  Articulated robots such as manipulators increasingly must operate in
uncertain and dynamic environments where interaction (with human coworkers, for
example) is necessary. In these situations, the capacity to quickly adapt to
unexpected changes in operational space constraints is essential. At certain
points in a manipulator's configuration space, termed singularities, the robot
loses one or more degrees of freedom (DoF) and is unable to move in specific
operational space directions. The inability to move in arbitrary directions in
operational space compromises adaptivity and, potentially, safety. We introduce
a geometry-aware singularity index, defined using a Riemannian metric on the
manifold of symmetric positive definite matrices, to provide a measure of
proximity to singular configurations. We demonstrate that our index avoids some
of the failure modes and difficulties inherent to other common indices.
Further, we show that this index can be differentiated easily, making it
compatible with local optimization approaches used for operational space
control. Our experimental results establish that, for reaching and path
following tasks, optimization based on our index outperforms a common
manipulability maximization technique and ensures singularity-robust motions.
