Search

Insect flight research is propelled by their unmatched flight capabilities. However, complex underlying aerodynamic phenomena make computational modeling of insect-type flapping flight a challenging task, limiting our ability in understanding insect flight and producing aerial vehicles exploiting same aerodynamic phenomena. To this end, novel mid-fidelity approach to modeling insect-type flapping vehicles is proposed. The approach is computationally efficient enough to be used within optimal design and optimal control loops, while not requiring experimental data for fitting model parameters, as opposed to widely used quasi-steady aerodynamic models. The proposed algorithm is based on Helmholtz–Hodge decomposition of fluid velocity into curl-free and divergence-free parts. Curl-free flow is used to accurately model added inertia effects (in almost exact manner), while expressing system dynamics by using wing variables only, after employing symplectic reduction of the coupled wing-fluid system at zero level of vorticity (thus reducing out fluid variables in the process). To this end, all terms in the coupled body-fluid system equations of motion are taken into account, including often neglected terms related to the changing nature of the added inertia matrix (opposed to the constant nature of rigid body mass and inertia matrix). On the other hand—in order to model flapping wing system vorticity effects— divergence-free part of the flow is modeled by a wake of point vortices shed from both leading (characteristic for insect flight) and trailing wing edges. The approach is evaluated for a numerical case involving fruit fly hovering, while quasi-steady aerodynamic model is used as benchmark tool with experimentally validated parameters for the selected test case. The results indicate that the proposed approach is capable of mid-fidelity accurate calculation of aerodynamic loads on the insect-type flapping wings.

Three recently approved space missions are headed towards Venus, to help answer major questions about Venus atmosphere and geology. However, many existing questions cannot be properly addressed without direct in situ measurements from Venus surface or within the atmosphere. To this end, flapping wing vehicle concept is selected, optimized for Venus atmospheric flight, and evaluated using energy efficiency as performance criteria. Flapping wing vehicle computational model is derived based on discrete variational mechanics and quasi-steady aerodynamics, with all relevant aerodynamic phenomena included. Flapping wing vehicle computational model is then embedded within optimization algorithm, which is utilized to obtain energy efficient flapping patterns for forward flight in Venus atmospheric surface conditions. Numerical optimization is performed for different neutrally buoyant configurations, with wingspan ranging from 10 mm to 1 m. Different forward velocities are used as well, where maximum velocity is limited by an advance ratio of 0.5. Bumblebee and hummingbird-sized vehicles, with a wingspan of 30 mm and 30 cm, are selected as the most representative test cases and thoroughly studied. It is proved that flapping wing propulsion is a feasible and effective concept for Venus exploration purposes. Finally, based on a comparison of the selected test cases general conclusions are drawn on the flapping wing dynamics and flight mechanics strategies in the Venus atmosphere. Significant difference in the propulsion mechanism has been observed, based on the aerial vehicle size. In order to maximize propulsive efficiency, the smaller vehicle mostly exploited aerodynamic forces related to the leading edge vortex, while the larger vehicle relied more on added mass and rotational forces.

Outstanding aerial capabilities that insects present in nature inspire researchers to undertake a challenge to develop a flapping aerial vehicle with performances unmatched by any manmade object. However, the complex aerodynamic phenomena crucial for the insect flight are not easily understood, let alone modeled and utilized for flight. The researchers managed to develop a quasi-steady aerodynamic model capable of capturing the most important aspects of a fruit fly-like insect flight, while still being efficient enough to allow for the usage in the flapping mechanism optimization loop. This experimentally justified quasi-steady model is used in the paper as a building block for creating a novel optimization algorithm, based on the discrete mechanics and optimal control framework. When compared to the conventional approaches to design optimization, this framework includes the natural description of the energy cost function, while incorporating the physical laws in the form of a discrete Lagrange–d’ Alembert equations inherently in optimization constraints. This leads to the discrete description of the inherently continuous problem, allowing the algorithm to search for the optimal solutions in the whole domain. In other words, in contrast to the conventional approaches involving the assumption on the function family and subsequent optimization on the parameters of that function type, this approach is not constrained by the user input and is capable of yielding any solution that respects the physical laws. As presented by the numerical test cases, optimizing the flapping patterns of a fruit fly-like aerial vehicle in standstill hovering leads to both effective and robust optimization tool.

