Zdravko Terze

Forward Dynamics of Open-Loop Multibody Mechanisms Using an Efficient Recursive Algorithm Based on Canonical Momenta

A new method for establishing the equations of motion of multibodymechanisms based on canonical momenta is introduced in this paper.In absence of constraints, the proposed forward dynamicsformulation results in a Hamiltonian set of 2n first order ODEsin the generalized coordinates q and the canonical momenta p.These Hamiltonian equations are derived from a recursiveNewton–Euler formulation. As an example, it is shown how, in thecase of a serial structure with rotational joints, an O(n)formulation is obtained. The amount of arithmetical operations isconsiderably less than acceleration based O(n) formulations.

Null Space Integration Method for Constrained Multibody Systems with No Constraint Violation

A method for integrating equations of motion of constrained multibodysystems with no constraint violation is presented. A mathematical model,shaped as a differential-algebraic system of index 1, is transformedinto a system of ordinary differential equations using the null-spaceprojection method. Equations of motion are set in a non-minimal form.During integration, violations of constraints are corrected by solvingconstraint equations at the position and velocity level, utilising themetric of the systems configuration space, and projective criterion to thecoordinate partitioning method. The method is applied to dynamicsimulation of 3D constrained biomechanical system. The simulation resultsare evaluated by comparing them to the values of characteristicparameters obtained by kinematic analysis of analyzed motion based onmeasured kinematic data.

On the choice of configuration space for numerical Lie group integration of constrained rigid body systems

Standard numerical integration schemes for multibody system (MBS) models in absolute coordinates neglect the coupling of linear and angular motions since finite positions and rotations are updated independently. As a consequence geometric constraints are violated, and the accuracy of the constraint satisfaction depends on the integrator step size. It is discussed in this paper that in certain cases perfect constraint satisfaction is possible when using an appropriate configuration space (without numerical constraint stabilization). Two formulations are considered, one where R^3 is used as rigid body configuration space and another one where rigid body motions are properly modeled by the semidirect product SE(3)=SO(3)@?R^3. MBS motions evolve on a Lie group and their dynamics is naturally described by differential equations on that Lie group. In this paper the implications of using the two representations on the constraint satisfaction within Munthe-Kaas integration schemes are investigated. It is concluded that the SE(3) update yields perfect constraint satisfaction for bodies constrained to a motion subgroup of SE(3), and in the general case both formulations lead to equivalent constraint satisfaction.

An Angular Momentum and Energy Conserving Lie-Group Integration Scheme for Rigid Body Rotational Dynamics Originating From Störmer–Verlet Algorithm

The paper presents two novel second order conservative Lie-group geometric methods for integration of rigid body rotational dynamics. First proposed algorithm is a fully explicit scheme that exactly conserves spatial angular momentum of a free spinning body. The method is inspired by the Stormer–Verlet integration algorithm for solving ordinary differential equations (ODEs), which is also momentum conservative when dealing with ODEs in linear spaces but loses its conservative properties in a nonlinear regime, such as nonlinear SO(3) rotational group. Then, we proposed an algorithm that is an implicit integration scheme with a direct update in SO(3). The method is algorithmically designed to conserve exactly both of the two “main” motion integrals of a rotational rigid body, i.e., spatial angular momentum of a torque-free body as well as its kinetic energy. As it is shown in the paper, both methods also preserve Lagrangian top integrals of motion in a very good manner, and generally better than some of the most successful conservative schemes to which the proposed methods were compared within the presented numerical examples. The proposed schemes can be easily applied within the integration algorithms of the dynamics of general rigid body systems.

A Constraint Stabilization Method for Time Integration of Constrained Multibody Systems in Lie Group Setting

The stabilization of geometric constraints is vital for an accurate numerical solution of the differential-algebraic equations (DAE) governing the dynamics of constrained multibody systems (MBS). Although this has been a central topic in numerical MBS dynamics using classical vector space formulations, it has not yet been sufficiently addressed when using Lie group formulations. A straightforward approach is to impose constraints directly on the Lie group elements that represent the MBS motion, which requires additional constraints accounting for the invariants of the Lie group. On the other hand, most numerical Lie group integration schemes introduce local coordinates within the integration step, and it is natural to perform the stabilization in terms of these local coordinates. Such a formulation is presented in this paper for index 1 formulation. The stabilization method is applicable to general coordinate mappings (canonical coordinates, Cayley-Rodriguez, Study) on the MBS configuration space Lie group. The stabilization scheme resembles the well-known vectors space projection and pseudo-inverse method consisting in an iterative procedure. A numerical example is presented and it is shown that the Lie group stabilization scheme converges normally within one iteration step, like the scheme in the vector space formulation.Copyright

CONSTRAINT GRADIENT PROJECTIVE METHOD FOR STABILIZED DYNAMIC SIMULATION OF CONSTRAINED MULTIBODY SYSTEMS

Constraint gradient projective method for stabilization of constraint violation during integration of constrained multibody systems is in the focus of the paper. Different mathematical models for constrained MBS dynamic simulation on manifolds are surveyed and violation of kinematical constraints is discussed. As an extension of the previous work focused on the integration procedures of the holonomic systems, the constraint gradient projective method for generally constrained mechanical systems is discussed. By adopting differentialgeometric point of view, the geometric and stabilization issues of the method are addressed. It is shown that the method can be applied for stabilization of holonomic and non-holonomic constraints in Pfaffian and general form.Copyright

