15 research outputs found
Homography-Based Positioning and Planar Motion Recovery
Planar motion is an important and frequently occurring situation in mobile robotics applications. This thesis concerns estimation of ego-motion and pose of a single downwards oriented camera under the assumptions of planar motion and known internal camera parameters. The so called essential matrix (or its uncalibrated counterpart, the fundamental matrix) is frequently used in computer vision applications to compute a reconstruction in 3D of the camera locations and the observed scene. However, if the observed points are expected to lie on a plane - e.g. the ground plane - this makes the determination of these matrices an ill-posed problem. Instead, methods based on homographies are better suited to this situation.One section of this thesis is concerned with the extraction of the camera pose and ego-motion from such homographies. We present both a direct SVD-based method and an iterative method, which both solve this problem. The iterative method is extended to allow simultaneous determination of the camera tilt from several homographies obeying the same planar motion model. This extension improves the robustness of the original method, and it provides consistent tilt estimates for the frames that are used for the estimation. The methods are evaluated using experiments on both real and synthetic data.Another part of the thesis deals with the problem of computing the homographies from point correspondences. By using conventional homography estimation methods for this, the resulting homography is of a too general class and is not guaranteed to be compatible with the planar motion assumption. For this reason, we enforce the planar motion model at the homography estimation stage with the help of a new homography solver using a number of polynomial constraints on the entries of the homography matrix. In addition to giving a homography of the right type, this method uses only \num{2.5} point correspondences instead of the conventional four, which is good \eg{} when used in a RANSAC framework for outlier removal
Coping with Algebraic Constraints in Power Networks
In the intuitive modelling of the power network, the generators and the loads are located at different subset of nodes. This corresponds to the so-called structure preserving model which is naturally expressed in terms of differential algebraic equations (DAE). The algebraic constraints in the structure preserving model are associated with the load dynamics. Motivated by the fact the presence of the algebraic constraints hinders the analysis and control of power networks, several aggregated models are reported in the literature where each bus of the grid is associated with certain load and generation. The advantage of these aggregated models is mainly due to the fact that they are described by ordinary differential equations (ODE) which facilitates the analysis of the network. However, the explicit relationship between the aggregated model and the original structure preserved model is often missing, which restricts the validity and applicability of the results. Aiming at simplified ODE description of the model together with respecting the heterogenous structure of the power network has endorsed the use of Kron reduced models; see e.g. [2]. In the Kron reduction method, the variables which are exclusive to the algebraic constraints are solved in terms of the rest of the variables. This results in a reduced graph, the (loopy) Laplaican matrix of which is the Schur complement of the (loopy) Laplacian matrix of the original graph. By construction, the Kron reduction technique restricts the class of the applicable load dynamics to linear loads. The algebraic constraints can also be solved in the case of frequency dependent loads where the active power drawn by each load consists of a constant term and a frequencydependent term [1],[3]. However, in the popular class of constant power loads, the algebraic constraints are âproperâ, meaning that they are not explicitly solvable. In this talk, first we revisit the Kron reduction method for the linear case, where the Schur complement of the Laplacian matrix (which is again a Laplacian) naturally appears in the network dynamics. It turns out that the usual decomposition of the reduced Laplacian matrix leads to a state space realization which contains merely partial information of the original power network, and the frequency behavior of the loads is not visible. As a remedy for this problem, we introduce a new matrix, namely the projected pseudo incidence matrix, which yields a novel decomposition of the reduced Laplacian. Then, we derive reduced order models capturing the behavior of the original structure preserved model. Next, we turn our attention to the nonlinear case where the algebraic constraints are not readily solvable. Again by the use of the projected pseudo incidence matrix, we propose explicit reduced models expressed in terms of ordinary differential equations. We identify the loads embedded in the proposed reduced network by unveiling the conserved quantity of the system