be characterized by preintegration and exact linearization with Jacobians independent of the linearization point. Non-biased IMU dynamics in navigation can be constructed into linear observed settings, leading to invariant filters with guaranteed behaviors such as local convergence and consistency. In this letter, we establish linear observed property for systems evolving on a smooth manifold through the connection structure endowed upon this space. Our key findings are the existence of linear observed systems on manifolds poses constraints on the curvature of the state space, beyond requiring the dynamics to be compatible with some connection-preserving transformations. Specifically, the flat connection case reproduces the characterization of linear observed systems on Lie groups, showing our theory is a true generalization.

On the Existence of Linear Observed Systems on Manifolds with Connection

TL;DR

This work generalizes the concept of linear observed systems from Euclidean spaces and Lie groups to dynamics on arbitrary smooth manifolds endowed with an affine connection . It formalizes two complementary properties—exact linearization with state-independent Jacobians and preintegration—and proves their equivalence under the manifold-connection framework; curvature constraints, such as or , become necessary conditions for existence. A key result shows that the flat Cartan connection recovers the Lie-group characterization of linear observed systems, thus linking the new theory to known group-affine observer results. An explicit non-Lie-group example on demonstrates the applicability of the framework to curved state spaces, while discussions clarify metric compatibility and the broader implications for invariant filtering in navigation and robotics. Overall, the paper provides a rigorous geometric foundation for designing linear-like observers on curved manifolds with practical implications for GNSS/IMU-based estimation and related navigation problems.”

Abstract

Linear observed systems on manifolds are a special class of nonlinear systems whose state spaces are smooth manifolds but possess properties similar to linear systems. Such properties can be characterized by preintegration and exact linearization with Jacobians independent of the linearization point. Non-biased IMU dynamics in navigation can be constructed into linear observed settings, leading to invariant filters with guaranteed behaviors such as local convergence and consistency. In this letter, we establish linear observed property for systems evolving on a smooth manifold through the connection structure endowed upon this space. Our key findings are the existence of linear observed systems on manifolds poses constraints on the curvature of the state space, beyond requiring the dynamics to be compatible with some connection-preserving transformations. Specifically, the flat connection case reproduces the characterization of linear observed systems on Lie groups, showing our theory is a true generalization.
Paper Structure (20 sections, 12 theorems, 7 equations, 1 figure)

This paper contains 20 sections, 12 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Theory
  4. Dynamics on Smooth Manifolds
  5. Vector Error Representation
  6. Linear Observed Property from Exact Linearization
  7. Linear Observed Property from Preintegration
  8. Existence of Linear Observed Systems
  9. $\mathcal{R}$=0 Classifies Linear Observed Systems on Groups
  10. Discussions
  11. An Example from Navigation
  12. Conclusions
  13. Proof of Theorem \ref{['theorem::a']}
  14. Proof of Theorem \ref{['theorem::b']}
  15. Proof of Theorem \ref{['theorem::c']}
  16. ...and 5 more sections

Key Result

Theorem 1

If $p$ and $\hat{p}$ are in some normal neighborhood $U$ of a complete manifold $M$, where any point pair in $U$ can be joined by a geodesic, then the local vector-error representation $(\Phi^t,\mathfrak{er},u_t)$ induces a smooth vector field $E$ on $\Phi^t(\hat{p})$ by $\mathfrak{er}(\Phi^t(p),\Ph

Figures (1)

  • Figure 1: An intuitive picture of a self-similar flow field patch and its related vector fields $E$ and $T$.

Theorems & Definitions (29)

  • Definition 1: Dynamics on Manifold
  • Definition 2: Flow Line
  • Definition 3: Local Vector-error Representation
  • Theorem 1
  • Definition 4: Exact Linearization Independent of State
  • Theorem 2
  • Definition 5: Self-Similar Flow Field
  • Theorem 3
  • Definition 6: Preintegration
  • Remark 1
  • ...and 19 more