Constrained Multibody Dynamics With Python: From Symbolic Equation Generation to Publication

2013 ◽  
Author(s):  
Gilbert Gede ◽  
Dale L. Peterson ◽  
Angadh S. Nanjangud ◽  
Jason K. Moore ◽  
Mont Hubbard
Keyword(s):  
Open Source ◽  
Undergraduate Students ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Academic Research ◽  
General Purpose ◽  
Engineering Software ◽  
Symbolic Equations Of Motion ◽  
High Level

Symbolic equations of motion (EOMs) for multibody systems are desirable for simulation, stability analyses, control system design, and parameter studies. Despite this, the majority of engineering software designed to analyze multibody systems are numeric in nature (or present a purely numeric user interface). To our knowledge, none of the existing software packages are 1) fully symbolic, 2) open source, and 3) implemented in a popular, general, purpose high level programming language. In response, we extended SymPy (an existing computer algebra system implemented in Python) with functionality for derivation of symbolic EOMs for constrained multibody systems with many degrees of freedom. We present the design and implementation of the software and cover the basic usage and workflow for solving and analyzing problems. The intended audience is the academic research community, graduate and advanced undergraduate students, and those in industry analyzing multibody systems. We demonstrate the software by deriving the EOMs of a N-link pendulum, show its capabilities for LATEX output, and how it integrates with other Python scientific libraries — allowing for numerical simulation, publication quality plotting, animation, and online notebooks designed for sharing results. This software fills a unique role in dynamics and is attractive to academics and industry because of its BSD open source license which permits open source or commercial use of the code.

Trajectory Tracking of Underactuated Multibody Systems

2011 ◽  
Author(s):  
Stefan Reichl ◽  
Wolfgang Steiner
Keyword(s):  
Trajectory Tracking ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
State Variables ◽  
Algebraic Equations

This work presents three different approaches in inverse dynamics for the solution of trajectory tracking problems in underactuated multibody systems. Such systems are characterized by less control inputs than degrees of freedom. The first approach uses an extension of the equations of motion by geometric and control constraints. This results in index-five differential-algebraic equations. A projection method is used to reduce the systems index and the resulting equations are solved numerically. The second method is a flatness-based feedforward control design. Input and state variables can be parameterized by the flat outputs and their time derivatives up to a certain order. The third approach uses an optimal control algorithm which is based on the minimization of a cost functional including system outputs and desired trajectory. It has to be distinguished between direct and indirect methods. These specific methods are applied to an underactuated planar crane and a three-dimensional rotary crane.

scholarly journals Integration of Nonlinear Models of Flexible Body Deformation in Multibody System Dynamics

2013 ◽  
Vol 9 (1) ◽  
Author(s):  
Martin Schulze ◽  
Stefan Dietz ◽  
Bernhard Burgermeister ◽  
Andrey Tuganov ◽  
Holger Lang ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Multibody System ◽  
Nonlinear Models ◽  
Equations Of Motion ◽  
Flexible Structures ◽  
General Purpose ◽  
Cosserat Rod ◽  
Rod Model ◽  
Flexible Deformation ◽  
Run Up

Current challenges in industrial multibody system simulation are often beyond the classical range of application of existing industrial simulation tools. The present paper describes an extension of a recursive order-n multibody system (MBS) formulation to nonlinear models of flexible deformation that are of particular interest in the dynamical simulation of wind turbines. The floating frame of reference representation of flexible bodies is generalized to nonlinear structural models by a straightforward transformation of the equations of motion (EoM). The approach is discussed in detail for the integration of a recently developed discrete Cosserat rod model representing beamlike flexible structures into a general purpose MBS software package. For an efficient static and dynamic simulation, the solvers of the MBS software are adapted to the resulting class of MBS models that are characterized by a large number of degrees of freedom, stiffness, and high frequency components. As a practical example, the run-up of a simplified three-bladed wind turbine is studied where the dynamic deformations of the three blades are calculated by the Cosserat rod model.

Variable-Gain Constraint Stabilization for General Multibody Systems with Applications

2004 ◽  
Vol 10 (9) ◽  
pp. 1335-1357
Author(s):  
Takashi Nagata
Keyword(s):  
Message Passing ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Message Passing Interface ◽  
Distributed Processing ◽  
General Purpose ◽  
Space Structure ◽  
Variable Gain ◽  
Matrix Operations ◽  
Gain Error

This paper presents a general and efficient formulation applicable to a vast variety of rigid and flexible multibody systems. It is based on a variable-gain error correction with scaling and adaptive control of the convergence parameter. The methodology has the following distinctive features. (i) All types of holonomic and non-holonomic equality constraints as well as a class of inequalities can be treated in a plain and unified manner. (ii) Stability of the constraints is assured. (iii) The formulation has an order Ncomputational cost in terms of both the constrained and unconstrained degrees of freedom, regardless of the system topology. (iv) Unlike the traditional recursive order Nalgorithms, it is quite amenable to parallel computation. (v) Because virtually no matrix operations are involved, it can be implemented to very simple general-purpose simulation programs. Noting the advantages, the algorithm has been realized as a C++ code supporting distributed processing through the Message-Passing Interface (MPI). Versatility, dynamical validity and efficiency of the approach are demonstrated through numerical studies of several particular systems including a crawler and a flexible space structure.

