Mechanical singularity

In engineering, a mechanical singularity is a position or configuration of a mechanism or a machine where the subsequent behaviour cannot be predicted, or the forces or other physical quantities involved become infinite or nondeterministic.
When the underlying engineering equations of a mechanism or machine are evaluated at the singular configuration (if any exists), then those equations exhibit mathematical singularity.
Examples of mechanical singularities are gimbal lock and in static mechanical analysis, an under-constrained system.
Types of singularities.
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. The relationship between the input and output velocities of the mechanism are defined by the following general equation:
formula_1
where formula_2is the output velocities, formula_3is the input velocities, formula_4is the direct-kinematics Jacobians, and formula_5is the inverse-kinematics Jacobian.
Type-I: Inverse-kinematics singularities.
This first kind of singularities occurs when:
formula_6
Type-II: Direct-kinematics singularities.
This second kind of singularities occurs when:
formula_7
Type-III: Combined singularities.
This kind of singularities occurs when for a particular configuration, both formula_8and formula_9become singular simultaneously. 