Mechanical singularity

There are three types of singularities that can be found in mechanisms: direct-kinematics singularities, inverse-kinematics singularities, and combined singularities. These singularities occur when one or both Jacobian matrices of the mechanisms becomes singular of rank-deficient.^[1] The relationship between the input and output velocities of the mechanism are defined by the following general equation:

${\displaystyle {\textbf {A}}{\dot {\textbf {x}}}+{\textbf {B}}{\dot {\textbf {q}}}={\textbf {0}}}$

where ${\displaystyle {\dot {\textbf {x}}}}$ is the output velocities, ${\displaystyle {\dot {\textbf {q}}}}$ is the input velocities, ${\displaystyle {\textbf {A}}}$ is the direct-kinematics Jacobians, and ${\displaystyle {\textbf {B}}}$ is the inverse-kinematics Jacobian.

This first kind of singularities occurs when:

${\displaystyle \det({\textbf {B}})=0}$

This second kind of singularities occurs when:

${\displaystyle \det({\textbf {A}})=0}$

This kind of singularities occurs when for a particular configuration, both ${\displaystyle {\textbf {A}}}$ and ${\displaystyle {\textbf {B}}}$ become singular simultaneously.