Dynamics of Serial Multibody Systems Using the Decoupled Natural Orthogonal Complement Matrices

1999 ◽  
Vol 66 (4) ◽  
pp. 986-996 ◽  
Author(s):  
S. K. Saha
Keyword(s):  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Kinematic Chain ◽  
Rigid Bodies ◽  
Dynamic Equations ◽  
Orthogonal Complement ◽  
Forward Dynamics ◽  
Velocity Constraints

Constrained dynamic equations of motion of serial multibody systems consisting of rigid bodies in a serial kinematic chain are derived in this paper. First, the Newton-Euler equations of motion of the decoupled rigid bodies of the system at hand are written. Then, with the aid of the decoupled natural orthogonal complement (DeNOC) matrices associated with the velocity constraints of the connecting bodies, the Euler-Lagrange independent equations of motion are derived. The De NOC is essentially the decoupled form of the natural orthogonal complement (NOC) matrix, introduced elsewhere. Whereas the use of the latter provides recursive order n—n being the degrees-of-freedom of the system at hand—inverse dynamics and order n3 forward dynamics algorithms, respectively, the former leads to recursive order n algorithms for both the cases. The order n algorithms are desirable not only for their computational efficiency but also for their numerical stability, particularly, in forward dynamics and simulation, where the system’s accelerations are solved from the dynamic equations of motion and subsequently integrated numerically. The algorithms are illustrated with a three-link three-degrees-of-freedom planar manipulator and a six-degrees-of-freedom Stanford arm.

Semi-Symbolical Equations of Motion for Flexible Multibody Systems

1993 ◽  
Author(s):  
Frank Melzer
Keyword(s):  
Data Model ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
General Purpose ◽  
Flexible Multibody Systems ◽  
Dynamical Analysis ◽  
Lightweight Structures ◽  
Elastic Bodies ◽  
Flexible Multibody ◽  
Increasing Demand

Abstract The need for computer aided engineering in the analysis of machines and mechanisms led to a wide variety of general purpose programs for the dynamical analysis of multibody systems. In the past few years the incorporation of flexible bodies in this methodology has evolved to one of the major research topics in the field of multibody dynamics, due to the use of more lightweight structures and an increasing demand for high-precision mechanisms such as robots. This paper presents a formalism for flexible multibody systems based on a minimum set of generalized coordinates and symbolic computation. A standardized object-oriented data model is used for the system matrices, describing the elastodynamic behaviour of the flexible body. Consequently, the equations of motion are derived in a form independent of the chosen modelling technique for the elastic bodies.

Design and Development of H∞ and μ-Synthesis Controllers for Nanorobotic Manipulation and Grasping Applications

2007 ◽  
Author(s):  
Reza Saeidpourazar ◽  
Beshah Ayalew ◽  
Nader Jalili
Keyword(s):  
Degrees Of Freedom ◽  
Controller Design ◽  
Equations Of Motion ◽  
Dynamic Equations ◽  
Robust Controller ◽  
Accuracy And Precision ◽  
Highly Nonlinear ◽  
Linearized Model ◽  
High Level ◽  
Robust Controller Design

This paper presents the development of H∞ and μ-synthesis robust controllers for nanorobotic manipulation and grasping applications. Here a 3 DOF (Degrees Of Freedom) nanomanipulator with RRP (Revolute Revolute Prismatic) actuator arrangement is considered for nanomanipulation purposes. Due to the sophisticated complexity, and expected high level of accuracy and precision (of the order of 10−7 rad in revolute actuators and 0.25 nm in the prismatic actuator) of the nanomanipulator, there is a need to design a suitable controller to guarantee an accurate manipulation process. However, structure of the nanomanipulator employed here, namely MM3A, is such that the dynamic equations of motion of the nanomanipulator are highly nonlinear and complicated. Linearizing these dynamic equations of the nanomanipulator simplifies the controller design process significantly. However, linearization could suppress some critical information about the system dynamics. In order to achieve the precise motion of the nanomanipulator utilizing the simple linearized model, H∞ and μ-synthesis robust controller design approaches are proposed. Following the development of the controllers, numerical simulations of the proposed controllers on the nanomanipulator are used to verify the positioning performance.

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