Manipulability ellipsoid

The manipulability ellipsoid is a concept in robotics that represents the manipulability of a robotic system in a graphical form. Here, the manipulability of a robot refers to its ability to alter the position of the end effector based on the joint configuration. A higher manipulability measure signifies a broader range of potential movements in that specific configuration. When the robot is in a singular configuration the manipulability measure diminishes to zero.

Definition

The manipulability ellipsoid is defined as the set^[1]

$\{\xi :\xi ^{\operatorname {T} }(J(q)J^{\operatorname {T} }(q))\xi \leq 1\}$

where q is the joint configuration of the robot and J is the robot Jacobian relating the end-effector velocity with the joint rates.

Geometric Interpretation

A geometric interpretation of the manipulability ellipsoid is that it includes all possible end-effector velocities normalized for a unit input at a given robot configuration. The axis of the ellipsoid can be computed by using the singular value decomposition of the robot Jacobian.^[1]^[2]

References

^ ^a ^b Spong, M.W.; Hutchinson, Seth; Vidyasagar, M. (2005). Robot Modeling and Control. Wiley. Wiley. ISBN 9780471765790.
^ "5.4. Manipulability – Modern Robotics". modernrobotics.northwestern.edu. Northwestern University. Retrieved 18 October 2023.