Observability for 3-D Range-only Positioning System

Three-dimensional range-only positioning system (3-DRPS) is characteristic of nonlinearity and measurement incompleteness and so on, and involved in observability. In order to attain the observable conditions for 3-DRPS an analytic expression of Gram matrix determinant of a linear system (LS) derived from the 3-DRPS is precisely deduced. After the observable conditions of the LS are attained, we prove a sufficient and necessary condition of the observability for the 3-DRPS which are simply expressed by the algebraical expression of observation point coordinates and measurements. This may be prone to being used as observability criteria for 3-DRPS. It is geometrically showed that the 3-DRPS is observable if and only if there are two observation points that are collinear with target, or there are three non-collinear observation points that are coplanar with target, or there are four non-coplanar observation points. The results have special reference to the application to control of observer points and choice of beacon position and satellite.