Direct and Adjoint Sensitivity Analysis of Ordinary Differential Equation Multibody Formulations

2014 ◽  
Vol 10 (1) ◽  
Author(s):  
Daniel Dopico ◽  
Yitao Zhu ◽  
Adrian Sandu ◽  
Corina Sandu
Keyword(s):  
Sensitivity Analysis ◽  
Multibody Dynamics ◽  
Multibody Systems ◽  
Third Party ◽  
Adjoint Sensitivity ◽  
Numerical Errors ◽  
The Third ◽  
Sensitivity Equations ◽  
Perform Sensitivity Analysis ◽  
Analytical Approaches

Sensitivity analysis of multibody systems is essential for several applications, such as dynamics-based design optimization. Dynamic sensitivities, when needed, are often calculated by means of finite differences. This procedure is computationally expensive when the number of parameters is large, and numerical errors can severely limit its accuracy. This paper explores several analytical approaches to perform sensitivity analysis of multibody systems. Direct and adjoint sensitivity equations are developed in the context of Maggi's formulation of multibody dynamics equations. The approach can be generalized to other formulations of multibody dynamics as systems of ordinary differential equations (ODEs). The sensitivity equations are validated numerically against the third party code fatode and against finite difference solutions with real and complex perturbations.

scholarly journals Dynamic Response Optimization of Complex Multibody Systems in a Penalty Formulation Using Adjoint Sensitivity

2015 ◽  
Vol 10 (3) ◽  
Author(s):  
Yitao Zhu ◽  
Daniel Dopico ◽  
Corina Sandu ◽  
Adrian Sandu
Keyword(s):  
Sensitivity Analysis ◽  
Multibody Dynamics ◽  
Multibody Systems ◽  
Real Life ◽  
Design Parameters ◽  
Vehicle Model ◽  
Adjoint Sensitivity ◽  
Penalty Formulation ◽  
Sensitivity Approach ◽  
Dynamic Performance Analysis

Multibody dynamics simulations are currently widely accepted as valuable means for dynamic performance analysis of mechanical systems. The evolution of theoretical and computational aspects of the multibody dynamics discipline makes it conducive these days for other types of applications, in addition to pure simulations. One very important such application is design optimization for multibody systems. In this paper, we focus on gradient-based optimization in order to find local minima. Gradients are calculated efficiently via adjoint sensitivity analysis techniques. Current approaches have limitations in terms of efficiently performing sensitivity analysis for complex systems with respect to multiple design parameters. To improve the state of the art, the adjoint sensitivity approach of multibody systems in the context of the penalty formulation is developed in this study. The new theory developed is then demonstrated on one academic case study, a five-bar mechanism, and on one real-life system, a 14 degree of freedom (DOF) vehicle model. The five-bar mechanism is used to validate the sensitivity approach derived in this paper. The full vehicle model is used to demonstrate the capability of the new approach developed to perform sensitivity analysis and optimization for large and complex multibody systems with respect to multiple design parameters with high efficiency.

MBSVT: Software for Modeling, Sensitivity Analysis, and Optimization of Multibody Systems at Virginia Tech

2014 ◽  
Author(s):  
Yitao Zhu ◽  
Daniel Dopico ◽  
Corina Sandu ◽  
Adrian Sandu
Keyword(s):  
Sensitivity Analysis ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Virginia Tech ◽  
Benchmark Problems ◽  
Adjoint Sensitivity ◽  
Normal Contact ◽  
Penalty Formulation ◽  
Sensitivity Equations ◽  
External Forces

This paper introduces MBSVT (Multibody Systems at Virginia Tech), as a software library for the kinematic and dynamic simulation of multibody systems, with forward kinematics and dynamics, direct and adjoint sensitivity analysis, and optimization capabilities. The MBSVT software was developed in Fortran 2003 as a collection of Fortran modules and it was tested on several different platforms using multiple compilers. The kinematic library includes dot-1 constraint, revolute, spherical, Euler, and translational joints, as well as distance and coordinates driving constraints. The forward dynamics uses the penalty formulation to write the equations of motion and both explicit and implicit Runge-Kutta numerical integrators are implemented to integrate the equations. The library implements external forces, such as translational spring-damper-actuator, bump stop, linear normal contact, and basic tire force. Direct and adjoint sensitivity equations are implemented for the penalty formulation. The L-BFGS-B quasi-Newton optimization algorithm [1] is integrated with the library, to carry out the optimization tasks. MBSVT also provides a connection with Matlab by means of the Matlab engine. 3D rendering is available via the graphic library MBSVT-viz based on OpenSceneGraph. The collection of benchmark problems provided includes a crank-slider mechanism, 2D and 3D excavators models, a vehicle suspension, and full vehicle model. The distribution includes a Cmake list, gfortran make files, MSV2010 project files, and a collection of training problems. Detailed doxygen documentation for the MBSVT library is available in html and pdf formats.

