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constraint stabilization of mechanical systems in ordinary differential equations form

PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART K-JOURNAL OF MULTI-BODY DYNAMICS 2011年第K1期225卷 12-33页

作者： Masarati, P. Politecn Milan Dipartimento Ingn Aerospaziale I-20156 Milan Italy

This work discusses a simple means to add kinematic constraints to existing mechanical problems formulated in terms of ordinary differential equations. The constraints are expressed by algebraic relationships between the co-ordinates of the unconstrained problem. A solution projection approach ensures compliance of the solution with the derivatives of holonomic constraint equations up to second order within the desired accuracy. The structure of the unconstrained problem is not altered, resulting in a simple, little intrusive, yet effective means to enforce kinematic constraints into existing formulations and implementations originally intended to address differential problems, without the complexity of solving differential-algebraic problems or resorting to implicit numerical integration schemes and without altering the number and type of equations of the original unconstrained problem. The proposed formulation is compared with known approaches. Numerical applications of increasing complexity illustrate its distinguishing aspects.

关键词： constrained system dynamics constraint stabilization numerical drift

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Performance investigation and constraint stabilization approach for the orthogonal complement-based divide-and-conquer algorithm

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MECHANISM AND MACHINE THEORY 2013年 67卷 111-121页

作者： Khan, I. M. Anderson, K. S. Rensselaer Polytech Inst Computat Dynam Lab Dept Mech Aerosp & Nucl Engn Troy NY 12180 USA

The introductory paper on the orthogonal complement-based divide-and-conquer algorithm (ODCA) lacked in properly characterizing the growth rate of the constraint violation error and the singularity handling capabilities of the algorithm [1]. In this paper, we investigate the performance of the ODCA with regards to constraint error growth and singularity handling capabilities. Moreover we introduce a new inverse dynamics-based constraint stabilization approach. The proposed method is applicable to general multibody systems with arbitrary number of closed kinematic loops. We compare the ODCA with augmented [2] and reduction [3] methods. Our results indicate that the error growth rate for the ODCA falls between these two traditional techniques. Moreover, using benchmark numerical problems, we illustrate the effectiveness of the stabilization scheme. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Constrained multibody systems Orthogonal complement Divide-and-conquer algorithm Logarithmic complexity constraint stabilization

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Projective constraint stabilization for a Power Series Forward Dynamics Solver

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JOURNAL OF DYNAMIC SYSTEMS MEASUREMENT AND CONTROL-TRANSACTIONS OF THE ASME 2013年第3期135卷 031004页

作者： Milenkovic, Paul Univ Wisconsin Dept Elect & Comp Engn Madison WI 53706 USA

A power series expression for the forward dynamics of a closed kinematic chain provides an explicit time-step update of the system state. The resulting numerical differential equation solver applies kinematic constraints to the power series terms for acceleration and higher derivatives of motion. Integrating acceleration determines velocity and position time histories that approximate the constraints to a high degree of precision when using a high order of the expansion. When high precision is not required, a lower order achieves shorter computation times, but that condition results in violation of the constraints in the absence of any correction. Projecting the velocities and positions onto the constraint manifold after each time step produces step changes. This paper determines which choices of linear subspace for this projection give step changes that are equal to the residues of truncating the power series solution for the kinematic portion of the problem. The limit of that power series gives position and velocity time histories that approximate the dynamics while giving an exact kinematic solution. Thus projection onto the constraints in this procedure determines sample values of an underlying solution for the motion trajectories, where that underlying solution is continuous in both velocity and position and also satisfies the kinematic constraints at all times. This property is confirmed by numerical simulation of a Clemens constant-velocity coupling.

关键词： kinematic differential equation closed kinematic chain power series Taylor series screw theory forward-dynamics problem differential-algebraic equation (DAE) constraint stabilization projection explicit time step

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Efficient large multibody flight dynamic simulation through improved nonlinear control constraint stabilization

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PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART G-JOURNAL OF AEROSPACE ENGINEERING 2023年第13期237卷 3032-3044页

作者： Wachlin, Jake Butler, Cason Costello, Mark Earthly Dynam Corp Atlanta GA USA Georgia Inst Technol Sch Aerosp Engn Atlanta GA USA Georgia Inst Technol Coll Engn Daniel Guggenheim Sch Aerosp Engn 270 Ferst Dr Atlanta GA 30313 USA

Some air vehicles are configured such that major components exhibit relative motion with respect to one another where this relative motion significantly affects vehicle motion. In these cases, multibody flight dynamics is needed to adequately model system dynamics. This paper reports on a numerically efficient and versatile method for multibody flight dynamic simulation where the air vehicle is idealized as a collection of rigid bodies connected together by a set of joints. The method uses constrained coordinates with a constraint stabilization method based on a nonlinear control framework. The key innovation lies in relating the connections of the rigid bodies to an undirected graph and its adjacency matrix. By reordering connections, the bandwidth of the adjacency matrix can be minimized leading to substantial computational improvements. These improvements are applied to several air vehicle systems to highlight the computational benefits of the proposed technique.

