On condition numbers and algorithms for determining a rigid body movement. BIT

Abstract Using a set of landmarks to represent a rigid body, a rotation of the body can be determined in least squares sense as the solution of an orthogonal Procrustes problem. We discuss some geometrical properties of the condition number for the problem of determining the orthogonal matrix representing the rotation. It is shown that the condition number critically depends on the configuration of the landmarks. The problem is also reformulated as an unconstrained nonlinear least squares problem and the condition number is related to the geometry of such problems. In the common 3-D case, the movement can be represented by using a screw axis. Also the condition numbers for the problem of determining the screw axis representation are shown to closely depend on the configuration of the landmarks. The condition numbers are finally used to show that the used algorithms are stable

Algorithms For Constrained And Weighted Nonlinear Least Squares

. A hybrid algorithm consisting of a Gauss-Newton method and a second order method for solving constrained and weighted nonlinear least squares problems is developed, analyzed and tested. One of the advantages of the algorithm is that arbitrary large weights can be handled and that the weights in the merit function do not get unnecessary large when the iterates diverge from a saddle point. The local convergence properties for the Gauss-Newton method is thoroughly analyzed and simple ways of estimating and calculating the local convergence rate for the Gauss-Newton method are given. Under the assumption that the constrained and weighted linear least squares subproblems attained in the Gauss-Newton method are not too ill-conditioned, global convergence towards a first order KKT point is proved. Key words. nonlinear least squares, optimization, parameter estimation, weights AMS subject classifications. 65K, 49D 1. Introduction. Assume that f : R n ! R m is a twice continuously diffe..