In order to study dynamics of multibody system (MBS) moving in ambient fluid, we adopt geometric modeling approach of fully coupled MBS-fluid system, incorporating boundary integral method and time integrator in Lie group setting. By assuming inviscid and incompressible fluid, the configuration space of the MBS-fluid system is reduced by eliminating fluid variables via symplectic reduction without compromising any accuracy. Consequently, the equations of motion for the submerged MBS are formulated without explicitly incorporating fluid variables, while effect of the fluid flow to MBS overall dynamics is accounted for by ‘added mass’ effect to the submerged bodies. In such approach, the ‘added masses’ are expressed as boundary integral functions of the fluid density and the flow velocity potential. In order to take into account additional viscous effects and include fluid vorticity and circulation in the system dynamics, vortex shedding and evolution mechanism is incorporated in the overall model by unsteady potential flow method, enforcing Kutta conditions on MBS sharp edges. In summary, presented approach exhibits significant computational advantages in comparison to the standard numerical procedures that - most commonly - comprise finite volume discretization of the whole fluid domain and (loosely coupled) solving fluid and MBS dynamics on different meshes.

In order to study the dynamics of a multibody system (MBS) moving in ambient fluid, the geometric modeling approach of a fully coupled multibody-fluid system is adopted, incorporating the boundary integral method and time integrator in Lie group setting. The configuration space of the multibody-fluid system is reduced by eliminating fluid variables via symplectic reduction without compromising any accuracy, if the fluid is assumed to be inviscid and incompressible. Consequently, the equations of motion for the submerged MBS are formulated without explicitly incorporating fluid variables, while the effect of the fluid flow on overall MBS dynamics is accounted for by added mass effect on the submerged bodies. By following this approach, the added masses can be computed by the boundary integral functions of the fluid density and the flow velocity potential. Vortex shedding and evolution mechanism is incorporated in the approach, to describe additional viscous effects and include fluid vorticity and circulation in the system dynamics. For vortex modeling, the unsteady potential flow method is utilized, enforcing the Kutta condition on sharp edges of the MBS. In summary, the presented approach exhibits significant computational advantages in comparison to the standard numerical procedures that — most commonly — comprise finite volume discretization of the whole fluid domain and (loosely coupled) separate solvers for fluid and MBS dynamics. The model implementation is demonstrated on the example of the three-body multibody chain.

Unit quaternion representation is widely used in flight simulation to overcome the limitations of the standard numerical ordinary-differential-equations (ODEs) based on three-parameters rotation variables (such as Euler angels), as they may impose kinematic singularities during aircraft's attitude reconstruction. However, these benefits do not come without a price, since the classical way of integrating rotational quaternions includes solving of differential-algebraic equations (DAEs) that requires post-integration numerical stabilization of the additional algebraic constraint enforcing the quaternion unitary norm. This can pose a problem in the case of longer flight simulations since improper numerical treatment of the quaternion-normalization constraint may induce numerical drift into the simulation results. As a remedy, the proposed novel algorithm circumvents DAE problem of quaternion integration by shifting update-integration-process from configuration manifold to the local tangential level of the incremental rotations (reducing thus integration to standard three ODEs problem at tangential Lie algebra level). This can be done due to the isomorphism of the Lie algebras of the rotational SO ( 3 ) group and the configuration manifold unit quaternion Sp ( 1 ) group. Besides avoiding DAE formulation by reducing integration process to standard three ODEs problem, the proposed algorithm also exhibits numerical advantages as it is discussed in the presented example.

