On the metrics and co-ordinate-system-induced sensitivity in computational kinematics

This paper is an attempt to investigate the sensitivity to change of units and co-ordinate systems in computational kinematics when it involves both orientation and displacement in three dimensional space. The focus is on the behaviour of the NewtonÐRaphson iterative technique for solving customary system of equations for kinematic loop closure. It is shown that with the change of units or co-ordinate systems for some initial points the method does not converge to the same solution. Such behaviour is attributed to the shift of the boundaries of the so-called basins of attraction which play an important role in the theory of Chaos and Fractals. A number of numerical examples have been investigated and presented in tabulated form. To reduce the e¤ect of sensitivity two procedures are suggested. One of them, Non-linear Elimination, is a recent development based on numerical elimination of variables in a system of equations. Investigation presented in this paper is the Þrst of its kind and it is hoped that it will initiate further research to treat the problem of sensitivity in computational kinematics