Conference Paper

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

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Abstract

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.

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... Exudyn models are created solely using the Python language. Python is also available in other multibody codes, such as ProjectChrono [18] or PyDy [8]. As a difference from the latter codes, Exudyn includes a large set of annotated examples. ...
... The expectation of the tests is to obtain a Python code from the LLM that can be directly processed in Python using the Exudyn package. 8 Tests are evaluated based on correct code syntax and on correct modeling of the dynamic system. In order to clearly distinguish between the two error types, syntax errors (counted by e syn ) are all errors that raise a Python error when the code is executed, except for Exudyn's solver failures due to modeling errors. ...
... However, looking at larger datasets available on HuggingFace [28], we find typical datasets 9 that are 7 https://platform.openai.com/tokenizer. 8 Due to the widely backwards compatibility of Exudyn with the version from September 2021, all tests are evaluated with Exudyn version 1.7.0. We did not observe syntax errors in the LLM outputs related to a change of version. ...
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Computational models are conventionally created with input data, script files, programming interfaces, or graphical user interfaces. This paper explores the potential of expanding model generation, with a focus on multibody system dynamics. In particular, we investigate the ability of Large Language Model (LLM), to generate models from natural language. Our experimental findings indicate that LLM, some of them having been trained on our multibody code Exudyn, surpass the mere replication of existing code examples. The results demonstrate that LLM have a basic understanding of kinematics and dynamics, and that they can transfer this knowledge into a programming interface. Although our tests reveal that complex cases regularly result in programming or modeling errors, we found that LLM can successfully generate correct multibody simulation models from natural-language descriptions for simpler cases, often on the first attempt (zero-shot). After a basic introduction into the functionality of LLM, our Python code, and the test setups, we provide a summarized evaluation for a series of examples with increasing complexity. We start with a single mass oscillator, both in SciPy as well as in Exudyn, and include varied inputs and statistical analysis to highlight the robustness of our approach. Thereafter, systems with mass points, constraints, and rigid bodies are evaluated. In particular, we show that in-context learning can levitate basic knowledge of a multibody code into a zero-shot correct output.
... We implemented an open-source version in Python using SymPy (SymPy, 2021), leveraging its mechanical toolbox. Alternative symbolic frameworks found in the literature are usually limited to rigid bodies (Verlinden et al., 2005;Kurz and Eberhard, 2009;Gede et al., 2013;Docquier et al., 2013) or are closed-source or using proprietary software (Reckdahl and Mitiguy, 1996;Kurtz et al., 2010;MotionGenesis, 2016;Lemmer, 2018). ...
... 2.3). We were also inspired by the package PyDy (Gede et al., 2013), which is a convenient tool to export the equations of motion to executable code and directly visualize the bodies in 3D. The core of our work consisted of implementing a class to define flexible bodies (FlexibleBody) and the corresponding Kane method for this class (Sect. ...
... In this section, we present different wind energy applications of the symbolic framework. We focus on models with at least one flexible body because the rigid body formulation of SymPy has been well verified (Gede et al., 2013). For each example, the equations of motion are given and their results are compared with OpenFAST (Jonkman et al., 2021) simulations. ...
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The article presents a symbolic framework (also called computer algebra program) that is used to obtain, in symbolic mathematical form, the linear and nonlinear equations of motion of a mid-fidelity multibody system including rigid and flexible bodies. Our approach is based on Kane's method and a nonlinear shape function representation for flexible bodies. The shape function approach does not represent the state of the art for flexible multibody dynamics but is an effective trade-off to obtain mid-fidelity models with few degrees of freedom, taking advantage of the separation of space and time. The method yields compact symbolic equations of motion with implicit account of the constraints. The general and automatic framework facilitates the creation and manipulation of models with various levels of complexity by adding or removing degrees of freedom. The symbolic treatment allows for analytical gradients and linearized equations of motion. The linear and nonlinear equations can be exported to Python code or dedicated software. There are multiple applications, such as time domain simulation, stability analyses, frequency domain analyses, advanced controller design, state observers, and digital twins. In this article, we describe the method we used to systematically generate the equations of motion of multibody systems and present the implementation of the framework using the Python package SymPy. We apply the framework to generate illustrative land-based and offshore wind turbine models. We compare our results with OpenFAST simulations and discuss the advantages and limitations of the method. The Python implementation is provided as an open-source project.
... Also, the use of generalized speeds to define the kinematics appears more natural for rotating systems rather than generalized coordinates. The method of generalized speeds is also suitable for implementation in a Computer Algebra System (CAS), assisting the bookkeeping of the equations, minimizing the risk for manual error and maintaining repeatability through coding which is highly desirable in an industrialized simulation process [14,15,16]. ...
