“Closed Form Order(n) Acceleration Equation for Rigid Multibody Systems”

Abstract:
Efficient algorithms to compute the equations of motion for mechanism are crucial in the study of control and dynamics of mechanical systems. Generally, the latter are second order differential equations in the generalized coordinates of the system based on physical principles. These are also called the acceleration equation of the system. To that end, this talk presents a derivation of a closed form O(N) acceleration equation for tree configured N-body systems. O(N) process means that the computational effort for the process is proportional to N. Standard closed form acceleration equation for such systems requires O(N3) effort to compute because of having to invert the system mass matrix. Faster O(N) recursive algorithms to compute joint accelerations have been published, but the pertinent equations are not in closed form. Rodriguez-Jain-Kreutz(RJK) showed that closed form O(N) acceleration equations for the considered systems can be obtained using their ‘spatial operator algebra’. They derived the O(N) inverse system mass matrix in factored form from the system mass matrix first and obtained the closed form O(N) acceleration equations as a consequence. In contrast, this paper derives the closed form O(N) acceleration equations from the acceleration equations used in the recursive algorithms. This derivation is accomplished using parent-child incidence matrices. The factored inverse system mass matrix and a corresponding alternate system mass matrix follow as a result.
The two incidence matrices are precisely the two principal spatial operators used by RJK. The difference is that the former are defined explicitly with arguments that are block diagonal matrices, whereas, the spatial operators are defined implicitly and their arguments are triangular matrices. The parent-child incidence matrices render the proofs involved in the O(N) acceleration equation derivation more straight-forward than previously done. Analysis presented here also shows that the derived closed form acceleration equation is faster in execution than the recursive algorithms.
Biosketch:

Mr. Tong is the Principal Director at Concurrent Dynamics International. He has been in the aerospace industry for over thirty years working on various aspects of the control systems for satellites and launch vehicles. He has been an Engineering Specialist in the Guidance and Control Subdivision of The Aerospace Corporation. He managed their real-time flight software testing laboratory. He had firsthand experience in testing the Remote Manipulator Arm control system on the Space Shuttle. Before all this, he was at the Hughes Aircraft Company specializing in target tracking software design and testing for
airborne radar systems. His research interests lie in multibody dynamics formulations, control and estimation theory. He graduated in 1974 from Case Western Reserve University