On the placement of open-loop robotic manipulators for reachability

Criteria for and implementation of the placement of robot manipulators with the objective of reaching specified target points are herein addressed. Placement of an open-loop robotic manipulator in a working environment is characterized by defining the position and orientation of the manipulator's base with respect to a fixed reference frame. The problem has become important in both the medical and manufacturing fields, where a robot arm must be appropriately placed with respect to targets (tasks) that cannot be moved. An applicable numerical formulation is presented. While other methods have used inverse kinematics solutions in their formulation for defining a locality for the manipulator base, this type of solution is difficult to implement because of the inherent complexities in determining all inverse kinematic solutions. The approach used in this work is based on characterizing the placement, forcing a cost function to impel the workspace envelope in terms of surface patches towards the target points and subject to functionality constraints, but not requiring the computation of inverse kinematics. The formulation and experimental code are demonstrated using a number of examples.