关键词： multibody dynamics constraint stabilization dynamic simulation computational efficiency flight dynamic modeling

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Analysis of Possible Solutions of Some Inverse Dynamical Problem with Regard for constraint stabilization

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LOBACHEVSKII JOURNAL OF MATHEMATICS 2024年第1期45卷 472-477页

作者： Mukharlyamov, R. G. Kaspirovich, I. E. Univ Russia RUDN Univ Patrice Lumumba Peoples Friendship Univ Moscow Russia

Methods for constructing systems of second-order ordinary differential equations whose solutions satisfy the constraint stabilization conditions are outlined. Modified Helmholtz conditions are used to bring the system to the form of the Lagrange equations. The expression of the central force is determined, which ensures the stabilization of the points movement trajectory.

关键词： dynamical system system of differential equations Helmholtz conditions constraint stabilization

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Application of constraint stabilization to Nonholonomic Mechanics 2

Application of Constraint Stabilization to Nonholonomic Mech...

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2nd International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM)

作者： Kaspirovich, I. E. Peoples Friendship Univ Russia Moscow 117198 Russia

ISBN: (纸本)9781509013227

This paper compares some methods of solving problems of nonholonomic systems. Appropriate choice of generalized coordinates allows setting up the Chaplygin equations and separating the motion equations from constraints. Another common method of solving such problems is the method of the Lagrange multipliers. However, using numerical integration in this case leads to error accumulation caused by deviations from the constraints equations and, as a result, to the solution instability in relation to the constraints equations. constraint stabilization can be applied to remove this instability using numerical integration. In this paper, this method of constraint stabilization is applied to generate a stable solution of motion equations with the Lagrange multipliers. The Euler and Runge-Kutta methods are used for numerical integration. In addition, at solving stabilization problem, an appropriate set of parameters can be selected to minimize the difference between the solution of the Chaplygin equations and that of equations with the Lagrange multipliers.

关键词： nonholonomic constraint constraint stabilization Chaplygin equations Lagrange multipliers

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Application of Baumgarte constraint stabilization to Inverse Dynamical Problem

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LOBACHEVSKII JOURNAL OF MATHEMATICS 2024年第1期45卷 416-425页

作者： Kaspirovich, K. Z. Mukharlyamov, R. G. Kaspirovich, I. E. Patrice Lumumba Peoples Friendship Univ Russia RU Moscow Russia

Dynamical systems are often being described with the help of the ordinary differential equations. But only a few problems can be solved analytically. Basically, the quantity theory of ordinary differential equations or numerical methods are used to analyze the properties of the solution. But most of the difference schemes can lead to the numerical instability with the regard for the constraints. To avoid the accumulation of errors, J. Baumgarte suggested to use the method of constraint stabilization. In this paper, the methods of solving inverse dynamical problems with regard for constraint stabilization are investigated.

关键词： dynamical system system of differential equations inverse dynamical problem dissipative function constraint stabilization

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A constraint stabilization METHOD FOR TIME INTEGRATION OF CONSTRAINED MULTIBODY SYSTEMS IN LIE GROUP SETTING

A CONSTRAINT STABILIZATION METHOD FOR TIME INTEGRATION OF CO...

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ASME Design Engineering Technical Conferences and Computers and Information in Engineering Conference (DETC)

作者： Mueller, Andreas Terze, Zdravko Shanghai Jiao Tong Univ Univ Michigan Joint Inst Shanghai Peoples R China Univ Zagreb Fac Mech Engn & Naval Arch Zagreb 41000 Croatia

ISBN: (纸本)9780791846391

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.

关键词： Multibody systems dynamics differential-algebraic equations (DAE) constraint stabilization Lie group integration

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constraint stabilization Application to Chaplygin Systems

Constraint Stabilization Application to Chaplygin Systems

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International Russian Automation Conference

作者： I. E. Kaspirovich R. G. Mukharlyamov RUDN University Peoples' Friendship University of Russia Moscow Russia

ISBN: (纸本)9781538649398

The paper considers the problem of the dynamics control of nonholonomic Chaplygin systems. The control is executed by introducing additional forces that correspond to the problem goals given by both holonomic and nonholonomic constraints. The determination of the expressions for the control actions by means of Lagrange multipliers leads to the instability of the numerical solution of the dynamic equations of a closed system. An introduction of extra variables estimating the deviations from the constraint equations and a construction of the extended system allows solving the problem of the constraint stabilization.

关键词： Chaplygin system constraint stabilization Lagrange multipliers Numerical integration

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The Embedded Projection Method: A general index reduction procedure for constrained system dynamics

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2006年第5-51期195卷 6974-6992页

作者： Borri, Marco Trainelli, Lorenzo Croce, Alessandro Politecn Milan Dipartimento Ingn Aerospaziale I-20156 Milan Italy

The 'Embedded Projection Method' (EPM) is an attempt to eliminate most of the difficulties related to the solution of high-index differential-algebraic equations (DAEs), as those typical of the dynamics of constrained systems. The intimate coupling between the algebraic and the differential parts lies at the origin of these difficulties, therefore we uncouple these components, introduce an unconstrained, modified state variable, and complement the resulting ordinary differential equation (ODE) by additional algebraic variables. As a result, a general, consistent index reduction from arbitrarily high values to one is obtained, allowing the use of any suitable ODE integration algorithm, without the need for a specialized DAE solver. The EPM solution is intrinsically endowed with greater accuracy and stability than conventional counterparts, thus balancing the higher complexity of the procedure. We illustrate the EPM including some applications of holonomic and non-holonomic mechanical systems. (c) 2005 Elsevier B.V. All rights reserved.

关键词： differential-algebraic equations index reduction constraint stabilization constrained dynamical systems embedded projection

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