... Even though Kane's method is suitable to implement in a Computer Algebra System (CAS), such as in Refs. [15,16], the equations of motion of the rotor pendulum system depicted in Figure 1 are manually derived to show the procedure of the method. ...
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The multiple-order response of a rotor equipped with a centrifugal pendulum vibration absorber (CPVA) is investigated in this study. CPVAs have been used in the abatement of torsional vibrations since the 1930s and have been extensively researched but seldom with a focus on the full multiple-harmonic response. Although the firing order in reciprocating engines dominates in amplitude, the higher-order harmonics have impact on driveline-related noise issues. This investigation is conducted in order to understand if an higher-order amplification can stem from the design of the CPVA or if this phenomenon is due an interaction between other powertrain components. Therefore, an isolated rotor-CPVA system is studied. The investigated rotor is subjected to an oscillating torque consisting of single-and multiple-order harmonics. The equations of motion are derived by Kane's method and the multiple-harmonic response of the system is analyzed. The linear steady-state response is compared with numerical time integration of the equations of motion.
... the Euler-Lagrange equations of the three-link planar robot in Figure 1 can be written, with the aid of the symbolic multibody dynamics package PyDy [12], as M (θ, p) ∈ R 3×3 , M (θ, p) 0 is the symmetric mass matrix of the system with entries M 11 = I 1 + I 2 + I 3 + l 2 c1 m 1 + m 2 l 2 1 + 2l 1 l c2 c 2 + l 2 c2 + m 3 l 2 1 + 2l 1 l 2 c 2 + 2l 1 l c3 c 23 + l 2 2 + 2l 2 l c3 c 3 + l 2 c3 M 12 = I 2 + I 3 + l c2 m 2 (l 1 c 2 + l c2 ) + m 3 l 1 l 2 c 2 + l 1 l c3 c 23 + l 2 2 + 2l 2 l c3 c 3 + l 2 c3 M 13 = I 3 + l c3 m 3 (l 1 c 23 + l 2 c 3 + l c3 ) ...
... The parameter uncertainties lie within the interval [p, p] ⊆ R 12 in Table 1, which was calculated for a fluctuation of ±5% of the nominal weight of the user with anthropometric data from [21]. With a set P b ⊂ [p, p] of 500 parameters drawn from a Latin Hypercube, the first step in the analysis is to numerically solve the sensitivity equation (12) over the time horizon [0, 3.5] for all p ∈ P b . According to the sampling approach of the previous section, the sensitivity bounds S x , S x : [0, 3.5] → R 6×12 are then estimated by minimizing/maximizing the entries of the matrices (10) for each t ∈ T s as in (13). ...
Preprint
A sensitivity-based approach for computing over-approximations of reachable sets, in the presence of constant parameter uncertainties and a single initial state, is used to analyze a three-link planar robot modeling a Powered Lower Limb Orthosis and its user. Given the nature of the mappings relating the state and parameters of the system with the inputs, and outputs describing the trajectories of its Center of Mass, reachable sets for their respective spaces can be obtained relying on the sensitivities of the nonlinear closed-loop dynamics in the state space. These over-approximations are used to evaluate the worst-case performances of a finite time horizon linear-quadratic regulator (LQR) for controlling the ascending phase of the Sit-To-Stand movement.
... 1 Due in part to this focus, it has become a popular language for scientific computing and data science, with a broad ecosystem of libraries (Oliphant, 2007). SymPy is itself used as a dependency by many libraries and tools to support research within a variety of domains, such as SageMath (The Sage Developers, 2016) (pure and applied mathematics), yt (Turk et al., 2011) (astronomy and astrophysics), PyDy (Gede et al., 2013) (multibody dynamics), and SfePy (Cimrman, 2014) (finite elements). ...
... Multibody Dynamics with Python (Gede et al., 2013). ...
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SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.
... This example is chosen from PyDy project [44]. A double pendulum is a pendulum with another pendulum attached to its end. ...
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... PyDy [30] was chosen as the framework for building the planar robot simulation as it uses a symbolic math backend that expresses equations of motion in a human-readable format. The kino-dynamic model of the planar robot was built using Kane's method [31]: a process for defining equations of motion with forces in different frames acting on rigid bodies with constraints. ...
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... Exudyn models are created solely using the Python language. Python is also available in other multibody codes, such as ProjectChrono [18] or PyDy [8]. As a difference to the latter codes, Exudyn includes a large set of annotated examples. ...
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... A different selection of system states would result in distinct expressions for joint moments and would also lead to Equations of Motion (EoMs) of different shape. Symbolic EoMs were obtained using Kane's Method (Kane and Levinson [5]) implementation of SymPy (Meurer et al. [6]) and evaluated numerically with the use of PyDy (Gede et al. [7]). ...
