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工作总结/周报/25.6.27汇总/一体化仿真软件组周报（第八十三期）.docx
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工作总结/周报/25.6.27汇总/一体化仿真软件组周报（第八十三期）.docx
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Research Publications at Politecnico di Milano
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# Post-Print
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This is the accepted version of:
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F. Gonzalez, P. Masarati, J. Cuadrado, M.A. Naya
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Assessment of Linearization Approaches for Multibody Dynamics Formulations
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Journal of Computational and Nonlinear Dynamics, Vol. 12, N. 4, 2017, 041009 (7 pages)
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doi:10.1115/1.4035410
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The final publication is available at https://doi.org/10.1115/1.4035410
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Access to the published version may require subscription.
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# When citing this work, cite the original published paper.
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$\circledcirc$ 2017 by ASME. This manuscript version is made available under the CC-BY 4.0 license http://creativecommons.org/licenses/by/4.0/
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# ASME Accepted Manuscript Repository
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Institutional Repository Cover Sheet
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<html><body><table><tr><td>Pierangelo</td><td>Masarati</td></tr><tr><td>First</td><td>Last</td></tr><tr><td></td><td>ASME Paper Title:Assessment of Linearization Approaches for Multibody Dynamics Formulations</td></tr><tr><td></td><td></td></tr><tr><td>Authors:</td><td></td></tr><tr><td></td><td>Gonzalez, Francisco; Masarati, Pierangelo; Cuadrado, Javier; Naya, Miguel A.</td></tr><tr><td></td><td>ASME Journal Title: Journal of Computational and Nonlinear Dynamics</td></tr><tr><td></td><td></td></tr><tr><td>Volume/Issue__12/4</td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td>https://asmedigitalcollection.asme.org/computationalnonlinear/article/doi/10.1115/1</td></tr><tr><td></td><td>ASMEDigital CollectionURL:2/Assessment-of-Linearization-Approaches-for</td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td>DOI:</td><td></td></tr><tr><td></td><td>10.1115/1.4035410</td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr></table></body></html>
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# Assessment of Linearization Approaches for Multibody Dynamics Formulations
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# Assessment of Linearization Approaches for Multibody Dynamics Formulations 线性化方法的多体动力学公式评估
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Francisco Gonz´alez∗
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Juan de la Cierva Post-Doctoral Fellow
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@ -54,6 +30,10 @@ Email: minaya@udc.es
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Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear Differential Algebraic Equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analysis are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics.
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A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.
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采用多体动力学方法推导机械系统动力学方程通常会导致一组高度非线性的微分代数方程 (DAEs)。虽然这种运动方程的形式适用于广泛的实际应用，但在某些情况下，需要访问线性化的系统动力学。当需要进行稳定性分析和模态分析时，以及为某些控制算法和状态估计器定义植物和系统模型时，都需要线性化的动力学表达式。
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文献中可以找到许多多体动力学线性化方法。这些方法在处理运动方程的方式和呈现结果的方式上存在差异，而这些差异又取决于用于描述机械系统的广义坐标的选择。这种选择与处理系统运动学约束的方式密切相关。可以区分出三种主要方法，可用于对迄今为止发表的大多数线性化方法进行分类。在这项工作中，我们将展示每种方法在静态平衡系统的线性化中的特性，并通过研究两个具有代表性的例子加以说明。
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# 1 Introduction
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@ -63,10 +43,19 @@ Such guidelines are also necessary to select linearization methods for those app
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Several methods have been proposed in the multibody literature to arrive at linearized forms of the dynamics equations. They present different features, depending mainly on the selection of the coordinates with which they describe the mechanical system. Some of them are recursive algorithms for robotic systems [11], others are built on Maggi-Kane’s coordinates [12]. Lagrange multipliers are a popular way to represent constrained systems and methods to linearize the resulting equations exist as well [6, 10, 13]. This paper categorizes linearization methods for multibody dynamics into three main groups, defined by the selection of generalized coordinates. Their properties are demonstrated in the linearization of the dynamics of mechanical systems in static equilibrium, for which Linear Time Invariant (LTI) problems are guaranteed to be obtained. Representative methods of each category were applied to the linearization of test problems and compared in terms of efficiency, ease of use, and accuracy. Moreover, in [14] two types of linearization problems were identified: heavily constrained systems in which the number of degrees of freedom is much lower than the number of kinematic variables, and systems (typically flexible mechanisms) in which both numbers are of similar magnitude. The behavior of the linearization methods was studied with an example of each type. Practical guidelines for the selection of the most appropriate technique for each case are presented to the reader as conclusions.
