TY - JOUR
T1 - A new factorization of the mass matrix for optimal serial and parallel calculation of multibody dynamics
AU - Fijany, Amir
AU - Featherstone, Roy
PY - 2013/2
Y1 - 2013/2
N2 - This paper describes a new factorization of the inverse of the joint-space inertia matrix M. In this factorization, M-1 is directly obtained as the product of a set of sparse matrices wherein, for a serial chain, only the inversion of a block-tridiagonal matrix is needed. In other words, this factorization reduces the inversion of a dense matrix to that of a block-tridiagonal one. As a result, this factorization leads to both an optimal serial and an optimal parallel algorithm, that is, a serial algorithm with a complexity of O(N) and a parallel algorithm with a time complexity of O(logN) on a computer with O(N) processors. The novel feature of this algorithm is that it first calculates the interbody forces. Once these forces are known, the accelerations are easily calculated. We discuss the extension of the algorithm to the task of calculating the forward dynamics of a kinematic tree consisting of a single main chain plus any number of short side branches. We also show that this new factorization of M -1 leads to a new factorization of the operational-space inverse inertia, Λ-1, in the form of a product involving sparse matrices. We show that this factorization can be exploited for optimal serial and parallel computation of Λ -1, that is, a serial algorithm with a complexity of O(N) and a parallel algorithm with a time complexity of O(logN) on a computer with O(N) processors.
AB - This paper describes a new factorization of the inverse of the joint-space inertia matrix M. In this factorization, M-1 is directly obtained as the product of a set of sparse matrices wherein, for a serial chain, only the inversion of a block-tridiagonal matrix is needed. In other words, this factorization reduces the inversion of a dense matrix to that of a block-tridiagonal one. As a result, this factorization leads to both an optimal serial and an optimal parallel algorithm, that is, a serial algorithm with a complexity of O(N) and a parallel algorithm with a time complexity of O(logN) on a computer with O(N) processors. The novel feature of this algorithm is that it first calculates the interbody forces. Once these forces are known, the accelerations are easily calculated. We discuss the extension of the algorithm to the task of calculating the forward dynamics of a kinematic tree consisting of a single main chain plus any number of short side branches. We also show that this new factorization of M -1 leads to a new factorization of the operational-space inverse inertia, Λ-1, in the form of a product involving sparse matrices. We show that this factorization can be exploited for optimal serial and parallel computation of Λ -1, that is, a serial algorithm with a complexity of O(N) and a parallel algorithm with a time complexity of O(logN) on a computer with O(N) processors.
KW - Multibody dynamics, Robot dynamics, Mass matrix factorization, Parallel computation
UR - http://www.scopus.com/inward/record.url?scp=84878501088&partnerID=8YFLogxK
U2 - 10.1007/s11044-012-9313-z
DO - 10.1007/s11044-012-9313-z
M3 - Article
SN - 1384-5640
VL - 29
SP - 169
EP - 187
JO - Multibody System Dynamics
JF - Multibody System Dynamics
IS - 2
ER -