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This paper presents a modeling approach using lumped element multibody systems to describe aeroe-lastic dynamics of aircraft. The model is shown to describe geometric nonlinearities arising from large structural displacements and oscillations such as that of a very flexible wing bending under aerodynamic loads. The presented approach is applied to a Goland's wing to determine the flutter speed resulting from coupling the multibody structure to a quasi-steady aerodynamic model. Flutter velocities are comparable to those obtained using modal representations for structural dynamics. Finally, the modeling appoach is shown to be scalable, parametrized and modular, all of which could make it valuable in iterative environments such as during conceptual design phases of very flexible aircraft.
... The manual derivation of the equations of motion are given here. However, the process and bookkeeping may be facilitated by means of CAS such as SymPy [31] . ...
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... Kane's dynamical equation is of the formF r +F * r = 0, whereF r andF * r represents generalized active forces and generalized inertia forces, respectively, as described in Kane and Levinson (1985) (chapter 6, page 159). The equations in the form necessary for simulation is obtained using python libraries, the details of which are given in Gede et al. (2013). Symbols m 1 , m 2 represent the mass of the body and swing leg, respectively, as shown in Figure 2. The length of both the legs is represented by l. ...
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... The following Kane's method integrator for equations of motion is due to VanderPlas [5] and Gede et al. [6]. Thus, we implement a Simulation() method which will apply the aforementioned procedure to determine the average oscillation period for each pendulum bob and then take an average over all masses. ...
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... Dallali [445] studied compliant humanoid robot dynamics by a symbolic modeling method. Gede [446] extended the SymPy computer algebra system to deduce symbolic equations of motion for constrained MSs with many DOFs. Hall [447] proposed a graph-theoretic symbolic multibody modeling environment of MapleSim. ...
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... the Euler-Lagrange equations of the three-link planar robot in Fig. 1 can be written, with the aid of the symbolic multibody dynamics package PyDy [5], as Fig. 1: Three-link planar robot for modeling a powered lower limb orthosis and the interaction with its user during a Sit-to-Stand (STS) movement. ...
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... We have a closed form expression for the position as a function of the curvatures q and of the curvilinear coordinates s = (s 1 , s 2 ): r = r(q, s). Since this expression is relatively complex we, at first, used Sympy and PyDy [120] to generate symbolic expressions for M (q), ∂M (q) ∂q i and F g (q). Then from the symbolic expressions C code is generated using the codegen module of the PyDy package (around 8000 lines of C code are generated). ...
Thesis
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... This example is chosen from PyDy project [44]. A double pendulum is a pendulum with another pendulum attached to its end. ...
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Chapter
This book is ideal for teaching students in engineering or physics the skills necessary to analyze motions of complex mechanical systems such as spacecraft, robotic manipulators, and articulated scientific instruments. Kane's method, which emerged recently, reduces the labor needed to derive equations of motion and leads to equations that are simpler and more readily solved by computer, in comparison to earlier, classical approaches. Moreover, the method is highly systematic and thus easy to teach. This book is a revision of Dynamics: Theory and Applications by T. R. Kane and D. A. Levinson and presents the method for forming equations of motion by constructing generalized active forces and generalized inertia forces. Important additional topics include approaches for dealing with finite rotation, an updated treatment of constraint forces and constraint torques, an extension of Kane's method to deal with a broader class of nonholonomic constraint equations, and other recent advances.
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This book takes a traditional approach to the development of the methods of analytical dynamics. After a review of Newtonian dynamics, the basic concepts of analytical dynamics - classification of constraints, classification of forces, virtual displacements, virtual work and variational principles - are introduced and developed. Next, Langrange's equations are derived and their integration is discussed. The Hamiltonian portion of the book covers Hamilton's canonical equations, contact transformations, and Hamilton-Jacobi theory. Also included are chapters on stability of motion, impulsive forces, and the Gibbs-Appell equation. Two types of examples are used throughout the book. The first type is intended to illustrate key results of the theoretical development, and these are deliberately kept as simple as possible. The other type is included to show the application of the theoretical results to complex, real-life problems. These examples are often quite lengthy, comprising an entire chapter in some cases. © 2005 Kluwer Academic / Plenum Publishers. All rights reserved.
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I anatomize a successful open-source project, fetchmail, that was run as a deliberate test of some theories about software engineering suggested by the history of Linux. I discuss these theories in terms of two fundamentally different development styles, the "cathedral" model, representing most of the commercial world, versus the "bazaar" model of the Linux world. I show that these models derive from opposing assumptions about the nature of the software-debugging task. I then make a sustained argument from the Linux experience for the proposition that "Given enough eyeballs, all bugs are shallow," suggest productive analogies with other self-correcting systems of selfish agents, and conclude with some exploration of the implications of this insight for the future of software.
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