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多体系统 (MBS) 动力学在过去几十年里经历了显著增长，这得益于计算机体系结构和软件的进步。MBS 动力学最具吸引力的特征之一是研究人员和分析师在处理特定问题时可以选择的各种建模和公式化方法[1]。一般来说，系统可以用一组独立的广义坐标进行建模，这会导致一组常微分方程 (ODEs)，或者使用相关的依赖坐标，其运动由一个或多个运动学约束相关联。在后一种情况下，动力学方程用一组 DAEs 表示，必须使用适当的算法来处理，例如 [2,3]。MBS 研究社区不断提出和开发新的公式和方法，这使得对它们的性能进行基准测试和表征变得必要。这项工作的最终目标是提供指导方针，帮助分析师在处理特定问题时选择最合适的公式。
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对于那些需要线性近似运动方程的应用，也需要选择线性化方法。这适用于某些经典的控制算法，它们需要用线性模型来表示要控制的系统[4]，以及一些状态和输入估计器，如卡尔曼滤波器[5]。即使不需要运动学方程的线性表达式，它的可用性可以使人们更容易了解系统的行为。使用从原始微分代数方程的线性化中提取的降阶模型，可以简化结构的和气弹的问题分析[6]；线性化模型在稳定性分析中也很有用[7–9]。模态分析和确定系统的固有频率和振动模态是线性化动力学的另一个重要应用[10]。
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在多体文献中，已经提出了几种方法来获得动力学方程的线性形式。它们具有不同的特征，主要取决于描述机械系统的坐标的选择。其中一些是用于机器人系统的递归算法[11]，另一些是基于 Maggi-Kane 坐标[12]。拉格朗日乘子是表示约束系统的常用方法，并且存在线性化所得方程的方法[6, 10, 13]。本文将多体动力学的线性化方法分为三大类，这些分类由广义坐标的选择定义。通过对静态平衡机械系统的动力学进行线性化，展示了它们的特性，从而保证获得线性时不变 (LTI) 问题。将每类方法的代表性应用于测试问题的线性化，并从效率、易用性和准确性方面进行比较。此外，在 [14] 中，确定了两种类型的线性化问题：高度约束的系统，其自由度数量远低于运动学变量的数量，以及（通常是柔性机构）自由度和运动学变量数量相似的系统。研究了线性化方法的行为，并对每种类型的系统给出了一个示例。文章最后向读者呈现了关于为每个情况选择最合适技术的实用指南。
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# 2 Coordinates Selection
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A multibody system can be described with a set of $n$ generalized coordinates $\mathbf{q}=\{q_{1},\dots,q_{n}\}^{\mathrm{~T~}}$ related by $m$ kinematic constraints $\Phi(\mathbf{q},t)=\mathbf{0}$ . For the purposes of this paper, these are assumed to be holonomic and linearly independent; accordingly, the $m\times n$ Jacobian matrix of the constraints $\Phi_{\mathbf{q}}$ can be assumed to have full row rank $m$ . The equations of motion can be expressed as a nonlinear system of DAEs as
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一个多体系统可以用一组 $n$ 个广义坐标 $\mathbf{q}=\{q_{1},\dots,q_{n}\}^{\mathrm{~T~}}$ 来描述，它们之间由 $m$ 个运动约束 $\Phi(\mathbf{q},t)=\mathbf{0}$ 相关联。为了本论文的目的，假设这些约束是全纯的且线性无关；因此，$m\times n$ 约束函数 $\Phi$ 关于广义坐标 $\mathbf{q}$ 的雅可比矩阵 $\Phi_{\mathbf{q}}$ 可以假设具有满行秩 $m$ 。运动方程可以表示为一组非线性DAE系统，如下所示：
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$$
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\begin{array}{c}{\mathbf{M}\ddot{\mathbf{q}}=\mathbf{f}+\mathbf{f}_{c}}\\ {\Phi\left(\mathbf{q},t\right)=\mathbf{0}}\end{array}
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$$
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@ -77,19 +66,32 @@ where $\mathbf{M}\left(\mathbf{q}\right)$ is the $n\times n$ mass matrix, $\math
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2. Redundant Coordinate Set (RCS), consisting of the $n$ coordinates and a set of $m$ Lagrange multipliers; and
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3. Unconstrained Coordinate Set (UCS), in which the system is modeled with the original $n$ coordinates of the unconstrained problem and the constraints are embedded in the formulation.