Mars exploration is currently in the focus of scientific community interests. The attempts for efficient exploration will probably include the unmanned aerial vehicle in the near future to explore the places inaccessible by rovers. Since the Mars atmospheric conditions render the conventional rotary and fixed wing aerial vehicles inefficient, the insect-type flapping concept emerged as a promising solution. This is due to the fact that insects on Earth fly efficiently at the same values of Reynolds number that the aerial vehicle for Mars exploration would exhibit. The paper proposes the novel design optimization algorithm for development of insect-type aerial vehicle capable of flight in Martian atmosphere. The optimization procedure utilizes the novel flapping pattern optimization based on a quasi-steady aerodynamic model, combined with the discrete mechanics and optimal control framework. A test case of a flapping vehicle with fruit fly-like wings, performing a standstill hovering on Mars, is analyzed in detail. The fruit fly wing is scaled with a wide range of uniform scaling factors and optimized for hovering on Mars in the conditions of bioinspired Reynolds number range. Apart from the single optimal combination for the standstill hovering with fruit fly-like wings, the algorithm also found different efficient flapping patterns for a wide range of scaling factors, providing directions for design of flapping aerial vehicles for Mars.

This paper describes a computationally efficient method for simulating dynamics of the coupled multibody-fluid system that utilizes symplectic and Lie-Poisson reductions in order to formulate fully coupled dynamical model of the multi-physical system by using solid variables only. The multibody system (MBS) dynamics is formulated in Lie group setting and integrated with the pertinent Lie group integration method that operates in MBS state space. The effects of fluid flow on MBS dynamics are accounted for by the added masses to the submerged bodies, calculated by boundary element method. The case study of coupled dynamics of three rigid ellipsoid (blunt) bodies in fluid flow without circulation is presented. In order to take into account additional viscous effects and include fluid vorticity and circulation in the system dynamics (when motion of the kinematical chain with sharp edges is considered), vortex shedding mechanism is incorporated in the overall model by numerically enforcing Kutta condition.

A novel ODE time stepping scheme for solving rotational kinematics in terms of unit quaternions is presented in the paper. This scheme inherently respects the unit-length condition without including it explicitly as a constraint equation, as it is common practice. In the standard algorithms, the unit-length condition is included as an additional equation leading to kinematical equations in the form of a system of differential-algebraic equations (DAEs). On the contrary, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form a system of ordinary differential equations (ODEs) on the Lie algebra $\mathit{so}(3)$ of the rotation group $\mathit{SO}(3)$ . This rotation vector defines an incremental rotation (and thus the associated incremental unit quaternion), and the rotation update is determined by the exponential mapping on the quaternion group. Since the kinematic ODE on $\mathit{so}(3)$ can be solved by using any standard (possibly higher-order) ODE integration scheme, the proposed method yields a non-redundant integration algorithm for the rotational kinematics in terms of unit quaternions, avoiding integration of DAE equations. Besides being ‘more elegant’—in the opinion of the authors—this integration procedure also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation. As presented in the paper, the numerical integration of three non-linear ODEs in terms of the rotation vector as canonical coordinates achieves a higher accuracy compared to integrating the four (linear in ODE part) standard-quaternion DAE system. In summary, this paper solves the long-standing problem of the necessity of imposing the unit-length constraint equation during integration of quaternions, i.e. the need to deal with DAE’s in the context of such kinematical model, which has been a major drawback of using quaternions, and a numerical scheme is presented that also allows for longer integration steps during kinematic reconstruction of large three-dimensional rotations.

Coordinate-free Lie-group integration method of arbitrary (and possibly higher) order of accuracy for constrained multibody systems (MBS) is proposed in the paper. Mathematical model of MBS dynamics is shaped as a DAE system of equations of index 1, whereas dynamics is evolving on the system state space modeled as a Lie-group. Since the formulated integration algorithm operates directly on the system manifold via MBS elements’ angular velocities and rotational matrices, no local rotational coordinates are necessary, and kinematical differential equations (that are prone to singularities in the case of three-parameter-based local description of the rotational kinematics) are completely avoided. Basis of the integration procedure is the Munthe–Kaas algorithm for ODE integration on Lie-groups, which is reformulated and expanded to be applicable for the integration of constrained MBS in the DAE-index-1 form. In order to eliminate numerical constraint violation for generalized positions and velocities during the integration procedure, a constraint stabilization projection method based on constrained least-square minimization algorithm is introduced. Two numerical examples, heavy top dynamics and satellite with mounted 5-DOF manipulator, are presented. The proposed Lie-group DAE-index-1 integration scheme is easy-to-use for an MBS with kinematical constraints of general type, and it is especially suitable for dynamics of mechanical systems with large 3D rotations where standard (vector space) formulations might be inefficient due to kinematical singularities (three-parameter-based rotational coordinates) or additional kinematical constraints (redundant quaternion formulations).

