Research Papers

Co-Design of an Active Suspension Using Simultaneous Dynamic Optimization

[+] Author and Article Information
James T. Allison

Assistant Professor
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: jtalliso@illinois.edu

Tinghao Guo

University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: guo32@illinois.edu

Zhi Han

Senior Software Developer
MathWorks, Inc.,
Natick, MA 01760
e-mail: zhi.han@mathworks.com

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 25, 2013; final manuscript received March 23, 2014; published online June 2, 2014. Assoc. Editor: Shinji Nishiwaki.

J. Mech. Des 136(8), 081003 (Jun 02, 2014) (14 pages) Paper No: MD-13-1099; doi: 10.1115/1.4027335 History: Received February 25, 2013; Revised March 23, 2014

Design of physical systems and associated control systems are coupled tasks; design methods that manage this interaction explicitly can produce system-optimal designs, whereas conventional sequential processes may not. Here, we explore a new technique for combined physical and control system design (co-design) based on a simultaneous dynamic optimization approach known as direct transcription, which transforms infinite-dimensional control design problems into finite-dimensional nonlinear programming problems. While direct transcription problem dimension is often large, sparse problem structures and fine-grained parallelism (among other advantageous properties) can be exploited to yield computationally efficient implementations. Extension of direct transcription to co-design gives rise to new problem structures and new challenges. Here, we illustrate direct transcription for co-design using a new automotive active suspension design example developed specifically for testing co-design methods. This example builds on prior active suspension problems by incorporating a more realistic physical design component that includes independent design variables and a broad set of physical design constraints, while maintaining linearity of the associated differential equations. A simultaneous co-design approach was implemented using direct transcription, and numerical results were compared with conventional sequential optimization. The simultaneous optimization approach achieves better performance than sequential design across a range of design studies. The dynamics of the active system were analyzed with varied level of control authority to investigate how dynamic systems should be designed differently when active control is introduced.

