Form Of Differential-algebraic Equations Research Articles

Обобщенная задача динамического уравновешивания и перспективные направления ее применения

The paper considers the generalized problem of the machines and mechanisms dynamic balance in terms of ensuring the given laws of altering reactions in the selected links. Representation of equations of the mechanical systems dynamics in the form of differential algebraic equations was used making it possible to obtain mathematical models of the nonlinear mechanical systems dynamics with the arbitrary structure of kinematic and force connections. With this approach, the constraint reactions are determined by algebraic equations from the system coordinates. The problem solution is based on changing the bonds selected reactions due to the impact on reactions in the other selected bonds by adding non-stationary terms to the selected bonds equation. Conditions for rigorous solution of the optimal control problem for a mechanical system are shown for the integral quality criterion not explicitly containing the control functions. The method is aimed at numerical models of the mechanical systems widely used in the programs for dynamic analysis of the coupled systems of bodies. Test examples are provided for manipulator, anthropomorphic robot and controlled suspension of a transport vehicle. The method was realized in the software package for simulating the controlled motion dynamics of the coupled systems of bodies.

Dynamic Modeling and Attitude Control of Large-Scale Flexible Parallel Multibody Spacecraft

In this paper, modeling and high-precision attitude control of large-scale flexible parallel multibody spacecraft (FPMS) are addressed. First, for describing the complex constraint relationship between each body of the multibody system, the dynamic model of flexible spacecraft in the form of differential-algebraic equations (DAEs) is established by applying the Lagrange equation of the first kind (augmented Lagrange method). Then, to facilitate numerical integration of the dynamic model and controller design, an independent coordinate method and block matrix operation-based transformation method are proposed to convert DAEs into ordinary differential equations. Based on this model, a two-timescale composite controller is designed by virtue of the singular perturbation theory to attain high-precision attitude control of the FPMS and suppress the vibration of flexible appendages. Finally, numerical simulations are conducted under two mission scenarios to validate the dynamic model and demonstrate that the accurate attitude control can be achieved.

A modified model parametrization algorithm for solving a special type of heat and mass transfer systems

A new method for solving nonlinear heat and mass transfer design tasks was considered. Systems using the Number of Transfer Units (NTU) method are a special type of mathematical model of heat and mass exchangers. It was observed, that the NTU models in a form of differential-algebraic equations (DAEs) cannot be directly solved with higher values of NTU. The requirements for consistent initial conditions, as well as numerical limitations of DAEs solvers, result, that the solution to the considered design problems that cannot be obtained by a classical direct shooting procedure. To overcome the presented difficulties, the αDAE model optimization algorithm was adjusted for solving NTU-based models. The new approach consists of 3 main steps: 1) task discretization by a multiple-shooting approach, 2) design an appropriate function fNTU(α) to effectively influence the variability of the state variables described by dynamical relations, 3) the iterative numerical optimization algorithm for the new parametrized system. Moreover, computations can be performed by a chosen numerical optimization approach, which can be communicated with an available outer procedure for solving differential-algebraic equations. The presented algorithm was implemented and applied to solve the design task with the NTU model of a counter-flow exchanger. The new approach was used to modify the system dynamics to influence the difficulty of the considered problem. Finally, the presented method enabled failure-free numerical computations for the higher values of the NTU parameter.

On stability, Bohl exponent and Bohl–Perron theorem for implicit dynamic equations

In this paper, we develop a stability theory for implicit dynamic equations which is a general form of differential-algebraic equations and implicit difference equations. We derive some results about robust stability of these equations subjected to Lipschitz perturbations. After that the so-called Bohl–Perron type stability theorems, which are known in the literature for regular explicit difference equations, are extended for implicit dynamic equations. Finally, the notion of Bohl exponent is introduced and we characterise the relation between the exponential stability and the Bohl exponent. Then, it is investigated that how the Bohl exponent with respect to dynamic perturbations and two-side perturbations depends on the system data.

Robust Stability of Implicit Dynamic Equations on Time Scales

This paper is concerned with the robust stability for implicit dynamic equations on time scales, which is a general form of differential algebraic equations and implicit difference equations. We show the reservation of exponential stability of these equations under small Lipschitz perturbations.

