Path Planning With Selective/Desriable Orientation in Finite Twist Mapping Space

The paper proposes a new approach for path planning based upon the finite twist mapping spaces and discusses the selective and desirable orientation in the planning. In the FT mapping space, positions and orientations are presented in conjunction with configuration space with the dexterous and partial dexterous workspaces. The path planning can hence be made with a range of dexterity of a manipulator to meet a desirable orientation. The work also extends to redundant manipulators.

Task-Oriented Direct Synthesis of Serial Manipulators Using Moment Invariants

This paper applies the finite twist mapping to spatial manipulators and gives a new approach for the direct synthesis of serial manipulators by using moment invariants and task decomposition. The task is represented in an image space as a cluster of a large number of points and is related to the finite twist of the manipulator. For a given task, the new approach applies a series of analysis to the data, and produces a set of invariant criteria. The type and disposition of the proximal joint are identified, and the remaining regional structure may then be designed. The approach is based upon the numerical analysis of a task without recourse to graphical analysis and is computer programmable. This new design procedure is illustrated with a flowchart and a numerical example is given.

Finite Twist Mapping and its Application to Planar Serial Manipulators with Revolute Joints

This paper reviews a method of representing the finite twist motion of a rigid body, and applies this concept to the problem of describing the motion capabilities of planar serial manipulators. In particular, this finite twist representation describes the position and orientation of the end-effector link relative to a defined reference position and orientation. The transformations associated with the joint motions are used to investigate the range of end-effector motion, including the effects of limits of joint travel. The finite twist representations are then extracted from such transformations and presented in an image space so that the motion of the manipulator is fully characterized. Further, the reachable and dexterous workspaces are predicted by the same general means. Examples of planar serial manipulators with two (RR) and three revolute (RRR) joints are given to illustrate the approach.

Parabolic fixed points, invariant curves and action-angle variables

A fixed point of an area-preserving mapping of the plane is called elliptic if the eigenvalues of its linearization are of unit modulus but not ±1; it is parabolic if both eigenvalues are 1 or −1. The elliptic case is well understood by Moser's theory. Here we study when is a parabolic fixed point surrounded by closed invariant curves. We approximate our mapping T by the phase flow of an Hamiltonian system. A pair of variables, closely related to the action-angle variables, is used to reduce T into a twist mapping. The conditions for T to have closed invariant curves are stated in terms of the Hamiltonian.

Monotone twist mappings and the calculus of variations

It is shown that a smooth area-preserving monotone twist mapping ϕ of an annulus A can be interpolated by a flow ϕt which is generated by a t-dependent Hamiltonian in ℝ × A having the period 1 in t and satisfying a Legendre condition. In other words, any such monotone twist mapping can be viewed as a section mapping for the extremals of variational problem on a torus:where F has period 1 in t and x and satisfies the Legendre condition Fẋẋ>0.