Configuration Space and Linkage Defects

The design of linkages presents many challenges.  Most notably the nonlinearity of the design space. However, another challenge is created by the presence of defects in linkage designs. The regularly studied linkage defects are named order, branch, and circuit. See Balli and Chand. An effective method to eliminate linkage designs that possess branch and circuit defects is to construct the complete configuration space of candidate mechanism designs. Constructing this space is a two part process:

  1. solve the forward kinematics equations for a set of input joint parameters to obtain all configurations at every input and
  2. sort all configurations into their proper solution branches and circuits.

Part (1) requires a solution method that finds all solutions to the forward kinematics equations. This can be accomplished by resultant elimination methods and continuation. Generally, a linkage’s forward kinematics equations are much simpler to solve than its synthesis equations. Part (2) involves sorting configurations (solutions to the fk equations) into branches and circuits. This is accomplished by tracing the branch curves with a Newton-Raphson based method. Particularly challenging is dealing with the singularities present in the configuration space, those are the points where solution branches would intersect.  A strategy for dealing with singular points is established in Plecnik and McCarthy.

Once the configuration space of a mechanism is constructed, one now simply checks whether the entirety of a motion requirement is contained on a single branch or circuit, depending on the application’s requirement. Here is an example:

The forward kinematics equations of a Watt II function generator are solved and solutions are sorted. The configuration space is constructed. Singularities are marked with purple points.

The motion requirement is a set of nine input-output points (in blue below). Taking a slice of the configuration space, we construct the mechanism’s input-output function.

We conclude that the analyzed mechanism is defective. Zooming in on the problematic area of the configuration space, we see two singular points and two mechanism circuits.

S. S. Balli and S. Chand, 2002. “Defects in link mechanisms and solution rectification,” Mechanism and Machine Theory, 37(9):851-876.

M. Plecnik and J. M. McCarthy, 2014. “Numerical synthesis of six-bar linkages for mechanical computation,” Journal of Mechanisms and Robotics.

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