A reference model for airborne wind energy systems for optimization and control

Airborne Wind Energy (AWE) is a promising new technology, and attracts a growing academic and industrial attention. Important research efforts have been deployed to develop prototypes in order to test the technology, generate control algorithms and optimize the efficiency of AWE systems. By today, a large set of control and optimization methods is available for AWE systems. However, because no validated reference model is available, there is a lack of benchmark for these methods. In this paper, we provide a reference model for pumping mode AWE systems based on rigid wings. The model describes the flight dynamics of a tethered 6 degrees of freedom (DOF) rigid body aircraft in form of differential-algebraic equations, based on Lagrange dynamics. With the help of least squares fitting the model is assessed using real flight data from the Ampyx Power prototype AP2. The model equations are smooth and have a low symbolic complexity, so as to make the model ideal for optimization and control. The information given in this paper aims at providing AWE researchers with a model that has been validated against flight data and that is well suited for trajectory and power output simulation and optimization.

A piston tribodynamic model with deterministic consideration of skirt surface grooves

The machining micro-grooves on the piston skirt surface are very common in internal combustion engines. To accurately investigate the influence of the grooves on piston tribodynamic performance, a coupling model of piston skirt-liner lubrication and crank-rod-piston multibody dynamics is built in this study. The lubrication model is solved by the Finite Element Method with deterministic treatment of the grooves. The multibody dynamics equations are formulated with Lagrange multipliers and constraint Jacobian matrix in the form of Differential Algebraic Equations (DAEs). The MEBDF approach is applied to a high efficient time integration of the DAEs. Based on the above model, the influences of grooves and some groove parameters on piston skirt-liner lubrication as well as piston slap are studied and revealed.

多体气动弹性系统超声速颤振分析

The supersonic flutter of aeroelastic multibody systems was investigated. Based on the theory of multibody dynamics and the piston theory for unsteady aerodynamic forces, the governing differential algebraic equations (DAEs) for the aeroelastic multibody system were established. The supersonic flutter analysis was conducted through solution of the eigenvalue problem in the form of DAEs to discuss the stability of the perturbed motion and the equilibrium of the multibody system. The approach was used for the supersonic flutter problems of a plate wing and a wing with control surfaces. The flutter speed of the wing was obtained with the control surfaces in different positions. The agreement between the simulation results and the experimental ones shows that, the present method has high computational accuracy, and is suitable for the flutter analysis of engineering structures consisting of multiple components.

A numerical method for the servo constraint problem of underactuated mechanical systems

AbstractServo constraints are used in inverse dynamics simulations of discrete mechanical systems, especially for trajectory tracking control problems [1], whose desired outputs are represented by state variables and treated as servo constraints [2]. Servo constraint problems can be classified into fully actuated and underactuated multibody systems, and the equations of motion take the form of differential algebraic equations (DAEs) including holonomic and servo constraints. For fully actuated systems, control inputs can be solved from the equations by model inversion, as the input distribution matrix is nonsingular and invertible. However, underactuated systems have more degrees of freedom than control inputs. The input distribution matrix is not invertible, and in contrast to passive constraints, the realization of servo constraints with the use of control forces can range from orthogonal to tangential [3]. Therefore, it is challenging for the determination of control inputs which force the underactuated system to realize the partly specified motion. For differentially flat underactuated systems, the differentiation index of DAEs may exceed three. Hence we need to apply specific index reduction techniques, such as the projection approach applied in [3], [4], and [6]. The present work applies index reduction by minimal extension [5] to differentially flat underactuated crane systems and shows that the index can be reduced from five to three and even to one. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dynamic Analysis of Stewart Platform by Bond Graphs

Dynamic model of Stewart platform was developed using bond graph technique in the paper. Stewart platform has simple, compact structure with high stiffness, high strength-to-weight ratio that make it very desirable in manufacturing technology. Due to closed loop structure and kinematic constraints, dynamic analysis of such systems is rather complicated. The paper proposes a general, object oriented modelling formulation based on bond graph component model approach to develop dynamic model of a hexapod system. Model is developed and simulations were carried out using software package BondSim. Dynamic model is generated in form of differential-algebraic equation (DAEs) and solved by BondSim.

