Abstract
A manipulator is termed kinematically redundant when the number of controllable inputs is greater than the dimension of the task space. This property increases the overall dexterity and hence application of the manipulator. Such applications include the avoiding of obstacles within the workspace, singular configurations and more generally the operation within complex environments and the performance of complex tasks.
The inverse kinematics represents an under-constrained transformation from task to joint space. The transformation being linear when the kinematics are formulated in terms of rates (or equivalently infinitesimal displacements) and non-linear when formulated in terms of positions. It is for this reason that the redundancy literature is virtually all formulated in terms of rates. This approach can cause problems of unpredictability of the manipulator’s configuration, an example of which is repeatability tasks. Such applications are better dealt with by an inverse position solution whenever possible.
The inverse position solution is under-constrained and involves highly non-linear transcendental functions and consequently requires a set of secondary equations for solution. In this paper, the secondary equations used to augment the position equations are derived from the equilibrium requirements of the mechanism. The equations, like the inverse rate methods are joint stiffness weighted. Joint stiffnesses play a major role upon the behaviour of the augmented equations and hence the ability to select desirable solutions. It effects such factors as the number of solutions, the numerical value of the solution and the usefulness of the solution. The effects of the stiffness matrix are discussed in this paper in order to gain a better understanding of the behaviour of the augmented equations.