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其中 $\mathbf{M}\left(\mathbf{q}\right)$ 是 $n\times n$ 质量矩阵，$\mathbf{f}$ 是广义施加力和速度相关力的数组，$\mathbf{f}_{c}$ 是由运动学约束引入的反应力。 方程 (1) 通常被转化为文献中已有的多体形式中的一种，这些形式可分为三大类 [14]：
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1. 最小坐标系 (MCS)，该形式的坐标数量等于系统的自由度数量，即 $n-m$；
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2. 冗余坐标系 (RCS)，由 $n$ 个坐标和一个包含 $m$ 个拉格朗日乘子的集合组成；
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3. 无约束坐标系 (UCS)，在该系统中，使用原始的 $n$ 个无约束问题的坐标，并将约束嵌入到形式化中。
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Velocity projection techniques, in which the system velocities are projected onto the subspace of admissible motion [15], fall into the MCS category. Graph-theory-based [16,17] and Transfer Matrix [18] multibody algorithms could also be included in this category.
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If a Lagrangian approach is followed, the constraint reactions are expressed in terms of the constraints Jacobian matrix as $\mathbf{f}_{c}=-\Phi_{\mathbf{q}}^{\mathrm{T}}\pmb{\lambda}$ , where $\lambda$ contains $m$ Lagrange multipliers. Using Baumgarte stabilization to remove constraints drift [19] one would directly arrive at a RCS method. Alternatively, eliminating the constraints delivers a UCS approach. UCS methods can also be obtained through penalty formulations [20] or with the force projection approach in [21].
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As mentioned, selecting one approach or another determines the output of the linearization methods, their efficiency and the way in which they convey information about the mechanical system. MCS formulations result in a linear problem with the exact spectrum of the constrained system; the eigenanalysis of this problem directly yields the $2\left(n-m\right)$ eigenvalues of the original system. RCS and UCS approaches may give rise to spurious or approximate eigenvalues; on the other hand, their use can be convenient for the sake of efficiency or ease of use. In the subsequent sections, representatives of each category will be compared in terms of effectiveness and efficiency, and their main features will be discussed. Two main factors, namely how easy it is to discriminate spurious eigenvalues and the simplicity of the linearized equations, also from the computational point of view, will be considered in the discussion.
