  We present an efficient algorithm for computing the closest singular
configuration to each non-singular pose of a 3-RPR planar manipulator
performing a 1-parametric motion. By considering a 3-RPR manipulator as a
planar framework, one can use methods from rigidity theory to compute the
singularity distance with respect to an intrinsic metric. Such a metric has the
advantage over any performance index used for indicating the closeness to
singularities, that the obtained value is a distance, which equals the radius
of a guaranteed singularity-free sphere in the joint space of the manipulator.
The proposed method can take different design options into account as the
platform/base can be seen as a triangular plate or as a pin-jointed triangular
bar structure. Moreover, we also allow the additional possibility of pinning
down the base/platform triangle to the fixed/moving system thus it cannot be
deformed. For the resulting nine interpretations, we compute the corresponding
intrinsic metrics based on the total elastic strain energy density of the
framework using the physical concept of Green-Lagrange strain. The global
optimization problem of finding the closest singular configuration with respect
to these metrics is solved by using tools from numerical algebraic geometry.
The proposed algorithm is demonstrated based on an example, which is also used
to compare the obtained intrinsic singularity distances with the corresponding
extrinsic ones.
