In
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, a mechanical singularity is a position or configuration of a
mechanism
Mechanism may refer to:
* Mechanism (engineering), rigid bodies connected by joints in order to accomplish a desired force and/or motion transmission
*Mechanism (biology), explaining how a feature is created
*Mechanism (philosophy), a theory that ...
or a
machine
A machine is a physical system using Power (physics), power to apply Force, forces and control Motion, movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to na ...
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
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity.
For exam ...
.
Examples of mechanical singularities are
gimbal lock
Gimbal lock is the loss of one degree of freedom in a three-dimensional, three-gimbal mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate t ...
and in static mechanical analysis, an under-constrained system.
Mechanical engineering
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.
References
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