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.

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).

Quaternions are favorable parameters to describe spatial rotations of rigid bodies since they give rise to simple equations governing the kinematics and attitude dynamics in terms of simple algebraic equations. Dual quaternions are the natural extension to rigid body motions. They provide a singularity-free purely algebraic parameterization of rigid body motions, and thus serve as global parameters within the so-called absolute coordinate formulation of MBS. This attractive feature is owed to the inherent redundancy of these parameters since they must satisfy two quadratic conditions (unit condition and Plcker condition). Formulating the MBS kinematics in terms of dual quaternions leads to a system of differential-algebraic equations (DAE) with index 3. This is commonly transformed to an index 1 DAE system by replacing the algebraic constraints with their time derivative. This leads to the well-known problem of constraint violation. A brute force method, enforcing the unit constraint of quaternions, is to normalize them after each integration step. Clearly this correction affects the overall solution and the dynamic consistency. Moreover, for unit dual quaternions the two conditions cannot simply be enforced in such a way. In this paper a non-redundant formulation of the motion equations in terms of dual quaternions is presented. The dual quaternion constraints are avoided by introducing a local canonical parameterization. The key to this formulation is to treat dual unit quaternions as Lie group. The formulation can be solved with any standard integration scheme. Examples are reported displaying the excellent performance of this formulation regarding the constraint satisfaction as well as the solution accuracy.

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 subdomain methods relying on boundary-fitted mesh. The technique is based on the partition of unity (PU) method and it enables fluid subdomain FSI interface mesh of different components to be smoothly connected. In this 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 generalized Hermite radial basis function interpolation (GHRBFI) which utilizes 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.

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 Störmer–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.

The influence of factors which can lead to incorrect prediction of dynamical stability of the periodic response of oscillators which contain a non-linear restoring characteristic with discontinuous or steep first derivative is considered in this paper. For that purpose, a simple one degree-of-freedom system with a piecewise-linear force-displacement relationship subjected to a harmonic excitation is analysed. Stability of the periodic response obtained in the frequency domain by the incremental harmonic balance method is determined by using the Floquet–Liapounov theorem. Responses in the time domain are obtained by digital simulation. The accuracy of determining the eigenvalues of the monodromy matrix (in the considered example) significantly depend on the corrective vector norm ‖ { r } ‖ , the accuracy ɛ of numerical determination of the times when the system undergoes a stiffness change, and on the number of step functions M (used in the Hsu's procedure), only for ‖ { r } ‖ > 1 × 10 − 5 , ɛ > 1 × 10 − 5 and M 2000 . Otherwise, except if the maximum modulus of the eigenvalues of the monodromy matrix is very close to unity, their influence on estimation of dynamical stability is minor. On the contrary, neglecting very small harmonic terms of the actual time domain response can cause a very large error in the evaluation of the eigenvalues of the monodromy matrix, and so they can lead to incorrect prediction of the dynamical stability of the solution, regardless of whether the maximum modulus of the eigenvalues of the monodromy matrix is close to unity or not.

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.

Differential-geometric characteristics and structure of optimized generalized coordinates partitioned vectors for generally constrained multibody systems are discussed. Generalized coordinates partitioning is well-known procedure that can be applied in the framework of numerical integration of DAE systems. However, although the procedure proves to be a very useful tool, it is known that an optimization algorithm for coordinates partitioning is needed to obtain the best performance. After short presentation of differential-geometric background of optimized coordinates partitioning, the structure of optimally partitioned vectors is discussed on the basis of gradient analysis of separate constraint submanifolds at configuration and velocity level when holonomic and non-holonomic constraints are present in the system. While, in the case of holonomic systems, the vectors of optimally partitioned coordinates have the same structure for generalized positions and velocities, when non-holonomic constraints are present in the system, the optimally partitioned coordinates generally differ at configuration and velocity level and separate partitioned procedure has to be applied. The conclusions of the paper are illustrated within the framework of the presented numerical example.Copyright © 2009 by ASME