Dynamic characteristics of a 3-RPR planar parallel manipulator with flexible intermediate links

SUMMARYThis paper presents the dynamic modeling of a 3-RPR planar parallel manipulator with three flexible intermediate links in order to investigate the effects of the intermediate links flexibility on the undesired vibrations of the end-effector. For this purpose, the intermediate links are modeled as Euler--Bernoulli beams with two types of fixed-pinned and fixed-free boundary conditions based on the assumed mode method (AMM). The equations of motion of the 3-RPR manipulator are formulated using the augmented Lagrange multipliers method in the form of differential algebraic equations (DAEs) by incorporating the elastic and rigid coordinates in the set of generalized coordinates. After defining the initial conditions and imposing external forces to the manipulator, the equations are then solved numerically using the Modified Extended Backward-Differentiation Formula Implicit (MEBDFI) approach. Comparison of the simulation results for two different boundary conditions shows clearly the effects of flexibility of the intermediate links on the vibration of the end-effector trajectory. Results of this work can be used for the dynamic modeling of other manipulators or to design a controller for reducing the undesired vibrations.

Qualitative analysis of an Ivlev-type bio-economic system

The paper considers a bio-economic system with Ivlev-type functional response. Formally, our bio-economic system takes the form of differential-algebraic equations. We investigate Hopf bifurcation and center stability of the proposed bio-economic system. Some easily verifiable criteria are obtained on these issues. Finally, a numerical example is supplied to illustrate the effectiveness of our criteria.

Modeling and Simulation of Hydraulic Actuated Multibody Systems by Bond Graphs

This paper deals with modeling and simulation of multibody systems consisting of rigid and flexible bodiesactuated by hydraulic motors. To develop the corresponding modelbond graph technique is used thatprovides a systematic component oriented methodology. The proposed procedure is explained on the example of arotary crane.The simulation model was developed and simulation conducted using BondSim© program. The model developed is based on physical modeling philosophy by systematically decomposing it into the components and using the models from theprogram library. Mathematical model of the system may be machine generated in the form of differential algebraic equations (DAEs) and solved using a solver capable of solving such models up to semi implicit index two models.

Stability case study of the ACROBOTER underactuated service robot

The dynamics of classical robotic systems are usually described by ordinary differential equations via selecting a minimum set of independent generalized coordinates. However, different parameterizations and the use of a nonminimum set of (dependent) generalized coordinates can be advantageous in such cases when the modeled device contains closed kinematic loops and/or it has a complex structure. On one hand, the use of dependent coordinates, like natural coordinates, leads to a different mathematical representation where the equations of motion are given in the form of differential algebraic equations. On the other hand, the control design of underactuated robots usually relies on partial feedback linearization based techniques which are exclusively developed for systems modeled by independent coordinates. In this paper, we propose a different control algorithm formulated by using dependent coordinates. The applied computed torque controller is realized via introducing actuator constraints that complement the kinematic constraints which are used to describe the dynamics of the investigated service robotic system in relatively simple and compact form. The proposed controller is applied to the computed torque control of the planar model of the ACROBOTER service robot. The stability analysis of the digitally controlled underactuated service robot is provided as a real parameter case study for selecting the optimal control gains.

Physical Modelling in MvStudium

In this paper, we describe “physical” modeling technology in MvStudium that is a graphical environment for modeling and simulation of complex dynamic systems. MvStudium.6 modeling language (MVL) is based on open hybrid automata or Behavior-Charts (for short BCharts), functional diagrams, and supports object-oriented modeling. B-Chart is an extension of UML state machine with do-activities in the form of differential-algebraic equations. External variables of MVL classes may be in the form of “Inputs-Outputs” (I/O) or “Contacts-Flows” (C/F). C/F components in MvStudium are analogous to Modelica components but in MVL there is no limitation on type (NAE, ODE, DAE), dimension, and form (explicit, semi-explicit, implicit) for a current solved system of equations prescribed to a current state of a B-Chart. MvStudium compiler automatically builds an executable model. It may run under the environment or it may be built in the form of DLL and used even in real-time applications.

