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Jacobian and Singularity

The Jacobian relates the joint velocities to the tool's velocity: a small change in the joint angles produces a corresponding change in the tool's position, scaled by $J$ . A singularity is a configuration where $J$ loses rank, so that even a small desired tool velocity requires unrealistically large joint velocities.

\dot{\mathbf x}=J(\mathbf q)\,\dot{\mathbf q}

the Jacobian relates joint velocity to tool velocity

Symbols

$J(\mathbf q)$	the Jacobian
$\dot{\mathbf q}$	joint velocities
$\dot{\mathbf x}$	the tool's velocity

Example

A 6-axis arm with 6 task degrees of freedom:

$J$ is $6\times6$ ; if it loses rank, a singularity occurs.

Near a singularity the joint velocities become unrealistically large even for a small tool velocity.

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Part of Robotics: Kinematics.