— This paper presents the analytic solution of a fundamental open problem in the framework of state esti-mation/nonlinear observability, which is the Unknown Input Observability problem (UIO problem). The problem consists in deriving the analytic criterion that allows us to automatically obtain the state observability in presence of disturbances (or unknown inputs). In other words, the problem is to extend the well known observability rank condition to the case when the dynamics are also driven by unknown inputs. Enunciated in the seventies by the control theory community, this problem was only solved in the linear case. The solution here provided holds for nonlinear systems in presence of a single unknown input. The first part of the paper presents this analytic solution. Very surprisingly, the complexity of the overall analytic criterion is comparable to the complexity of the observability rank condition. The second part of the paper applies this analytic criterion to a robotics system when its dynamics are affected by an external disturbance (e.g., due to the presence of wind). To corroborate the results of our observability analysis we perform extensive simulations and we show that, a simple estimator based on an Extended Kalman Filter, provides results that agree with what we could expect from the observability analysis.