Before revision:  Utilizing the \boxplus \backslash \boxminus operations and further defining a \oplus operation on the respective manifold, we propose a canonical representation of the robotic system, which enables us to separate the manifold structures from the system descriptions in each step of the Kalman filter, ultimately leading to a generic , symbolic and manifold-embedding Kalman filter framework . This proposed Kalman filter framework can be used by only casting the system model into the canonical form without going through the cumbersome hand-derivation of the on-manifold Kalman filter. This is particularly useful when the robotic system is of high dimension.
After revision:  Utilizing the \boxplus \backslash \boxminus operations and further defining a \oplus operation on the respective manifold, we propose a canonical representation of the robotic system, which enables us to separate the manifold structures from the system descriptions in each step of the Kalman filter, ultimately leading to a generic , symbolic and manifold-embedding Kalman filter framework that are naturally evolving on the manifold.