Redundancy-free computational procedure for solving dynamics of rigid body by using quaternions as the rotational kinematic parameters will be presented in the paper. On the contrary to the standard algorithm that is based on redundant DAE-formulation of rotational dynamics of rigid body that includes algebraic equation of quaternions’ unit-length that has to be solved during marching-in-time, the proposed method will be based on the integration of a local rotational vector in the minimal form at the Lie-algebra level of the SO(3) rotational group during every integration step. After local rotational vector for the current step is determined by using standard (possibly higher-order) integration ODE routine, the rotational integration point is projected to Sp(1) quaternion-group via pertinent exponential map. The result of the procedure is redundancy-free integration algorithm for rigid body rotational motion based on the rotational quaternions that allows for straightforward minimal-form-ODE integration of the rotational dynamics.

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 © 2014 by ASME

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.

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 system's 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.

During numerical forward dynamics of discrete mechanical systems with constraints, a numerical violation of system kinematical constraints is the basic source of time-integration errors and frequent difficulty that analyst has to cope with. The stabilized time-integration procedure, whose stabilization step is based on projection of the integration results to the underlying constraint manifold via post-integration correction of the selected coordinates, is proposed in the paper. After discussing optimization of the partitioning algorithm, the geometric and stabilization issues of the method are addressed and it is shown that the projective stabilization algorithm can be applied for numerical stabilization of holonomic and non-holonomic constraints in Pfaffian and general form. As a continuation of the previous work, a further elaboration of the projective stabilization method applied on non-holonomic discrete mechanical systems is reported in the paper and numerical example is provided.Copyright © 2007 by ASME

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) × ℝ3. 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) × ℝ3 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) × ℝ3 formulation should be used since the SE(3) formulation is numerically more complex, however.

Stormer-Verlet integration scheme has many attractive properties when dealing with ODE systems in linear spaces: it is explicit, 2nd order, linear/angular momentum preserving and it is symplectic for Hamiltonian systems. In this paper we investigate its application for numerical simulation of the multibody system dynamics (MBS) by formulating StormerVerlet algorithm for the constrained mechanical systems with the direct rotation group SO(3) upgrade in Lie-group setting. Starting from the investigations on the single rigid body rotational dynamics, the paper introduces modified RATTLE integration scheme with the SO(3) rotational upgrade that is designed via exponential map and utilization of the rotation group Lie-algebra so(3), which is determined from the canonical coordinate of Hamiltonian system during integration of the system dynamics. CONFIGURATION SPACE AND BASIC FORMULATION In the adopted approach, the configuration space of MBS comprising k bodies is modeled as a Lie-group

The paper surveys geometric mathematical framework for computational modeling of multibody system dynamics. Starting with the configuration space of rigid body motion and analysis of it’s Lie group structure, the elements of respective Lie algebra are addressed and basic relations pertinent to geometrical formulations of multibody system dynamics are surveyed. Dynamical model of multibody system on manifold introduced, along with the outline of geometric characteristics of holonomic and non‐holonomic kinematical constraints.

The generalized coordinates partitioning is a well-known procedure that can be applied in the framework of a numerical integration of the DAE systems. However, although the procedure proves to be a very useful tool, it is known that an optimization algorithm for the coordinates partitioning is needed to obtain the best performance. In the paper, the optimized partitioning of the generalized coordinates is revisited in the context of a numerical forward dynamics of the holonomic and non-holonomic multibody systems. After a short presentation of the geometric background of the optimized coordinates partitioning, a structure of the optimally partitioned vectors is discussed on the basis of a gradient analysis of the separate constraint sub-manifolds at the configuration and the velocity levels when holonomic and non-holonomic constraints are present in the system. It is shown that, for holonomic systems, the vectors of optimally partitioned coordinates have the same structure for the generalized positions and velocities. On the contrary, in the case of non-holonomic systems, the optimally partitioned coordinates generally differ at the configuration and the velocity levels. The conclusions of the paper are illustrated within the framework of the presented numerical example.

