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.