This paper investigates the problem of non linear observability when part (or even all) of the system inputs is unknown. A new definition of indistinguishable states is first provided. Then, in order to separate the effect of the known inputs from the effect of the unknown inputs on the system outputs, the state is augmented. This allows us to obtain the extension of basic properties, which hold in the case of known inputs. Starting from these properties, the paper analyzes the state observability and provides sufficient conditions that allow us to establish if a given system is weakly locally observable. On the other hand, the proposed method computes a codistribution defined in the augmented space. This makes the computational cost dependent on the dimension of the augmented state. In the case of a single unknown input, the paper proposes a method to operate a separation on the previous codistribution. Thanks to this separation, it is possible to derive the observability properties by computing a codistribution that is defined in the original space. As it will be seen, the analytic derivations required to perform this separation are complex and we are currently extending them to the multiple unknown inputs case. The proposed approach is used to derive the observability properties of many systems, starting from very simple ones. The last application is a very complex unknown input observability problem. Specifically, we derive the observability properties for the visual-inertial structure from problem in the case when part of the inertial inputs are missing.