Kinematic Redundancy

Having more joints than a task strictly needs, so the same end-effector pose can be reached by infinitely many joint configurations.

If a robot has more joints, n, than the dimensionality of the task space, m, it's said to be kinematically redundant for that task, and the extra n−m degrees of freedom are called the redundancy. A full end-effector pose is 6-dimensional, so a 7-axis arm has 1 redundant degree of freedom: with the end effector held fixed, the elbow can still move through a range of positions. This kind of internal motion that doesn't affect the end effector is called self-motion, and corresponds to the null space of the Jacobian matrix. Redundancy means a single target has infinitely many joint-angle solutions, and redundancy resolution is choosing one of them by some criterion: the most common approach, the Jacobian pseudoinverse, gives the solution with the smallest sum of squared joint velocities, and secondary objectives — avoiding obstacles, staying away from joint limits or singular configurations — can be layered into the null space on top of that. The cost is that inverse kinematics no longer has a unique answer and needs extra optimization. Redundancy is relative to the task: even a 6-axis arm counts as redundant if all that matters is a 3-dimensional end-effector position.

ExampleKeeping your palm flat on a table without moving it, you can still raise or lower your elbow — that's the redundant degree of freedom in a human arm. A 7-axis robot arm can likewise adjust its elbow position while the end effector stays put, to steer around a nearby obstacle.

Also called

Redundancy Resolution, Redundant Manipulator

Related

Null Space · Jacobian Pseudoinverse · 7-DoF Robot Arm · Swivel Angle · Null-Space Control · Inverse Kinematics (IK)

Sources

Modern Robotics (Lynch & Park) preprint PDF, Example 4.7 与 Ch. 5–6 (Chinese)