On the Existence of Linear Observed Systems on Manifolds with Connection

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.