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：我们提出PV-OSIMr算法，这是一种用于计算运动链德拉苏斯矩阵（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中已知的最低阶。PV-OSIMr通过利用力和运动传播子的组合性优化波波夫-维列沙金（PV）求解器计算而推导得出。其计算复杂度为O(n + m²)，相较于原始PV-OSIM算法的O(n + m²d)和扩展力传播子算法（EFPA）的O(n+md+m²)，其中n为关节数，m为约束数，d为运动链深度。由于德拉苏斯矩阵计算需构建m×m规模的矩阵并至少遍历所有n个关节一次，PV-OSIMr的渐近计算复杂度已达到最优。我们进一步对该算法进行基准测试，发现在实际应用中其效率通常优于PV-OSIM和EFPA。