Incremental Forward Kinematics of Wire-Suspended Parallel Mechanical System (A Many-Worlds Interpretation Approach)

In this paper, we discuss an incremental forward kinematics of wire-suspended mechanical systems. In order to deal with a mechanical system of this type, we have to take into account the characteristic property of a wire, in that it cannot take any compressive force. We give a general formulation and its solution of the incremental forward kinematics problem taking account of the slack condition of a wire. We also consider the influence of displacement of the mass center position of the suspended platform on the kinematic behavior of the system. An interval arithmetic approach is proposed to deal with the uncertainty in the kinematic parameters. One of the important points is that the number of possible solutions of the formulated incremental kinematics problem is often more than one. We introduce a kind of parallelism, referred to as the “many-worlds interpretation” taken from the quantum mechanics theory, to this problem and offer an approach to deal with plural possible kinematic states simultaneously. The developed approach is based on basic equations in general form and is applicable to various wire-suspended mechanical systems. The feasibility of the proposed incremental kinematics approach is demonstrated by example calculations of three-, six-, and eight-wire systems. On the basis of the example forward kinematics calculation results, we conclude the following. The influence of displacement of mass center position of the platform is not insignificant. The number of possible kinematic states becomes large in the case of the neighborhood of singular configuration. In spite of an incremental kinematics based on linearization, the required computational cost of the proposed parallelism approach is considerable; it is, however, demonstrated that the proposed approach is still fairly practical from the viewpoint of computation. The developed approach cannot deal with drastic change in kinematic configuration in a single incremental step, since it is based on linearization of the kinematic relation; however, the approach can distinguish such discontinuity.