Distributed Operational Space Formulation of Serial Manipulators

This paper presents a new parallel algorithm for the operational space dynamics of unconstrained serial manipulators, which outperforms contemporary sequential and parallel algorithms in the presence of two or more processors. The method employs a hybrid divide and conquer algorithm (DCA) multibody methodology which brings together the best features of the DCA and fast sequential techniques. The method achieves a logarithmic time complexity (⁠$O(log(n)$⁠) in the number of degrees of freedom (⁠$n$⁠) for computing the operational space inertia (⁠$Λe$⁠) of a serial manipulator in presence of $O(n)$ processors. The paper also addresses the efficient sequential and parallel computation of the dynamically consistent generalized inverse (⁠$J¯e$⁠) of the task Jacobian, the associated null space projection matrix (⁠$Ne$⁠), and the joint actuator forces (⁠$τnull$⁠) which only affect the manipulator posture. The sequential algorithms for computing $J¯e$⁠, $Ne$⁠, and $τnull$ are of $O(n)$⁠, $O(n2)$⁠, and $O(n)$ computational complexity, respectively, while the corresponding parallel algorithms are of $O(log(n))$⁠, $O(n)$⁠, and $O(log(n))$ time complexity in the presence of $O(n)$ processors.