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

Abstract

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.

Asymptotic Tracking for Systems with Structured and Unstructured Uncertainties

“…The RISE method has also been used as an identification technique. For example, the method has been applied to identify friction (e.g., [14] and [15]), for range identification in perspective and paracatadioptric vision systems (e.g., [5] and [6]), and for fault detection and identification (e.g., [17]). …”

Asymptotic Tracking for Aircraft via Robust and Adaptive Dynamic Inversion Methods

“…The positive definite function satisfies the inequality (26) provided the sufficient condition introduced in (21) is satisfied. In (26), the continuous, positive definite functions are defined as (27) where , and .…”

“…The inequalities in (26) and (32) can be used to show that in ; hence in . Given that in , standard linear analysis methods can be used to prove that in from (4).…”

Lyapunov-Based Range Identification For Paracatadioptric Systems

“…to propose a range identification method based on a sequence of linear approximation-based observers. More recently, Gupta et al constructed a nonlinear observer in [20], based on the work in [8], to asymptotically identify the range for the nonaffine paracatadioptric system considered in [19].…”

“…The contribution of the current paper (and our preliminary work in [20], [21]) is the development of a nonlinear estimator to extract the range information (and hence, the Euclidean coordinates) from a paracatadioptric system where the image dynamics are expressed in affine or nonaffine forms. The contributions in this paper are that the developed nonlinear observer can be applied to both affine and nonaffine systems; can be applied to both the fixed and moving camera configurations; is continuous; and leads to an asymptotic result.…”