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.

翻译：本文提出PV-OSIMr，一种用于计算运动树Delassus矩阵（亦称逆操作空间惯性矩阵）的高效算法，其计算复杂度为文献已知最低。该算法通过利用力传播器与运动传播器的组合性优化Popov-Vereshchagin（PV）求解器计算而导出。其计算复杂度为O(n + m²)，而原始PV-OSIM算法为O(n + m²d)，扩展力传播器算法（EFPA）为O(n + md + m²)；其中n为关节数，m为约束数，d为运动树深度。由于Delassus矩阵计算需要构建一个m×m维矩阵，且必须至少遍历所有n个关节一次，PV-OSIMr的渐近计算复杂度是最优的。我们进一步对算法进行基准测试，发现其在实践中通常比PV-OSIM和EFPA更为高效。