Kinematically redundant manipulators, those with more than the minimum number of degrees of freedom, are currently being discussed by many researchers because of their potential to optimize some objectives while satisfying the primary task of end-effector trajectory tracking. Among these objectives, a consideration for the use of the redundancy of a robot manipulator is to make the manipulator as dexterous as possible. The optimality constraints-based methods(OCBMs), which resolve the redundancy by introducing optimality constraints of necessary conditions for optimizing a dexterous measure, usually produce conservative joint trajectories.
With the advantage of conservative property, the algorithms of inverse kinematics based on optimality constraints have some problems because those are based only on necessary conditions for optimality. One of the problems is a switching, i.e., an undesirable configuration change from a maximum value of a performance measure to a minimum value may occur and cause an inverse kinematic solution to be unstable. Therefore, it is needed to analyze the characteristics of optimal solutions for redundant manipulators and to study the methods of overcoming the limitations.
In the thesis, optimality conditions for the kinematic control of a redundant manipulator are derived. Based on the conditions, the topological property of the configuration space is analyzed, and method of assuring the conservative property of joints are proposed. In addition that, the evaluation of several dexterity measures using the loci satisfying optimality conditions on the configuration space is performed. Through some examples of cyclic tasks, the analysis and the techniques are illustrated for a planar three degree-of-freedom manpulator.
First, sufficient conditions for the optimal solution are derived. The conditions consist of both the equilibrium condition, which is used as optimality constraints in the OCBMs, and the definitness of the projected Hessian matrix. Using these conditions, the explicit forms of the switching condition in the OCBMs are obtained. The conditions also show that the configuration at which switching occurs is equivalent to an algorithmic singularity in the extended Jacobian method. Especially, the definitness of the projected Hessian gives information on potentially good configurations (PGC).
Second, a method assuring the repeatability of joints is proposed. In order to make the inverse kinematic solution be repeatable, an optimization technique within the PGC region is formulated by introducing an artificial penalty function of sigmoid. A proper selection of a slope parameter of the sigmoid function assures the repeatibility of the robot joint trajectories.
Finally, a performance measure for the quantification of the dexterity of manipulators, which is the product of the determinants of the Jacobians of the newly defined sub-link systems, is proposed. The measure shows how far kinematically redundant manipulators are from singularity. The measure assures that the inverse kinematic solution remains in the same category of configurations. Also, the characteristics of several dexterity measures are compared with the new measure. The new measure is useful in designing the optimal link lengths of an arm.