A formulation for the kinematics of multibody systems is presented, which uses Lie group concepts. With line coordinates the kinematics is parameterized in terms of the screw coordinates of the joints. Thereupon, the Lagrangian motion equations are derived, and explicit expressions are given for the objects therein. It is shown how the kinematics and thus the motion equations can be expressed without the introduction of body-fixed reference frames. This admits the processing of CAD data, which refers to a single (world) frame. For constrained multibody systems, the Lagrangian motion equations are projected to the constraint manifold, which yields the equations of Woronetz. The mathematical models for numerical integration routines of MBS are surveyed and constraint gradient projective method for stabilization of constraint violation is presented., U radu je prikazano matematičko modeliranje kinematike diskretnih mehaničkih sustava s kinematičkim vezama pomoću Lievih grupa. Kinematika sustava parametarizirana je koristeći vijčane koordinate zglobova kinematičkog lanca. Nastavljajući se na takav kinematički model, Lagrangeove dinamičke jednadžbe gibanja sustava izvedene su u nastavku rada. Koristeći takav pristup, pokazano je kako se kinematički model, a također i dinamičke jednadžbe gibanja mehaničkog sustava, mogu izvesti bez upotrebe lokalnih koordinatnih sustava vezanih za pojedina tijela kinematičkog lanca. Takvo matematičko modeliranje omogućava izravnu upotrebu CAD podataka koji se, u pravilu, izražavaju u jedinstvenom koordinantnom sustavu. U slučaju dodatnih kinematičkih ograničenja narinutih na sustav, jednadžbe gibanja izvedene su projiciranjem Lagrangeovih jednadžbi na višestrukost ograničenja, čime se model izražava u obliku jednadžbi Woronetza. U radu su, nadalje, prikazane formulacije matematičkih modela koji se koriste kao podloga numeričkih algoritama za vremensko integriranje jednadžbi dinamike, a također je, uz izrađeni numerički primjer, opisana i metoda stabilizacije numeričkih rješenja na višestrukosti ograničenja.

Numerical simulation procedures for landing dynamics of large transport aircraft are briefly presented. Developed numerical procedures allow for determination of dynamic response of landing aircraft for different flight and touch-down parameters. A non-linear dynamic model of landing aircraft, which serves as a basis for computational procedures, is synthesised by modelling of aircraft structural subsystems using a multibody dynamics approach. A dynamic model with variable kinematical structures includes discontinuous dynamics of landing gear oleo-pneumatic shock-absorber with friction and hydraulic/thermodynamic processes. Non-linear tire contact dynamics and unilateral dynamics of nose gear elastic leg assembly is modelled as well. The longitudinal and lateral aerodynamic loads are estimated by considering various aircraft system configurations (landing gears in “up’’ and “down’’ position, different control surfaces in active/inactive modes). A mathematical model is derived as a differential- algebraic (DAE) system. The developed numerical tools are modularly shaped and efficient numerical integration methods as well as original procedures for MBS constraint stabilization are applied for dynamic response determination. On the basis of the presented model, dynamic simulations of landing cases of large transport aircraft were performed for different initial descent velocities with focus on determination of dynamical loading of main landing gear assembly., U ovom radu ukratko su opisane numeričke simulacijske procedure za dinamiku slijetanja velikog transportnog zrakoplova. Razvijene numeričke procedure omogućavaju određivanje dinamičkog odziva zrakoplova prilikom slijetanja i to za različite parametre leta i slijetanja. Nelinearni dinamički model zrakoplova pri slijetanju, kao osnova računalnih procedura, dobiven je sintezom modela konstrukcijskih podsustava zrakoplova primjenom mehaničkih i matematičkih algoritama dinamike konstrukcijskih sustava. Dinamički model zrakoplova s varijabilnom kinematičkom topologijom obuhvaća diskontinuiranu dinamiku oleo-pneumatske elastične noge glavnog podvozja s uključenim termodinamičkim/hidrauličnim procesima, kontaktnu dinamiku gume te nelinearnu unilateralnu dinamiku elastične noge nosnog kotača. Uzdužna i bočna aerodinamička opterećenja procijenjena su za različite konfiguracije letjelice (uvučeno/izvučeno podvozje, izvučena/uvučena zakrilca te otkloni ostalih upravljačkih površina). Matematički model izveden je kao sustav diferencijalno-algebarskih jednadžbi (DAE). Razvijeni računalni alati oblikovani su modularno te su za potrebe određivanja dinamičkog odziva primijenjene efikasne metode numeričke integracije, kao i originalne procedure stabilizacije dinamičkih odziva konstrukcijskih sustava sa složenim kinematičkim ograničenjima. Temeljem izloženog modela provedene su dinamičke simulacije slijetanja velikog transportnog zrakoplova i to za različite brzine spuštanja s fokusom na određivanje dinamičkog opterećenja glavnog podvozja.

