Singularity analysis of robotic manipulators with velocity-constraints for minimally invasive surgery

This paper presents a novel framework for analyzing the mobility of constrained serial manipulators. We focus on the type of constraints that arise in minimally invasive surgery, where a long shaft is inserted into the human body through a incision point, or trocar. The trocar constraint will in this case change the mobility of the manipulator and the location and nature of the singularities. For minimally invasive surgery both the mobility and the singularities of the system need to be known to obtain safe and reliable operation. Velocity constraints on the chain will in general complicate the mobility analysis of the manipulator as conventional methods such as manipulability and other methods that require the manipulator Jacobian cannot be applied because these methods do not take the constraints into account. The main contribution of the paper is to find the end-effector velocity by adding the velocity at the constraint and the velocity of the joints after the constraint (often called the wrist) and observing that these velocities need to span the whole end-effector velocity space. We then use this new representation of the end-effector velocity to find the mobility of the constrained manipulators. The framework can be used both to determine the optimal manipulator geometry in the presence of chain constraints and, once the manipulator geometry is chosen, to control the robot such that the mobility is maintained high and singularities avoided. The first property of the presented framework can for example be used to find the optimal geometry of the wrist of a surgical robot subjects to incision point constraints, and the latter can be used to control the robot so that singularities are avoided. The singularities typically take a very different form than for the unconstrained case.