Manipulability

How freely an arm's end effector can move in each direction from its current pose, and how close it is to a singularity.

Manipulability was first given a quantitative definition by T. Yoshikawa in a 1985 IJRR paper. Given the Jacobian matrix J (which maps joint velocity to end-effector velocity), letting the joint velocity range over the unit sphere ‖θ̇‖ = 1 traces out an ellipsoid of corresponding end-effector velocities, called the manipulability ellipsoid: its principal axes point along the eigenvectors of JJᵀ, and the length of each semi-axis is the square root of the corresponding eigenvalue. The longer the ellipsoid is along some direction, the more easily the end effector can move that way; a direction where it's squashed flat signals the arm is approaching a singular pose (an orientation where it loses the ability to move in that direction), and the ellipsoid collapses into a line segment or a plane exactly at a singularity. The commonly used scalar index w = √det(JJᵀ) is proportional to the ellipsoid's volume and equals zero at a singularity. It's used to choose good working poses and to guide structural design, and is also a common optimization objective for null-space motion in redundant arms, steering the arm actively away from singularities.

ExampleFor a planar two-link arm (link lengths L1, L2), manipulability is w = L1·L2·|sin θ2|, where θ2 is the elbow angle: fully extended (θ2 = 0) gives w = 0, meaning the end effector can no longer extend further along the arm's direction; bent to 90° at the elbow gives the maximum w, where the end effector moves freely in every direction.

Also called

Manipulability Ellipsoid, Yoshikawa Manipulability

Related

Jacobian Matrix · Singular Configuration (Kinematic Singularity) · Kinematic Redundancy · Null Space · Jacobian Pseudoinverse

Sources

Lynch & Park, Modern Robotics（2017）5.4 节：Manipulability (Chinese)
Wikipedia: Manipulability ellipsoid
Buss 逆运动学综述（参考文献列出 Yoshikawa, Manipulability of robotic mechanisms, IJRR 1985） (Chinese)