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 passivitypreservation 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.

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 performace is compared with other standard methods in nonholonomic dynamics, and their merits verified in practice.

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We consider splitting methods for the numerical solution of differential equations with controls, arising from a marine vessel rigid body model. This model is used in simulations of offshore pipe-lay operations. The system of ordinary differential equations comprises a Coriolis centripetal term, a damping term, a term for environmental forces, a term for restoring forces and moments, and a control term. We propose a splitting into the conservative part, which is nonlinear, but completely integrable (i.e. solutions of the system can be obtained by quadrature), and the damping and control parts, which are linear and can be integrated exactly. The passivity of this system is also considered.