Workspace Analysis of Mechanical Manipulators Using Polynomial Discriminants

Abstract

A new method, for the investigation of manipulator workspace, based on polynomial displacement equations and their discriminants is proposed in this paper. The approach followed enables one to obtain (a) analytical expressions describing the workspace boundary surfaces in Cartesian coordinates, (b) the distribution of the number of ways to position the hand inside the workspace, and (c) the conditions on kinematic parameters when the motion of the hand degenerates. Joint limitations are incorporated in the method. In the present paper and a companion paper [17], the method is rigorously applied to eight types of manipulators with various combinations of revolute and prismatic pairs having the last three revolute axes intersect orthogonally at at point. A numerical example is presented for illustration.

A method for mapping the boundaries of collision-free reachable workspaces

“…(18) to the manipulator's inherent kinematic constraints Eq. (1), we will have an extended set of constraint equations which can address the collision-avoidance problem between two convex objects…”

“…(18), which is a system of 11 equations; therefore, the extended constraints Eq. (19) is a system of 13 equations.…”

Higher order singularities of regional manipulators

“…The contours between these regions are the boundaries given by 'P( R, z ) = 0. Kohli and Spanos (1984) show that region boundaries correspond to a repeated joint angle value in the inverse kinematics. Thus 'P( R, z ) = 0 is also the locus of all workspace points with a repeated joint value.…”

“…Four types of tangency can occur: a simple tangent, or two point contact ( Figure 3a); an osculating point, or three point contact ( Figure 3b); four point contact ( Figure 3c); or paired tangency (Figure 3d). Other workspace analysis schemes that are based on the inverse kinematics (e.g., Kohli and Spanos (1984)) do not distinguish between these types of contact.…”

“…Examples of such work can be found in Kohli and Spanos (1984), who used polynomial discriminants, as well as Pai (1989) and Burdick (1991), who used topology to determine properties of robot singularities. In such previous work, it has been shown that the region boundary surfaces of the workspace correspond to stationary configurations (i. e., singularities) of various configurations, or poses, of the robot.…”