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.
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.
The dynamics simulation of multibody systems (MBS) using spatial velocities (non-holonomic velocities) requires time integration of the dynamics equations together with the kinematic reconstruction ...equations (relating time derivatives of configuration variables to rigid body velocities). The latter are specific to the geometry of the rigid body motion underlying a particular formulation, and thus to the used configuration space (c-space). The proper c-space of a rigid body is the Lie group SE (3), and the geometry is that of the screw motions. The rigid bodies within a MBS are further subjected to geometric constraints, often due to lower kinematic pairs that define SE (3) subgroups. Traditionally, however, in MBS dynamics the translations and rotations are parameterized independently, which implies the use of the direct product group SO (3)×ℝ3 as rigid body c-space, although this does not account for rigid body motions. Hence, its appropriateness was recently put into perspective.
In this paper the significance of the c-space for the constraint satisfaction in numerical time stepping schemes is analyzed for holonomically constrained MBS modeled with the ‘absolute coordinate’ approach, i.e. using the Newton–Euler equations for the individual bodies subjected to geometric constraints. The numerical problem is considered from the kinematic perspective. It is shown that the geometric constraints a body is subjected to are exactly satisfied if they constrain the motion to a subgroup of its c-space. Since only the SE (3) subgroups have a practical significance it is regarded as the appropriate c-space for the constrained rigid body. Consequently the constraints imposed by lower pair joints are exactly satisfied if the joint connects a body to the ground. For a general MBS, where the motions are not constrained to a subgroup, the SE (3) and SO (3)×ℝ3 yield the same order of accuracy. Hence an appropriate configuration update can be selected for each individual body of a particular MBS, which gives rise to tailored update schemes. Several numerical examples are reported illustrating this statement.
The practical consequence of using SE (3) is the use of screw coordinates as generalized coordinates. To account for the inevitable singularities of 3-parametric descriptions of rotations, the kinematic reconstruction is additionally formulated in terms of (dependent) dual quaternions as well as a coordinate-free ODE on the c-space Lie group. The latter can be solved numerically with Lie group integrators like the Munthe-Kaas integration method, which is recalled in this paper.
•Numerical integration of constrained MBS in absolute coordinate formulation.•Kinematic reconstruction for rigid bodies in the Lie group setting and dual quaternions.•Constraint satisfaction when SE(3) or SO(3)xR³ as configuration space Lie group analyzed.•Constraints are satisfied by any numerical integrator if rigid body motions are constrained to subgroups of c-space Lie group.•Generically SE(3) represents the optimal configuration space.
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 surface atmospheric 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 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.
•Flapping wing vehicles can be used for Venus atmospheric flight and exploration.•Quasi-steady forward flight model can be used to model Venus flapping flight.•Flapping wing dynamics can be efficiently optimized using DMOC framework.•Added mass and rotational forces have strong impact on Venus flapping flight.
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).
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.
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.
Multibody systems are dynamical systems characterized by intrinsic symmetries and invariants. Geometric mechanics deals with the mathematical modeling of such systems and has proven to be a valuable ...tool providing insights into the dynamics of mechanical systems, from a theoretical as well as from a computational point of view. Modeling multibody systems, comprising rigid and flexible members, as dynamical systems on manifolds, and Lie groups in particular, leads to frame-invariant and computationally advantageous formulations. In the last decade, such formulations and corresponding algorithms are becoming increasingly used in various areas of computational dynamics providing the conceptual and computational framework for multibody, coupled, and multiphysics systems, and their nonlinear control. The geometric setting, furthermore, gives rise to geometric numerical integration schemes that are designed to preserve the intrinsic structure and invariants of dynamical systems. These naturally avoid the long-standing problem of parameterization singularities and also deliver the necessary accuracy as well as a long-term stability of numerical solutions. The current intensive research in these areas documents the relevance and potential for geometric methods in general and in particular for multibody system dynamics. This paper provides an exhaustive summary of the development in the last decade, and a panoramic overview of the current state of knowledge in the field.
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.
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.
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