During numerical forward dynamics of constrained multibody systems, a numerical violation of system kinematical constraints is the important issue that has to be properly treated. In this paper, the stabilized time-integration procedure, whose constraint stabilization step is based on the projection of integration results to underlying constraint manifold via post-integration correction of the selected coordinates is discussed. A selection of the coordinates is based on the optimization algorithm for coordinates partitioning. After discussing geometric background of the optimization algorithm, new formulae for optimized partitioning of the generalized coordinates are derived. Beside in the framework of the proposed stabilization algorithm, the new formulae can be used for other integration applications where coordinates partitioning is needed. Holonomic and non-holonomic systems are analyzed and optimal partitioning at the position and velocity level are considered further. By comparing the proposed stabilization method to other projective algorithms reported in the literature, the geometric and stabilization issues of the method are addressed. A numerical example that illustrates application of the method to constraint violation stabilization of non-holonomic multibody system is reported.

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 © 2003 by ASME

An general algorithm for deriving equations of motion of multibody mechanisms is presented and discussed. It is to be used for unconstrained multibody systems and the proposed forward dynamics formulation, based on a recursive Newton-Euler procedure, results in a Hamiltonian equations. An example of a serial structure system with rotational joints is provided.

A procedure for setting objective parameters (forward velocity, step length, jumping height, angular momentum, take-off and landing kinematic configuration) for a mechanical prototype of a hopping robot with articulated leg is presented. The goal is to obtain the robot's energy efficient locomotion on irregular terrain using torque and power limited electric actuators, as well as tuned passive actuation. The prototype weights 11.7 kg and consists of an upper body of 0.6 m and upper and lower leg of 0.3 m. The algorithm is based on a close examination of the machine's passive dynamic behaviour throughout a specific range of investigated locomotion patterns. By an adjustment of the robot's passive motion the desired objective parameters are attained using as small actuation as possible. This approach enables the control of the robot's motion while adjusting the machine step length up to 0.5 m and forward velocity up to 0.9 m/s, whereas the driving torques and power supply are limited to ą 40 Nm and 400 W, respectively. The maximum driving torques and the maximum power consumption is well within the range of the specifications of the prototype's built in actuators.

A method for the integration of the equations of motion of constrained multibody systems in descriptor form with no constraint violation is presented. The mathematical model, shaped as a differential-algebraic system of index 1 [1], is transformed to an ordinary differential equations' system (ODE) using a null-space projection method [2]. The dynamical equations are set in a non-minimal form. During integration, the violations of constraints are corrected by solving the constraint equations, utilising the metric of system's configuration space and an orthogonal-complement matrix of the system's Jacobian [3]. Since the constraint violation is one of the basic sources of errors of an integration procedure, the method allows for accurate and robust dynamical simulation of the multibody system's motion. It is stable and compatible with most frequently used ODE solvers.

A method for establishing the equations of motion of multibody mechanisms based on canonical momenta is described and discussed. In the case of unconstrained systems, the proposed formulation based on recursive Newton-Euler algorithm results in a Hamiltonian set of first order ODE's in the adopted generalized coordinates and the cannonical momenta.