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, 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:
where is the output velocities, is the input velocities, is the direct-kinematics Jacobians, and is the inverse-kinematics Jacobian.

Type-I: Inverse-kinematics singularities

This first kind of singularities occurs when:

Type-II: Direct-kinematics singularities

This second kind of singularities occurs when:

Type-III: Combined singularities

This kind of singularities occurs when for a particular configuration, both and become singular simultaneously.