Kinematic singularity

A stretched arm loses a motion direction#

For a planar arm with its links fully aligned, the joints can initially move the tip perpendicular to the arm but cannot produce every tip-velocity direction. The Modern Robotics singularity lesson demonstrates this rank loss with two- and three-joint arms.

The limitation is instantaneous. It does not mean the mechanism can never move inward or reach another configuration after first bending its joints.

Rank matters more than a determinant#

A robot Jacobian need not be square. Comparing its rank with the maximum achievable for the selected task handles redundant and lower-mobility robots as well as square Jacobians.

A robot designed for a two-dimensional task is not automatically singular because it cannot perform arbitrary six-dimensional motion. Singularity concerns a loss relative to that mechanism's normal capability.

Near a singularity also matters#

Even before exact rank loss, some task velocities can require very large joint velocities. MIT's manipulation notes connect this behavior to small singular values and discuss constrained differential IK.

This is different from gimbal lock in Euler angles, which is a coordinate-representation singularity. Switching to quaternions fixes that representation issue but cannot restore a motion direction lost through the robot's joint geometry.