Simultaneous Optimal System and Controller Design for Multibody Systems with Joint Friction using Direct Sensitivities

Real-world multibody systems are often subject to phenomena like friction, joint clearances, and external events. These phenomena can significantly impact the optimal design of the system and its controller. This work addresses the gradient-based optimization methodology for multibody dynamic systems with joint friction using a direct sensitivity approach for gradient computation. After a thorough review of various friction models developed over the years, the Brown McPhee model has been found to be the most suitable for the study due to its accuracy for dynamic simulation and its compatibility with sensitivity analysis. The methodology supports co-design of the system and its controller, which is especially relevant for applications like robotics and servo-mechanical systems where the actuation and the design are highly dependent on each other. Numerical results are obtained using a new implementation of the MBSVT (Multi-Body Systems at Virginia Tech) software package; MBSVT 2.0 is reprogrammed in Julia for ease of implementation while maintaining high computational efficiency. Three case studies are provided to demonstrate the attractive properties of simultaneous optimal design and control approach for certain applications.

[1]  J. Ambrósio,et al.  Contact-impact events with friction in multibody dynamics: Back to basics , 2023, Mechanism and Machine Theory.

[2]  G. Steinebach.  Construction of Rosenbrock–Wanner method Rodas5P and numerical benchmarks within the Julia Differential Equations package , 2023, BIT Numerical Mathematics.

[3]  Adrian Sandu,et al.  Computation of Direct Sensitivities of Spatial Multibody Systems with Joint Friction , 2022, Journal of Computational and Nonlinear Dynamics.

[4]  Paweł Malczyk,et al.  Adjoint method for optimal control of multibody systems in the Hamiltonian setting , 2021 .

[5]  H. Elmqvist,et al.  Modia - Equation Based Modeling and Domain Specific Algorithms , 2021, Proceedings of 14th Modelica Conference 2021, Linköping, Sweden, September 20-24, 2021.

[6]  Alan Edelman,et al.  Composing Modeling And Simulation With Machine Learning In Julia , 2021, 2022 Annual Modeling and Simulation Conference (ANNSIM).

[7]  Viral B. Shah,et al.  High-performance symbolic-numerics via multiple dispatch , 2021, ACM Commun. Comput. Algebra.

[8]  Suyong Kim,et al.  Stiff Neural Ordinary Differential Equations , 2021, Chaos.

[9]  Viral B. Shah,et al.  ModelingToolkit: A Composable Graph Transformation System For Equation-Based Modeling , 2021, ArXiv.

[10]  Daniel Dopico,et al.  Augmented Lagrangian index-3 semi-recursive formulations with projections , 2020, Multibody System Dynamics.

[11]  Ehsan Noroozinejad Farsangi,et al.  Simultaneous optimization approach for combined control–structural design versus the conventional sequential optimization method , 2020, Structural and Multidisciplinary Optimization.

[12]  Paweł Malczyk,et al.  Hamiltonian direct differentiation and adjoint approaches for multibody system sensitivity analysis , 2020, International Journal for Numerical Methods in Engineering.

[13]  Michael J. Alexander-Ramos,et al.  Simultaneous Combined Optimal Design and Control Formulation for Aircraft Hybrid-Electric Propulsion Systems , 2020 .

[14]  G. Wellein,et al.  A Recursive Algebraic Coloring Technique for Hardware-efficient Symmetric Sparse Matrix-vector Multiplication , 2020, ACM Trans. Parallel Comput..

[15]  Ali Ramadhan,et al.  Universal Differential Equations for Scientific Machine Learning , 2020, ArXiv.

[16]  Alan Edelman,et al.  Sparsity Programming: Automated Sparsity-Aware Optimizations in Differentiable Programming , 2019 .

[17]  Vaibhav Dixit,et al.  A Comparison of Automatic Differentiation and Continuous Sensitivity Analysis for Derivatives of Differential Equation Solutions , 2018, 2021 IEEE High Performance Extreme Computing Conference (HPEC).

[18]  Filipe Marques,et al.  Modeling and analysis of friction including rolling effects in multibody dynamics: a review , 2018, Multibody System Dynamics.

[19]  Moritz Diehl,et al.  CasADi: a software framework for nonlinear optimization and optimal control , 2018, Mathematical Programming Computation.

[20]  Edward J. Haug,et al.  Simulation of spatial multibody systems with friction , 2018 .

[21]  Qiang Tian,et al.  A comprehensive survey of the analytical, numerical and experimental methodologies for dynamics of multibody mechanical systems with clearance or imperfect joints , 2018 .

[22]  Adrian Sandu,et al.  Adjoint sensitivity analysis of hybrid multibody dynamical systems , 2018, Multibody System Dynamics.

[23]  Adrian Sandu,et al.  Modeling and sensitivity analysis methodology for hybrid dynamical system , 2017, Nonlinear Analysis: Hybrid Systems.

[24]  John McPhee,et al.  A Continuous Velocity-Based Friction Model for Dynamics and Control With Physically Meaningful Parameters , 2016 .

