Auto-triangulation and auto-trilateration - Part 2: Three-dimensional experimental verification

In the companion paper, [Prec Eng 2002, in press] we introduced an approach to extending the idea of self-calibration to triangulation and trilateration. The advantages of this approach over the conventional approach to self-calibration (that uses a calibration artifact) include the facts that it only requires a linear transducer instead of an artifact, it extends naturally to higher dimensional self-calibration problems (three-dimensional self-calibration), and it is capable of producing any calibration map instead of a predetermined discrete calibration map. In the aforementioned companion paper, our discussion concentrated on describing the underlying concepts and the process for generating a self-calibration map. It also concentrated on proving that the approach could, in the limit, calibrate any point in the feasible calibration workspace (or any length in the measuring range of the linear transducer). All this was discussed in the context of a two-dimensional calibration problem. This paper extends the mathematical formulation to three-dimensional self-calibration or auto-trilateration. First, the basic formulation to produce the auto-calibration equations for trilateration is developed. Next, a three-dimensional verification experiment is described and its results discussed. Auto-trilateration allows for cheap and efficient calibration of three-axis machining centers and can promote regular and cost-effective calibration in the NC machining industry.