Coordinate-free formulation of a 3-2-1 wire-based tracking device using Cayley-Menger determinants

This paper deals with the problem of estimating the pose of a rigid moving object by measuring the length of six wires attached to it. Among all possible locations for the attachments on the moving object, the "3-2-1" configuration exhibits the highest number of favorable properties. A closed-form coordinate-free solution to the forward kinematics of this particular configuration is given in terms of Cayley-Menger determinants. The proposed formulation is mathematically more tractable compared to previous ones because all terms are determinants with geometric meaning. This accommodates a more thorough investigation of the properties of the device and leads to formulas whose numerical conditioning is independent from the chosen reference frames.