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速度投影技术，其中系统速度被投影到允许运动的子空间 [15]，属于 MCS 类别。基于图论 [16,17] 和传递矩阵 [18] 的多体算法也应包含在这个类别中。
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如果采用拉格朗日方法，约束反作用力可以表示为约束雅可比矩阵的函数，即 $\mathbf{f}_{c}=-\Phi_{\mathbf{q}}^{\mathrm{T}}\pmb{\lambda}$ ，其中 $\lambda$ 包含 $m$ 个拉格朗日乘子。使用 Baumgarte 稳定化来消除约束漂移 [19] 将直接得到一种 RCS 方法。或者，消除约束会得到一种 UCS 方法。UCS 方法也可以通过惩罚公式 [20] 或 [21] 中的力投影方法获得。
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如前所述，选择一种方法或另一种方法决定了线性化方法的输出、效率以及它们传递关于机械系统信息的方式。MCS 格式化结果是一个具有约束系统精确谱的线性问题；对该问题的特征分析可以直接得到原始系统的 $2\left(n-m\right)$ 个特征值。RCS 和 UCS 方法可能会产生虚假或近似特征值；另一方面，它们的使用可以为了效率或易用性而方便。在后续部分，将从有效性和效率方面比较每个类别的代表，并讨论其主要特征。两个主要因素，即区分虚假特征值的难易程度以及线性化方程的简单性，也将从计算的角度来考虑。
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# 3 Linearization Methods
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In this section three methods to linearize Eqs. (1) are introduced. The first one is a MCS formulation obtained via velocity projections [22]. The second directly linearizes a RCS description of the dynamics [6]. The last one is a UCS formulation obtained replacing the kinematic constraints with penalty systems [20].
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本节介绍三种线性化 Eqs. (1) 的方法。第一种是基于速度投影得到的 MCS 格式化 [22]。第二种直接线性化了 RCS 对动态的描述 [6]。最后一种是基于 UCS 格式化，通过用惩罚系统替代运动学约束得到 [20]。
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# 3.1 MCS Formulation: Velocity Projection
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The system dynamics can be expressed in terms of a set of $n-m$ independent velocities $\dot{\mathbf{z}}$ using the transformation
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系统动力学可以表示为一组 $n-m$ 个独立的速率 $\dot{\mathbf{z}}$，使用变换：
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$$
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{\dot{\mathbf{q}}}=\mathbf{R}{\dot{\mathbf{z}}}
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where $\mathbf{R}\left(\mathbf{q}\right)$ is an $n\times(n-m)$ velocity transformation matrix that verifies $\mathbf{R}^{\mathrm{T}}\boldsymbol{\Phi}_{\mathbf{q}}^{\mathrm{T}}=\mathbf{0}$ . Matrix $\mathbf{R}$ can be determined via several methods: e.g., coordinate selection (splitting $\mathbf{q}$ into independent and dependent variables), the zero eigenvalue theorem [23], Singular Value Decomposition (SVD) [24,25], QR decomposition [26], and Gram-Schmidt orthonormalization [27, 28]. All these methods determine the subspace $\mathbf{R}$ of the velocities that complies with the constraints through operations on the Jacobian $\Phi_{\mathbf{q}}$ . Applying the transformation in Eq. (2), the system dynamics become a system of $(n-m)$ ODEs
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其中 $\mathbf{R}\left(\mathbf{q}\right)$ 是一个 $n\times(n-m)$ 速度变换矩阵，它满足 $\mathbf{R}^{\mathrm{T}}\boldsymbol{\Phi}_{\mathbf{q}}^{\mathrm{T}}=\mathbf{0}$ 。矩阵 $\mathbf{R}$ 可以通过多种方法确定：例如，坐标选择（将 $\mathbf{q}$ 分解为独立变量和相关变量），零特征值定理 [23]，奇异值分解 (SVD) [24,25]，QR 分解 [26]，以及 Gram-Schmidt 正交化 [27, 28]。所有这些方法都通过对雅可比矩阵 $\Phi_{\mathbf{q}}$ 进行操作，来确定满足约束条件的的速度子空间 $\mathbf{R}$ 。应用公式 (2) 中的变换，系统动力学变为 $(n-m)$ 阶常微分方程组。
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$$
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\mathbf{H}_{1}=\mathbf{R}^{\mathrm{T}}\mathbf{M}\mathbf{R}\ddot{\mathbf{z}}-\mathbf{R}^{\mathrm{T}}\left(\mathbf{f}-\mathbf{M}\dot{\mathbf{R}}\dot{\mathbf{z}}\right)=\mathbf{0}