Natural Coordinates in the Optimal Control of Multibody Systems

The formulation of multibody dynamics in terms of natural coordinates (NCs) leads to equations of motion in the form of differential-algebraic equations (DAEs). A characteristic feature of the natural coordinates approach is a constant mass matrix. The DAEs make possible (i) the systematic assembly of open-loop and closed-loop multibody systems, (ii) the design of state-of-the-art structure-preserving integrators such as energy-momentum or symplectic-momentum schemes, and (iii) the direct link to nonlinear finite element methods. However, the use of NCs in the optimal control of multibody systems presents two major challenges. First, the consistent application of actuating joint-forces becomes an issue since conjugate joint-coordinates are not directly available. Second, numerical methods for optimal control with index-3 DAEs are still in their infancy. The talk will address the two aforementioned issues. In particular, a new energy-momentum consistent method for the optimal control of multibody systems in terms of NCs will be presented.

A modeling of sliding joint on one-dimensional flexible medium

The dynamic modeling of a sliding joint on a one-dimensional medium, such as a cable or a beam, is studied in this paper. The sliding joint is implemented by positioning it at a moving node on the one-dimensional medium, which is realized by variable-length elements at either side of the joint. The variable-length element is established with an absolute nodal coordinate formulation (ANCF) in the framework of the Arbitrary Lagrange–Euler (ALE) description. The sliding of the joint is described by the increasing of the length on one side of the one-dimensional medium and a corresponding decreasing of the other side. In order to capture the discontinuity of the slopes at the position of the sliding joint, the moving node has two slopes as generalized coordinates which are equal to each other in the case of a beam but not in the case of a cable, and in order to avoid the addition–deletion constraint, the node adjacent to the moving node is added or deleted if the element is too long or too short. The governing equations for the coupled system are derived in terms of D’Alembert’s principle and the resulting equations of motion are formulated in the standard form of differential algebraic equations of multibody systems. Numerical examples are presented to validate the method proposed by comparing with analytical results which are available or are made possible by simplifying the model.

512 修正されたハミルトンの原理に基づくマルチボディシステムの解析 : 零空間を用いたラグランジュ乗数の消去(解析法,制御,安定性解析,OS-19 マルチボディダイナミクス,総合テーマ「伝統を,未来へ!」)

The governing equations for multibody systems are often formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. As a method to solve the DAEs by eliminating the Lagrange multipliers, there is a method called the null space method. In a previous report, one of the authors presented procedure to formulate the governing equations of multibody systems by using the variational principle called the modified Hamilton's principle. The governing equation obtained by the presented procedure involves the Lagrange multipliers. In this report, use of the null space method in the modified Hamilton's principle is discussed. Then, a new method to obtain the null space matrix is presented. Finally, numerical examples are shown.

63964 ANALYSIS OF MULTIBODY SYSTEMS BASED ON THE MODIFIED HAMILTON'S PRINCIPLE : ELIMINATION OF THE LAGRANGE MULTIPLIERS BASED ON THE NULL SPACE METHOD(Modeling, Formalisms, and DAE Solution Method)

The governing equations for multibody systems are often formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. As a method to solve the DAEs by eliminating the Lagrange multipliers, there is a method called the null space method. In a previous report, one of the authors presented procedure to formulate the governing equations of multibody systems by using the variational principle called the modified Hamilton's principle. The governing equation obtained by the presented procedure involves the Lagrange multipliers. In this report, use of the null space method in the modified Hamilton's principle is discussed. Then, a new method to obtain the null space matrix is presented. Finally, numerical examples are shown.

Numerical integration of discrete mechanical systems with mixed holonomic and control constraints

The present work aims to incorporate control (or servo) constraints into finite-dimensional mechanical systems subject to holonomic constraints. In particular, we focus on underactuated systems, defined as systems in which the number of degrees of freedom exceeds the number of inputs. The corresponding equations of motion can be written in the form of differential-algebraic equations (DAEs) with a mixed set of holonomic and control constraints. Apart from closed-loop multibody systems, the present formulation accommodates the so-called rotationless formulation of multibody dynamics. The rotationless formulation has proven to be especially well-suited for the design of energy and momentum conserving schemes, which typically exhibit superior numerical stability properties (see [4, 7, 10]). Subsequent to the incorporation of the servo constraints, we deal with a reformulation of the underlying DAEs, which is amenable to a direct numerical discretization. To this end, we apply a specific projection method to the DAEs in terms of redundant coordinates. A similar projection approach has been previously developed in the framework of generalized coordinates by Blajer & Kolodziejczyk [12]. A numerical example is presented, which deals with a 3D rotary crane.