Analytical and Graphical Solutions to the Forward Position Problem of Two Robots Manipulating a Spatial Linkage Payload

Abstract

This paper presents a solution to the forward position problem of two PUMA-type robots manipulating a spatial four-bar linkage payload. To simplify the kinematic analysis, the Bennett linkage, which is a special geometry spatial four-bar, will be regarded as the payload. The orientation of a specified payload link is described by a sixth-order polynomial and a specified joint displacement in the wrist subassembly of one of the robots is described by a second-order polynomial. A solution technique, based on orthogonal transformation matrices with dual number elements, is used to obtain closed-form solutions for the remaining unknown joint displacements in the wrist subassembly of each robot. An important result is that, for a given set of robot input angles, twenty-four assembly configurations of the robot-payload system are possible. Repeated roots of the polynomials are shown to correspond to the stationary configurations of the system. The paper emphasizes that an understanding of the kinematic geometry of the system is essential to verify the number of possible solutions to the forward position problem. Graphical methods are also presented to provide insight into the assembly and stationary configurations. A numerical example of the two robots manipulating the Bennett linkage is included to demonstrate the importance of the polynomial and closed-form solutions.