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$$
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If the inputs are represented by $\mathbf{u}$ , the system can be linearized about an equilibrium configuration $\mathbf{z}_{0},\,\dot{\mathbf{z}}_{0},\,\ddot{\mathbf{z}}_{0},\,\mathbf{u}_{0}$ as follows
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如果输入表示为 $\mathbf{u}$，则系统可以围绕平衡构型 $\mathbf{z}_{0},\,\dot{\mathbf{z}}_{0},\,\ddot{\mathbf{z}}_{0},\,\mathbf{u}_{0}$ 进行线性化，如下所示：
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$$
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\begin{array}{c}{{{\bf{H}}_{1}\left({{{\bf{z}}_{0}}+\delta{{\bf{z}},{{\dot{\bf{z}}}}_{0}}+\delta{{\dot{\bf{z}}},{{\dot{\bf{z}}}}_{0}}+\delta{{\ddot{\bf{z}}},{{\bf{u}}_{0}}}+\delta{\bf{u}}\right)\cong}}\\ {{\left.{\frac{{\partial{\bf{H}}_{1}}}{{\partial{\bf{z}}}}}\right|_{0}\delta{\bf z}+\frac{{\partial{\bf{H}}_{1}}}{{\partial{\dot{\bf{z}}}}}\bigg\vert_{0}\delta{\dot{\bf z}}+\frac{{\partial{\bf{H}}_{1}}}{{\partial{\dot{\bf z}}}}\bigg\vert_{0}\,\delta{\dot{\bf z}}+\frac{{\partial{\bf{H}}_{1}}}{{\partial{\bf{u}}}}\bigg\vert_{0}\,\delta{\bf u}}\end{array}
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\begin{array}{c}{{{\bf{H}}_{1}\left({{{\bf{z}}_{0}}+\delta{{\bf{z}},{{\dot{\bf{z}}}}_{0}}+\delta{{\dot{\bf{z}}},{{\dot{\bf{z}}}}_{0}}+\delta{{\ddot{\bf{z}}},{{\bf{u}}_{0}}}+\delta{\bf{u}}\right)\cong}}\\ {{{\frac{{\partial{\bf{H}}_{1}}}{{\partial{\bf{z}}}}}|_{0}\delta{\bf z}+\frac{{\partial{\bf{H}}_{1}}}{{\partial{\dot{\bf{z}}}}}\bigg\vert_{0}\delta{\dot{\bf z}}+\frac{{\partial{\bf{H}}_{1}}}{{\partial{\dot{\bf z}}}}\bigg\vert_{0}\,\delta{\dot{\bf z}}+\frac{{\partial{\bf{H}}_{1}}}{{\partial{\bf{u}}}}\bigg\vert_{0}\,\delta{\bf u}}\end{array}
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$$
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The terms in Eq. (4) take the form
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@ -114,13 +117,13 @@ $$
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$$
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The expressions for the numerical evaluation of the partial derivatives of $\mathbf{R}$ are detailed in [29]. The linearized dynamics can then be written as
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数值评估偏导数 $\mathbf{R}$ 的表达式详见 [29]。线性化动力学方程可以写为：
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$$
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\mathbf{M}_{r}\delta{\ddot{\mathbf{z}}}+\mathbf{C}_{r}\delta{\dot{\mathbf{z}}}+\mathbf{K}_{r}\delta\mathbf{z}=\mathbf{F}_{r}\delta\mathbf{u}
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$$
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Equation (6) has $2\left(n-m\right)$ eigenvalues that represent the exact spectrum of the problem. Moreover, its leading matrix $\mathbf{M}_{r}$ is symmetric and positive-definite. As a consequence, Eq. (6) can be used in the solution of forward-dynamics problems with explicit integration schemes.
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方程 (6) 具有 $2\left(n-m\right)$ 个特征值，它们代表问题的精确谱。 此外，其首导矩阵 $\mathbf{M}_{r}$ 是对称正定矩阵。 因此，方程 (6) 可用于带有显式积分方案的逆动力学问题的求解。
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# 3.2 RCS Formulation: Generalized Eigenanalysis
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Following a Lagrangian approach, the dynamics equations (1) can be expressed in the form
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