The least squares fit of a hyperplane to uncertain data

Abstract

Sensor based robotic systems are an important emerging technology. Whenrobots are working in unknown or partially known environments, they need rangesensors that will measure the Cartesian coordinates of surfaces of objects intheir environment. Like any sensor, range sensors must be calibrated. The rangesensors can be calibrated by comparing a measured surface shape to a knownsurface shape. The most simple surface is a plane and many physical objects haveplanar surfaces. Thus, an important problem in the calibration of range sensorsis 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 leastsquares fit of a hyperplane to uncertain data. The first order necessaryconditions require the solution of an eigenvalue problem. We have shown that thesolution satisfies the second order conditions (the Hessian matrix is positivedefinite). Thus, our solution satisfies the sufficient conditions for a localminimum. We have performed numerical experiments that demonstrate that oursolution is superior to alternative methods.