Forward position problem of two PUMA-type robots manipulating a planar four-bar linkage payload

This paper presents a closed-form solution to the forward position problem of two PUMA-type robots manipulating a planar four-bar linkage payload. The orientation of a specified payload link is described, in the plane of the payload, by a sixth-order polynomial and the angular displacement of a specified link in a robot end-effector is described by a second-order polynomial. Closed-form solutions for the remaining unknown angular displacement are obtained using orthogonal transformation matrices with dual number elements. The paper shows that, for a given set of robot input angles, twenty-four assembly configurations of the robot-payload system are possible. The polynomials provide insight into these configurations, and also reveal stationary configurations of the system. The paper emphasizes that an understanding of the kinematic geometry is important in the development of the closed-form solution. Graphical methods are presented which provide insight into the geometry, and a check of the analytical approach. For illustrative purposes, a numerical example of the two robots manipulating a four-bar linkage payload in a manufacturing process is presented