The least squares fit of a hyperplane to uncertain data

Abstract

Sensor based robotic systems are an important emerging technology. When robots are working in unknown or partially known environments, they need range sensors that will measure the Cartesian coordinates of surfaces of objects in their environment. Like any sensor, range sensors must be calibrated. The range sensors can be calibrated by comparing a measured surface shape to a known surface shape. The most simple surface is a plane and many physical objects have planar surfaces. Thus, an important problem in the calibration of range sensors is to find the best (least squares) fit of a plane to a set of 3D points. We have formulated a constrained optimization problem to determine the least squares fit of a hyperplane to uncertain data. The first order necessary conditions require the solution of an eigenvalue problem. We have shown that the solution satisfies the second order conditions (the Hessian matrix is positive definite). Thus, our solution satisfies the sufficient conditions for a local minimum. We have performed numerical experiments that demonstrate that our solution is superior to alternative methods.