Passivity-preserving splitting methods for rigid body systems

A rigid body model for the dynamics of a marine vessel, used in simulations of offshore pipe-lay operations, gives rise to a set of ordinary differential equations with controls. The system is input-output passive. We propose passivity-preserving splitting methods for the numerical solution of a class of problems which includes this system as a special case. We prove the passivity-preservation property for the splitting methods, and we investigate stability and energy behaviour in numerical experiments. Implementation is discussed in detail for a special case where the splitting gives rise to the subsequent integration of two completely integrable flows. The equations for the attitude are reformulated on $SO(3)$ using rotation matrices rather than local parametrizations with Euler angles.Comment: 27 pages, 4 figures. To be published in 'Multibody System Dynamics

Passivity-preserving splitting methods for rigid body systems

A rigid body model for the dynamics of a marine vessel, used in simulations of offshore pipe-lay operations, gives rise to a set of ordinary differential equations with controls. The system is input–output passive. We propose passivity-preserving splitting methods for the numerical solution of a class of problems which includes this system as a special case. We prove the passivity-preservation property for the splitting methods, and we investigate stability and energy behaviour in numerical experiments. Implementation is discussed in detail for a special case where the splitting gives rise to the subsequent integration of two completely integrable flows. The equations for the attitude are reformulated on SO(3) using rotation matrices rather than local parameterisations with Euler angles

Energy-preserving Integrators applied to nonholonomic systems

We introduce energy-preserving integrators for nonholonomic mechanical systems. We will see that the nonholonomic dynamics is completely determined by a triple (D, Π, H) , where D is the dual of the vector bundle determined by the nonholonomic constraints, Π is an almost-Poisson bracket (the nonholonomic bracket) and H: D→ R is a Hamiltonian function. For this triple, we can apply energy-preserving integrators, in particular, we show that discrete gradients can be used in the numerical integration of nonholonomic dynamics. By construction, we achieve preservation of the constraints and of the energy of the nonholonomic system. Moreover, to facilitate their applicability to complex systems which cannot be easily transformed into the aforementioned almost-Poisson form, we rewrite our integrators using just the initial information of the nonholonomic system. The derived procedures are tested on several examples: a chaotic quartic nonholonomic mechanical system, the Chaplygin sleigh system, the Suslov problem and a continuous gearbox driven by an asymmetric pendulum. Their performance is compared with other standard methods in nonholonomic dynamics, and their merits verified in practice.This work has been partially supported by MINECO (Spain) MTM2013-42870-P, MTM2015-69124-REDT, the ICMAT SeveroOchoa project SEV-2015-0554 and the Nils-Abel project 010-ABEL-CM-2014ANILS. It has also received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 691070. M. Farré has been financially supported by a FPU scholarship from MECD