Direct sensitivity analysis of multibody systems with holonomic and nonholonomic constraints via an index-3 augmented Lagrangian formulation with projections

Abstract

Optimizing the dynamic response of mechanical systems is often a necessary step during the early stages of product development cycle. This is a complex problem that requires to carry out the sensitivity analysis of the system dynamics equations if gradient-based optimization tools are used. These dynamics equations are often expressed as a highly nonlinear system of Ordinary Differential Equations (ODEs) or Differential-Algebraic Equations (DAEs), if a dependent set of generalized coordinates with its corresponding kinematic constraints is used to describe the motion. Two main techniques are currently available to perform the sensitivity analysis of a multibody system, namely the direct differentiation and the adjoint variable methods.In this paper, we derive the equations that correspond to the direct sensitivity analysis of the index-3 augmented Lagrangian formulation with velocity and acceleration projections. Mechanical systems with both holonomic and nonholonomic constraints are considered. The evaluation of the system sensitivities requires the solution of a Tangent Linear Model (TLM) that corresponds to the Newton-Raphson iterative solution of the dynamics at configuration level, plus two additional nonlinear systems of equations for the velocity and acceleration projections. The method was validated in the sensitivity analysis of a set of examples, including a five-bar linkage with spring elements, which had been used in the literature as 1 D. Dopico et al.benchmark problem for similar multibody dynamics formulations, a point-mass system subjected to nonholonomic constraints, and a full-scale vehicle model.

Dynamic Sensitivity Analysis of Deployable Flexible Space Structures With a Large Number of Design Parameters

“…The clause "default (shared)" indicates that the unspecified variables are shared among threads. This subsection shows a benchmark for the SA of rigid body dynamics [5,6]. As shown in Fig.…”

Hamiltonian direct differentiation and adjoint approaches for multibody system sensitivity analysis

“…The most efficient algorithms designed for solving problems described by (11) require the knowledge of how MBS behavior will be affected due to the changes in model parameters. The process of studying such implications is known as sensitivity analysis and requires the calculation of the gradient of objective functions and constraint conditions with respect to the vector of design variables.…”

“…[4][5][6][7][8] The direct method has been used in sensitivity analysis of flexible MBS 9,10 as well as in problems involving nonholonomic constraints. 11 Moreover, it has recently been applied for sensitivity analysis of multibody systems described by the transfer matrix method. 12 The authors have also utilized the method in dynamic balancing of a four-bar linkage.…”

Adjoint sensitivity index-3 augmented Lagrangian formulation with projections

“…Sensitivity analysis has also applications to parameter estimation, although studies such as Blanchard, Sandu, and Sandu (2009, 2010a, 2010b, 2010c present parameter estimation and control approaches that do not use sensitivity analysis. Sensitivity analysis studies for multibody dynamical systems modeled by ordinary differential equations (ODE) were presented in Dopico et al (2015) and Dopico, Sandu, et al (2014), modeled by differential algebraic equations (DAE) in Cao et al (2003), Schaffer (2005Schaffer ( , 2006, Petzold et al (2006), Zhu et al (2018), Dopico et al (2018), Haug (2020), using Maggi's formulation in Dopico et al (2013), and for hybrid systems in Sandu (2019,2020), and Serban and Recuero (2019).…”