Composed Fluid–Structure Interaction Interface for Horizontal Axis Wind Turbine Rotor

In this paper we propose a technique for high-fidelity Fluid-Structure Interaction (FSI) spatial interface reconstruction, of a Horizontal Axis Wind Turbine (HAWT) rotor model composed of an elastic blade mounted on a rigid hub. The technique is aimed at enabling re-usage of existing blade Finite Element Method (FEM) models, now with high-fidelity fluid sub-domain methods relying on boundary-fitted mesh. The technique is based on the Partition of Unity method and it enables fluid sub-domain FSI interface mesh of different components to be smoothly connected. In the paper we use it to connect a beam FEM model to a rigid body, but the proposed technique is by no means restricted to any specific choice of numerical models for the structure components, or methods of their surface recoveries. To stress-test robustness of the connection technique we recover elastic blade surface from collinear mesh, and remark on repercussions of such a choice. For the HAWT blade recovery method itself we use Generalised Hermite Radial Basis Function Interpolation which utilises the interpolation of small rotations in addition to displacement data. Finally, for the composed structure we discuss consistent and conservative approaches to FSI spatial interface formulations.

Is there an optimal choice of configuration space for lie group integration schemes applied to constrained MBS

Recently various numerical integration schemes have been proposed for numerically simulating the dynamics of constrained multibody systems (MBS) operating. These integration schemes operate directly on the MBS configuration space considered as a Lie group. For discrete spatial mechanical systems there are two Lie group that can be used as configuration space: SE(3) and SO(3) × R 3 . Since the performance of the numerical integration scheme clearly depends on the underlying configuration space it is important to analyze the effect of using either variant. For constrained MBS a crucial aspect is the constraint satisfaction. In this paper the constraint violation observed for the two variants are investigated. It is concluded that the SE(3) formulation outperforms the SO(3) × R 3 formulation if the absolute motions of the rigid bodies, as part of a constrained MBS, belong to a motion subgroup. In all other cases both formulations are equivalent. In the latter cases the SO(3) × R 3 formulation should be used since the SE(3) formulation is numerically more complex, however.

Lie Group Forward Dynamics of Fixed-Wing Aircraft With Singularity-Free Attitude Reconstruction on SO(3)

This paper proposes an approach to formulation and integration of the governing equations for aircraft flight simulation that is based on a Lie group setting, and leads to a nonsingular coordinate-free numerical integration. Dynamical model of an aircraft is formulated in Lie group state space form and integrated by ordinary-differential-equation (ODE)-on-Lie groups Munthe-Kaas (MK) type of integrator. By following such an approach, it is assured that kinematic singularities, which are unavoidable if a three-angles-based rotation parameterization is applied for the whole 3D rotation domain, do not occur in the proposed noncoordinate formulation form. Moreover, in contrast to the quaternion rotation parameterization that imposes additional algebraic constraint and leads to integration of differential-algebraic equations (DAEs) (with necessary algebraic-equation-violation stabilization step), the proposed formulation leads to a nonredundant ODE integration in minimal form. To this end, this approach combines benefits of both traditional approaches to aircraft simulation (i.e., three angles parameterization and quaternions), while at the same time it avoids related drawbacks of the classical models. Besides solving kinematic singularity problem without introducing DAEs, the proposed formulation also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation and when aircraft motion pattern comprises steady rotational component of its 3D motion. This is due to the fact that a Lie group setting and applied MK integrator determine vehicle orientation on the basis of integration of local (tangent, nonlinear) kinematical differential equations (KDEs) that model process of 3D rotations (i.e., vehicle attitude reconstruction on nonlinear manifold SO(3)) more accurately than “global” KDEs of the classical formulations (that are linear in differential equations part in the case of standard quaternion models).

Modelling and Integration Concepts of Multibody Systems on Lie Groups

Lie group integration schemes for multibody systems (MBS) are attractive as they provide a coordinate-free and thus singularity-free approach to MBS modeling. The Lie group setting also allows for developing integration schemes that preserve motion integrals and coadjoint orbits. Most of the recently proposed Lie group integration schemes are based on variants of generalized alpha Newmark schemes. In this chapter constrained MBS are modeled by a system of differential-algebraic equations (DAE) on a configuration space being a subvariety of the Lie group \(SE(3)^{n}\). This is transformed to an index 1 DAE system that is integrated with Munthe-Kaas (MK) integration scheme. The chapter further addresses geometric integration schemes that preserve integrals of motion. In this context, a non-canonical Lie-group Stormer-Verlet integration scheme with direct \(SO(3)\) rotational update is presented. The method is 2nd order accurate and it is angular momentum preserving for a free-spinning body. Moreover, although being fully explicit, the method achieves excellent conservation of the angular momentum of a free rotational body and the motion integrals of the Lagrangian top. A higher-order coadjoint-preserving integration scheme on \(SO(3)\) is also presented. This method exactly preserves spatial angular momentum of a free body and it is particularly numerically efficient.