Systems and methods for tracking a path using the null-space

A system comprising:

• a manipulator arm configured for robotically moving a distal end effector relative to a proximal base, the manipulator arm having a plurality of joints between the distal end effector and the proximal portion coupled to the base, the plurality of joints together providing sufficient degrees of freedom to allow a range of joint states for an individual state of the distal end effector;
• an input for receiving a manipulation command to move the end effector with a desired end effector movement;
• and
• a processor coupling the input device to the manipulator arm, the processor configured to perform operations including:
• calculating an end effector displacing movement of the plurality of joints in response to the manipulation command by calculating joint velocities within a null-perpendicular-space of a Jacobian;
• defining a position-based constraint of one or more joints of the plurality of joints within a joint space of the plurality of joints, the position-based constraint including one or more paths corresponding to a desired movement of the one or more joints within the joint space of the plurality of joints;
• calculating a tracking movement of the plurality of joints so that a position of the one or more joints tracks at least one of the one or more paths, wherein calculating the tracking movement comprises calculating joint velocities of the one or more joints within a null-space of the Jacobian in a direction towards the at least one of the one or more paths, the null-space being orthogonal to the null-perpendicular space;
• and
• transmitting a command to the manipulator arm to drive the plurality of joints according to the calculated movements so as to effect the desired end effector displacing movement concurrently with the desired movement of the one or more joints.