Singularities in Inverse Kinematics: Detection and Handling

A 6-DOF articulated robot arm is following a Cartesian end-effector trajectory using resolved-rate (velocity-level) inverse kinematics, i.e. $\dot{q} = J^{-1}(q)\,\dot{x}$, where $J$ is the geometric Jacobian. As the arm approaches a certain configuration, the commanded joint velocities spike toward very large values even though the desired end-effector velocity $\dot{x}$ stays small and bounded.

Explain what is physically and mathematically happening, how you would *detect* this condition reliably, and what concrete techniques you would use to keep the controller stable and the motion safe. Discuss the trade-offs of your chosen mitigation(s).