[25]  Miles Lubin,et al.  Forward-Mode Automatic Differentiation in Julia , 2016, ArXiv.

[26]  Barak A. Pearlmutter,et al.  Automatic differentiation in machine learning: a survey , 2015, J. Mach. Learn. Res..

[27]  Barry Smith,et al.  Sparse Matrix-Matrix Products Executed Through Coloring , 2015, SIAM J. Matrix Anal. Appl..

[28]  Christopher G. Atkeson,et al.  Optimization based full body control for the atlas robot , 2014, 2014 IEEE-RAS International Conference on Humanoid Robots.

[29]  Adrian Sandu,et al.  MBSVT: Software for Modeling, Sensitivity Analysis, and Optimization of Multibody Systems at Virginia Tech , 2014 .

[30]  James T. Allison,et al.  Co-design of an active suspension using simultaneous dynamic optimization , 2014, DAC 2011.

[31]  Alfonso Callejo Goena,et al.  Dynamic Response Optimization of Vehicles through Efficient Multibody Formulations and Automatic Differentiation Techniques , 2014 .

[32]  J. G. Jalón,et al.  Multibody dynamics with redundant constraints and singular mass matrix: existence, uniqueness, and determination of solutions for accelerations and constraint forces , 2013 .

[33]  J. Fraczek,et al.  On the unique solvability of a direct dynamics problem for mechanisms with redundant constraints and Coulomb friction in joints , 2011 .

[34]  Karl-Erik Årzén,et al.  Modeling and optimization with Optimica and JModelica.org - Languages and tools for solving large-scale dynamic optimization problems , 2010, Comput. Chem. Eng..

[35]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[36]  E. Pennestrì,et al.  Multibody dynamics simulation of planar linkages with Dahl friction , 2007 .

[37]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

[38]  R. Serban,et al.  SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers , 2005, TOMS.

[39]  Shengtai Li,et al.  Sensitivity analysis of differential-algebraic equations and partial differential equations , 2005, Comput. Chem. Eng..

[40]  J. C. García Orden,et al.  Analysis of Joint Clearances in Multibody Systems , 2005 .

[41]  Elizabeth R. Jessup,et al.  A Technique for Accelerating the Convergence of Restarted GMRES , 2005, SIAM J. Matrix Anal. Appl..

[42]  Russ Tedrake,et al.  Efficient Bipedal Robots Based on Passive-Dynamic Walkers , 2005, Science.

[43]  J. Ambrósio,et al.  Dynamic Analysis for Planar Multibody Mechanical Systems with Lubricated Joints , 2004 .

[44]  John McPhee,et al.  A Regularized Contact Model with Asymmetric Damping and Dwell-Time Dependent Friction , 2004 .

[45]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[46]  Paul L. Fackler,et al.  Applied Computational Economics and Finance , 2002 .

[47]  James Kennedy,et al.  Particle swarm optimization , 2002, Proceedings of ICNN'95 - International Conference on Neural Networks.

[48]  L. Petzold,et al.  Software and algorithms for sensitivity analysis of large-scale differential algebraic systems , 2000 .

[49]  E. Hairer,et al.  Stiff differential equations solved by Radau methods , 1999 .

[50]  P. I. Barton,et al.  Efficient sensitivity analysis of large-scale differential-algebraic systems , 1997 .

[51]  J. Nocedal,et al.  A Limited Memory Algorithm for Bound Constrained Optimization , 1995, SIAM J. Sci. Comput..

[52]  Carlos Canudas de Wit,et al.  A new model for control of systems with friction , 1995, IEEE Trans. Autom. Control..

[53]  Alan C. Hindmarsh,et al.  Description and use of LSODE, the Livermore Solver for Ordinary Differential Equations , 1993 .

[54]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[55]  E. J. Haug,et al.  Computer aided kinematics and dynamics of mechanical systems. Vol. 1: basic methods , 1989 .

[56]  L. Petzold.  Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations , 1983 .

[57]  L. Petzold.  A description of dassl: a differential/algebraic system solver , 1982 .

[58]  Jorge J. Moré,et al.  User Guide for Minpack-1 , 1980 .

[59]  Adrian Sandu,et al.  Direct and Adjoint Sensitivity Analysis of Ordinary Differential Equation Multibody Formulations , 2014, Journal of Computational and Nonlinear Dynamics.

[60]  J. Ambrósio,et al.  Dynamics of Multibody Systems With Spherical Clearance Joints , 2005 .

[61]  Steven S. Muchnick,et al.  Advanced Compiler Design and Implementation , 1997 .

[62]  Gerd Steinebach,et al.  Order-reduction of ROW-methods for DAEs and method of lines applications , 1995 .

[63]  F. Potra,et al.  A Topology Based Approach for Exploiting Sparsity in Multibody Dynamics , 1995 .

[64]  C. Bischof,et al.  ADIFOR-Generating Derivative Codes from Fortran Programs , 1992 .

[65]  C. Kelley.  Solving Nonlinear Equations with Newton's Method , 1987 .

[66]  D. J. Bell,et al.  Numerical Methods for Unconstrained Optimization , 1979 .

[67]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..