Solving the burmester problem using kinematic mapping

Planar kinematic mapping is applied to the five-position Burmester problem for planar four-bar mechanism synthesis. The problem formulation takes the five distinct rigid body poses directly as inputs to generate five quadratic constraint equations. The five poses are on the fourth order curve of intersection of up to four hyperboloids of one sheet in the image space. Moreover, the five poses uniquely specify these two hyperboloids. So, given five positions of any reference point on the coupler and five corresponding orientations, we get the fixed revolute centres, the link lengths, crank angles, and the locations of the coupler attachment points by solving a system of five quadratics in five variables that always factor in such a way as to give two pairs of solutions for the five variables (when they exist)

Unified kinematic analysis of general planar parallel manipulators

A kinematic mapping of planar displacements is used to derive generalized constraint equations having the form of ruled quadric surfaces in the image space. The forward kinematic problem for all three-legged, three-degree-of-freedom planar parallel manipulators thus reduces to determining the points of intersection of three of these constraint surfaces, one corresponding to each leg. The inverse kinematic solutions, though trivial, are implicit in the formulation of the constraint surface equations. Herein the forward kinematic solutions of planar parallel robots with arbitral, mixed leg architecture are exposed completely, and in a unified way, for the first time. Copyrigh

Solving the forward kinematics of a planar three-legged platform with holonomic higher pairs

A practical solution procedure for the forward kinematics problem of a fully-parallel planar three-legged platform with holonomic higher pairs is presented. Kinematic mapping is used to represent distinct planar displacements of the end-effector as discrete points in a three dimensional image space. Separate motions of each leg trace skew hyperholoids of one sheet in this space. Therefore, points of intersection of the three hyperholoids represent solutions to the forward kinematics problem. This reduces the problem to solving three simultaneous quadratics. Applications of the platform are discussed and an illustrative numerical example is given

Singular configurations of wrist-partitioned 6R serial robots: A geometric perspective for users

In this paper the singular configurations of wrist-partitioned 6R serial robots in general, and the KUKA KR-15/2 industrial robot in particular, are analytically described and classified. While the results are not new, the insight provided by the geometric analysis for users of such robots is. Examining the problem in the joint axis parameter space, it is shown that when the end-effector reference point is taken to be the wrist centre the determinant of the associated Jacobian matrix splits into four factors, three of which can vanish. Two of the three potentially vanishing factors give a complete description of the positioning singularities and the remaining one a complete description of the orientation singularities, in turn providing a classification scheme

Extreme distance to a spatial circle

Determination of shortest distances in the three dimensional task space of robots is pertinent to pick-and-place operations, collision avoidance, and for impact prediction in dynamic simulation. The conventional approach is to find perpendicular distances between planar patches approximating body surfaces. In contrast, this paper treats four variants of shortest distance computations wherein one or both elements are circular edges. These three dimensional cases include circle and point, circle and plane, circle and line and two non coplanar circles. Solutions to these four fundamental problems are developed with elementary geometry. Examples are presented, and the closed form algebraic solutions are verified with descriptive geometric constructions