What does a decrease in the rank of the Jacobian indicate?

• A. The manipulator is approaching a kinematic singularity, risking large joint motions for small end-effector changes.
• B. The manipulator has unlimited reach.
• C. The end-effector can move in all directions without constraints.
• D. Joint velocities become impossible to compute due to numerical overflow alone.

Answer: (C)

A decrease in the Jacobian’s rank means the end-effector velocity space has fewer independent directions available. This signals a kinematic singularity: the manipulator can lose one or more degrees of freedom, so it cannot freely move in all directions. Near such configurations, small commanded end-effector motions can require very large joint motions to achieve, and the mapping from end-effector velocities to joint velocities becomes ill-conditioned or even impossible for some directions. The ideas of unlimited reach or full-direction freedom are incompatible with a reduced rank, and numerical overflow is a computer-niceness issue, not a fundamental kinematic property. So, a drop in rank indicates approaching a singularity with constrained motion and potential velocity amplification.