Graph-Theoretic Sensitivity Analysis of Multibody Systems

2014 ◽  
Vol 9 (4) ◽  
Author(s):  
Joydeep M. Banerjee ◽  
John J. McPhee
Keyword(s):  
Sensitivity Analysis ◽  
Multibody Systems ◽  
Software Implementation ◽  
Computational Costs ◽  
Graph Theoretic ◽  
Sensitivity Equations ◽  
Finite Difference Formulation ◽  
Perform Sensitivity Analysis ◽  
System Equations ◽  
Linear Graphs

A graph-theoretic formulation to perform sensitivity analysis on multibody systems is presented in this article. In this formulation, linear graphs are used to capture the system topologies from which a graph-theoretic formulation simultaneously generates the system equations and the sensitivity equations. This ensures the automated, accurate, and efficient generation of sensitivity equations. The basic formulation steps are outlined to illustrate the process of the generation of sensitivity equations. The salient aspects of multibody systems are presented along with a brief description of the software platform that has been used to implement the algorithm. A 3D pendulum and a double-wishbone suspension system are analyzed to demonstrate the application of the algorithm. The results are validated by using a finite difference formulation. Finally, the efficiency of the software implementation is assessed by comparing the computational costs associated with the proposed method and that of existing methods.

Direct and Adjoint Sensitivity Analysis of Multibody Systems Using Maggi’s Equations

2013 ◽  
Author(s):  
Daniel Dopico ◽  
Yitao Zhu ◽  
Adrian Sandu ◽  
Corina Sandu
Keyword(s):  
Sensitivity Analysis ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Third Party ◽  
Adjoint Sensitivity Analysis ◽  
Adjoint Sensitivity ◽  
Dynamics Of Multibody Systems ◽  
Dynamical Optimization ◽  
Analytical Expressions ◽  
Coordinate Partitioning

The importance of the sensitivity analysis of multibody systems for several applications is well known, concretely design optimization based on the dynamics of multibody systems usually requires the sensitivity analysis of the equations of motion. A broad range of methods for the dynamics of multibody systems include the state space formulations based on Maggis equations, nullspace methods or coordinate partitioning. Dynamic sensitivities, when needed, are often calculated by means of finite differences but, depending of the number of parameters involved, this procedure can be very demanding in terms of CPU time and the accuracy obtained can be very poor in many cases. In this paper, several ways to perform the sensitivity analysis are explored and analytical expressions for the direct and adjoint sensitivity analysis of multibody systems are presented, all of them based on Maggi’s formulations. Moreover, two different approaches to the adjoint sensitivity analysis of multibody systems are presented. Although particularized to one formulation, the general expressions provided in the paper, are intended to be easily generalized and applied to any other formulation that can be expressed as an ODE-like system of equations, including penalty formulations. Besides, to check the validity and correctness of the proposed equations, the solutions of all the methods proposed are compared: 1) between them, 2) with the third party code FATODE and 3) with the numerical solution using real and complex perturbations. Finally, all the techniques proposed are applied to the dynamical optimization of a multibody system.

Direct Sensitivity Analysis of Spatial Multibody Systems With Joint Friction Using Index-1 Formulation

2021 ◽  
Author(s):  
Adwait Verulkar ◽  
Corina Sandu ◽  
Daniel Dopico ◽  
Adrian Sandu
Keyword(s):  
Sensitivity Analysis ◽  
Dynamic Systems ◽  
Multibody Systems ◽  
Friction Model ◽  
Design Parameters ◽  
Adjoint Sensitivity ◽  
Joint Friction ◽  
Multibody Dynamic ◽  
Model Dynamics ◽  
Direct Sensitivity

Abstract Sensitivity analysis is one of the most prominent gradient based optimization techniques for mechanical systems. Model sensitivities are the derivatives of the generalized coordinates defining the motion of the system in time with respect to the system design parameters. These sensitivities can be calculated using finite differences, but the accuracy and computational inefficiency of this method limits its use. Hence, the methodologies of direct and adjoint sensitivity analysis have gained prominence. Recent research has presented computationally efficient methodologies for both direct and adjoint sensitivity analysis of complex multibody dynamic systems. The contribution of this article is in the development of the mathematical framework for conducting the direct sensitivity analysis of multibody dynamic systems with joint friction using the index-1 formulation. For modeling friction in multibody systems, the Brown and McPhee friction model has been used. This model incorporates the effects of both static and dynamic friction on the model dynamics. A case study has been conducted on a spatial slider-crank mechanism to illustrate the application of this methodology to real-world systems. Using computer models, with and without joint friction, effect of friction on the dynamics and model sensitivities has been demonstrated. The sensitivities of slider velocity have been computed with respect to the design parameters of crank length, rod length, and the parameters defining the friction model. Due to the highly non-linear nature of friction, the model dynamics are more sensitive during the transition phases, where the friction coefficient changes from static to dynamic and vice versa.

