Lyapunov-based range and motion identification for a nonaffine perspective dynamic system

Many applications require the interpretation of the Euclidean motion of features of a 3-dimensional (3D) object through 2D images. In this paper, the range and the Euclidean coordinates of an object undergoing general affine motion are determined for a paraboloid imaging system. Unlike image systems that are based on a planar image surface (or spherical or ellipsoidal surfaces), the perspective dynamic system resulting from the paraboloid projected image does not maintain an affine form. Because of the nonaffine form, existing range identification observers can not be directly utilized. The contribution of the current result is the development of a nonlinear state estimator that can be applied to the nonaffine perspective dynamic system to determine the range and Euclidean coordinates of an object feature without the use of linear approximations. The nonlinear estimator asymptotically determines the range information from a single camera provided some observability conditions are satisfied and that the Euclidean motion parameters are known. The proposed technique is developed through a Lyapunov-based design and stability analysis, and simulation results are provided that illustrate the performance of the state estimator