  We present PV-OSIMr, an efficient algorithm for computing the Delassus matrix
(also known as the inverse operational space inertia matrix) for a kinematic
tree, with the lowest order computational complexity known in literature.
PV-OSIMr is derived by optimizing the Popov-Vereshchagin (PV) solver
computations using the compositionality of the force and motion propagators. It
has a computational complexity of O(n + m^2 ) compared to O(n + m^2d) of the
original PV-OSIM algorithm and O(n+md+m^2 ) of the extended force propagator
algorithm (EFPA), where n is the number of joints, m is the number of
constraints and d is the depth of the kinematic tree. Since Delassus matrix
computation requires constructing an m x m sized matrix and must consider all
the n joints at least once, the asymptotic computational complexity of PV-OSIMr
is optimal. We further benchmark our algorithm and find it to be often more
efficient than the PV-OSIM and EFPA in practice.