Dynamics Study and Sensitivity Analysis of Flexible Multibody Systems With Interval Parameters

2016 ◽  
Author(s):  
Wang Zhe ◽  
Qiang Tian ◽  
Hiayan Hu
Keyword(s):  
Sensitivity Analysis ◽  
Surrogate Model ◽  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Flexible Multibody Systems ◽  
Computational Time ◽  
Interval Parameters ◽  
Computation Efficiency ◽  
Sensitivity Equations ◽  
Flexible Multibody

The dynamics of flexible multibody systems with interval parameters is studied based on a non-intrusive computation methodology. The Absolute Nodal Coordinate Formulation (ANCF) is used to model the rigid-flexible multibody system, including the finite elements of the ANCF and the ANCF Reference Nodes (ANCF-RNs). The Chebyshev sampling methods including Chebyshev tensor product (CTP) sampling method and Chebyshev collocation method (CCM), are utilized to generate the Chebyshev surrogate model for Interval Differential Algebraic Equations (IDAEs). For purpose of preventing the interval explosion problem and maintaining computation efficiency, the interval bounds of the IDAEs are determined by scanning the deduced Chebyshev surrogate model. To further improve the computation efficiency, OpenMP directives are also used to parallelize the solving process of the Differential Algebraic Equations (DAEs) by fixing the uncertain interval parameter at the given sampling points. The sensitivity analysis of flexible multibody systems with interval parameters is initially performed by using the direct differentiation method. The direct differentiation method differentiates the dynamic equations with respect to the design variable, which yields the system sensitivity equations governed by DAEs. The generalized alpha method is introduced to integrate the sensitivity DAEs. The sensitivity equations of flexible multibody systems with interval parameters are also described by the IDAEs. Based on the continuum mechanics, the computational efficient analytical formulations for the derivative items of the system sensitivity equations are deduced. Three examples are studied to validate the proposed methodology, including the complicated spatial rigid-flexible multibody systems with a large number of uncertain interval parameters, the flexible system with uncertain interval clearance size joint, and the first order sensitivity analysis of flexible multibody systems with interval parameters. Firstly, the dynamics analysis of a six-arm space robot with six interval parameters is performed. For this case study, the interval dynamics cannot be obtained by directly scanning the IDAEs because extremely huge sets of DAEs with deterministic samples have to be solved. The estimated total computational time for solving the scanned IDAEs will be 1850 days! However, the computational time for solving the scanned Chebyshev surrogate model is 9796.97 seconds. It shows the effectiveness of the proposed computation methodology. Then, the nonlinear dynamics of a planar slider-crank mechanism with uncertain interval clearance size joint is studied in this work. The kinetics model of the revolute clearance joints is formulated under the ANCF-RN framework. Moreover, the influence of the LuGre and the modified Coulomb’s friction force models on the system’s dynamic response is investigated. By analyzing the bounds of dynamic response, the bifurcation diagrams are observed. It must be highlighted that with increasing the size of clearance, it does not automatically lead to unstable behaviors. Finally, the first order sensitivity analysis of flexible multibody systems with interval parameters is also studied in this work. The third one of a flexible mechanism with interval parameters is used to perform the sensitivity analysis.

Computing Sensitivity Analysis of Vehicle Dynamics Based on Multibody Models

2013 ◽  
Author(s):  
Yitao Zhu ◽  
Daniel Dopico ◽  
Corina Sandu ◽  
Adrian Sandu
Keyword(s):  
Sensitivity Analysis ◽  
Vehicle Dynamics ◽  
Multibody Systems ◽  
Dynamics Simulation ◽  
Adjoint Sensitivity ◽  
Analysis And Design ◽  
Systems Analysis And Design ◽  
Dynamics Of Multibody Systems ◽  
Direct Sensitivity ◽  
Vehicle Suspension System

Vehicle dynamics simulation based on multibody dynamics techniques has become a powerful tool for vehicle systems analysis and design. As this approach evolves, more and more details are required to increase the accuracy of the simulations, to improve their efficiency, or to provide more information that will allow various types of analyses. One very important direction is the optimization of multibody systems. Sensitivity analysis of the dynamics of multibody systems is essential for design optimization. Dynamic sensitivities, when needed, are often calculated by means of finite differences but, depending of the number of parameters involved, this procedure can be very demanding in terms of time and the accuracy obtained can be very poor in many cases if real perturbations are used. In this paper, several ways to perform the sensitivity analysis of multibody systems are explored including the direct sensitivity approaches and the adjoint sensitivity ones. Finally, the techniques proposed are applied to the dynamical optimization of a five bar mechanism and a vehicle suspension system.

Sign in / Sign up

Export Citation Format

Share Document