THE LEAST-SQUARES FIT OF A HYPERPLANE TO UNCERTAIN DATA

Sensor based robotic systems are an important emerging technology. Whe n robots are working in unknown or partially known environments, they need range sensors that will measure the Cartesian coordinates of surf aces of objects in their environment. Like any sensor, range sensors m ust be calibrated. The range sensors can be calibrated by comparing a measured surface shape to a known surface shape. The most simple surfa ce 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 for mulated a constrained optimization problem to determine the least squa res fit of a hyperplane to uncertain data. The first order necessary c onditions 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 suffici ent conditions for a local minimum. We have performed numerical experi ments that demonstrate that our solution is superior to alternative me thods.