Finding the Position and Orientation of a Sensor on a Robot Manipulator Using Quaternions

The problem of finding the relative position and orientation between the reference frames of a link-mounted sensor and the link has been formulated as a kinematic equation of the form H l H x = H x H c in terms of homogeneous transformation matrices by Shiu and Ahmad (1987). In this article, normal ized quaternions (Euler parameters) are used to transform the kinematic equation into two simple and structured linear systems with rank-deficient coefficient matrices. Closed-form solutions to these systems are derived using the generalized inverse method with singular-value decomposition analysis. To obtain a unique solution, two distinct robot movements are required. This leads to an overdetermined system of equations. A criterion for selecting the independent set of equations is developed. A set of closed-form formulae for the solution of these equations are derived. The selection crite rion and the solution formulae can be easily incorporated in application programs that require the calculation of the rela tive position and orientation of the sensor.