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Forward dynamics of multibody systems: a recursive Hamiltonian approach

Friday, 16 September, 2005 - 15:00
Campus: Brussels Humanities, Sciences & Engineering campus
Faculty: Science and Bio-engineering Sciences
Joris Naudet
phd defence

The increase in processing power and the theoretical breakthroughs achieved in multibody
systems dynamics have improved the usefulness of dynamic simulations to such an extent that
the development of a whole range of applications has been triggered. Dynamic simulations
are used for the analysis of mechanisms, virtual prototyping, simulators, computer animation,
advanced control, etc. and are gaining in popularity. This thesis proposes a method for
formulating the equations of motion of multibody systems with the purpose of further
improving the efficiency of dynamic simulations. The proposed algorithm is recursive and is
based on a set of Hamiltonian equations.

The simulation of multibody systems comprises two essential steps: formulating the equations
of motion and solving the equations of motion. Most methods for formulating the equations of
motion of multibody systems are based on accelerations: whether the Newton-Euler
equations, the Lagrangian equations or the principle of virtual work or virtual power are used,
these methods provide a set of second order differential equations and the simulation
algorithms come down to calculating and integrating accelerations.

A set of Hamiltonian equations on the other hand does not contain accelerations, but is
expressed in terms of canonical momenta and their time derivatives instead, resulting in twice
as many first order differential equations. The motivation for using Hamiltonian equations can
be found in the numerical integration of constrained multibody systems. Adding algebraic
constraints to the differential equations results in a set of differential-algebraic equations with
differential index 2, while acceleration-based algorithms typically have index 3. It is a wellestablished
fact that index 3 DAEs are more difficult to solve than systems with a lower

The method proposed to obtain a Hamiltonian set of the equations of motion of a multibody
system results in a recursive O(n) algorithm. It is based on the well-known articulated body
method and introduces the concept of articulated momentum vectors. Closed-loop systems are
handled by performing a coordinate reduction to obtain a set of independent coordinates. The
resulting algorithm allows to compete with O(n) acceleration-based algorithms, while
exploiting the benefits of the lower differential index. It even outperforms the comparable
acceleration-based methods for formulating the equations of motion: counting the number of
arithmetical operations needed to establish the equations of motion reveals that the proposed
algorithm requires less computations than other acceleration based methods described in the