Singularity avoidance and the kinematics of an eight-revolute-joint manipulator

Abstract

Many general purpose robot manipulators consist of six serially connected actuators whose functional intent is to provide six Cartesian degrees-of-freedom to an end-effector. This kinematic mapping of actuator freedoms to Cartesian freedoms is valid only when the six actuator-screws are linearly independent. When the six actuator-screws become linearly dependent, the manipulator has lost full freedom and the end-effector is unable to follow arbitrary Cartesian trajectories with finite actuator speeds. We begin our study with a geometrically "optimum" six-revolute-joint robot manipulator, which is decoupled into a regional structure and an orientation structure. Since this 6R manipulator is kinematically decoupled, the regional and orientation structures comprise two separate screw systems, both of third order. When the order of either of these systems falls below three, the 6R manipulator will be at a configuration singularity. With the three actuator-screws of the orientation structure placed in a configuration singularity, we find the screw which is instantaneously reciprocal to this system. With this reciprocal screw we locate a fourth "redundant" actuator-screw "best" suited for returning freedom to the orientation structure. With the regional structure placed in its configuration singularity, we do likewise by adding to it the actuator-screw "best" suited for returning the lost freedom. Hence, the orientation and regional structures form two separate 4R structures, taken together forming an 8R manipulator. Conceptually, the algorithms to control both the 4R orientation and 4R regional structures begin in primary mode with three actuator-screws apiece. In primary mode we use the traditional position kinematics approach. Whenever either of the orientation or regional structures has difficulty following its Cartesian trajectory, control for that structure goes into secondary mode, where the redundant actuator is then called upon. The algorithms maintain a sixth-order screw system for an 8R manipulator with bounded actuator rates and well-behaved actuator values. The numerical computations for an 8R manipulator are comparable to those for the "optimum" 6R manipulator and can be implemented in real-time.