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Friedland, B., 1996, Advanced Control System Design, Prentice-Hall, Upper Saddle River, NJ.
Reyer, J. A., Fathy, H. K., Papalambros, P. Y., and Ulsoy, A. G., 2001, “Comparison of Combined Embodiment Design and Control Optimization Strategies Using Optimality Conditions,” The Proceedings of the 2001 ASME Design Engineering Technical Conferences, Pittsburgh, PA.
Roos, F., 2007, “Towards a Methodology for Integrated Design of Mechatronic Servo Systems,” Ph.D. dissertation, Royal Institute of Technology, Stockholm, Sweden.
Li, Q., Zhang, W., and Chen, L., 2001, “Design for Control—A Concurrent Engineering Approach for Mechatronic Systems Design,” IEEE/ASME Trans. Mechatronics, 6(2), pp. 161–169. [CrossRef]
Fathy, H. K., Reyer, J. A., Papalambros, P. Y., and Ulsoy, A. G., 2001, “On the Coupling Between the Plant and Controller Optimization Problems,” The Proceedings of the 2001 American Control Conference, Arlington, VA.
Peters, D. L., Papalambros, P. Y., and Ulsoy, A. G., 2009, “On Measures of Coupling Between the Artifact and Controller Optimal Design Problems,” The Proceedings of the 2009 ASME Design Engineering Technical Conferences, San Diego, CA.
Cervantes, A., and Biegler, L. T., 1999, “Optimization Strategies for Dynamic Systems,” Encycl. Optim., 4, pp. 216–227.
Biegler, L. T., 2007, “An Overview of Simultaneous Strategies for Dynamic Optimization,” Chem. Eng. Process.: Process Intensif., 46(11), pp. 1043–1053. [CrossRef]
Hargraves, C. R., and Paris, S. W., 1987, “Direct Trajectory Optimization Using Nonlinear Programming and Collocation,” J. Guid. Control Dynam., 10(4), pp. 338–342. [CrossRef]
Ozimek, M. T., Grebow, D. J., and Howell, K. C., 2008, “Solar Sails and Lunar South Pole Coverage,” The Proceedings of the 2008 AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu, HI.
Betts, J. T., 2010, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, SIAM, Philadelphia, PA.
Xiang, Y., Arora, J. S., and K.Abdel-Malek, 2010, “Physics-Based Modeling and Simulation of Human Walking: A Review of Optimization-Based and Other Approaches,” Struct. Multidiscip. Optim., 42(1), pp. 1–23. [CrossRef]
Jung, E., Lenhart, S., and Feng, Z., 2002, “Optimal Control of Treatments in a Two-Strain Tuberculosis Model,” Discrete Contin. Dynam. Syst. Series B, 2(4), pp. 476–482. Available at: http://www.math.utk.edu/~lenhart/docs/tbfinal.pdf.
Brenan, K. E., Campbell, S. L., and Petzold, L. R., 1996, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia, PA.
Athans, M., and Falb, P., 1966, Optimal Control: An Introduction to the Theory and Its Applications, McGraw-Hill, New York.
Biegler, L. T., 2010, Nonlinear Programming Concepts, Algorithms, and Applications to Chemical Processes, SIAM, Philadelphia, PA.
Powell, D. M. J., 1978, “Algorithms for Nonlinear Constraints That Use Lagrangian Functions,” Math. Programming, 14(1), pp. 224–248. [CrossRef]
MathWorks, Inc., 2013, “Simulink Design Optimization Product Reference,” http://www.mathworks.com/products/sl-design-optimization/, retrieved Feb. 19.
Coleman, T. F., and More, J. J., 1983, “Estimation of Sparse Jacobian Matrices and Graph Coloring Problems,” SIAM J. Numer. Anal., 20(1), pp. 187–209. [CrossRef]
Rutquist, P. E., and Edvall, M. M., 2010, PROPT-MATLAB Optimal Control Software, Tomlab Optimization, Inc., Pullman, WA.
Sun, E. T., and Stadtherr, M. A., 1988, “On Sparse Finite-Difference Schemes Applied to Chemical Process Engineering Problems,” Comput. Chem. Eng., 12(8), pp. 849–851. [CrossRef]
Vasantharajan, S., and Biegler, L. T., 1990, “Simultaneous Strategies for Optimization of Differential-Algebraic Systems with Enforcement of Error Criteria,” Comput. Chem. Eng., 14(10), pp. 1083–1100. [CrossRef]
Rao, A. V., Benson, D. A., Darby, C., Patterson, M. A., Francolin, C., Sanders, I., and Huntington, G. T., 2010, “Algorithm 902: GPOPS, A Matlab Software for Solving Multiple-Phase Optimal Control Problems Using the Gauss Pseudospectral Method,” ACM Trans. Math. Software, 37(2), p. 22. [CrossRef]
Laird, C. D., Wong, A. V., and Akesson, J., 2011, “Parallel Solution of Large-Scale Dynamic Optimization Problems,” The Proceedings of the 21st European Symposium on Computer Aided Process Engineering—ESCAPE 21, Sithonia, Chalkidiki, Greece.
Allison, J. T., and Herber, D. R., 2014, “Multidisciplinary Design Optimization of Dynamic Engineering Systems,” AIAA J., 52(4), pp. 691–710. [CrossRef]
Herman, A. L., and Conway, B. A., 1996, “Direct Optimization Using Collocation Based on High-Order Gauss-Lobatto Quadrature Rules,” J. Guid. Control Dynam., 19(3), pp. 592–599. [CrossRef]
Williams, P., 2009, “Hermite-Legendre-Gauss-Lobatto Direct Transcription in Trajectory Optimization,” J. Guid. Control Dynam., 32(4), pp. 1392–1395. [CrossRef]
Allison, J. T., Kokkolaras, M., and Papalambros, P. Y., 2007, “On Selecting Single-Level Formulations for Complex System Design Optimization,” ASME J. Mech. Des., 129(9), pp. 898–906. [CrossRef]
Cramer, E. J., Dennis, J. E., Jr., Frank, P. D., Lewis, R. M., and Shubin, G. R., 1994, “Problem Formulation for Multidisciplinary Optimization,” SIAM J. Optim., 4(4), pp. 754–776. [CrossRef]
Bertsekas, D. P., and Tsitsiklis, J. N., 1997, Parallel and Distributed Computation: Numerical Methods, Athena Scientific, Belmont, MA.
Smith, M. C., and Walker, G. W., 2005, “Interconnected Vehicle Suspension,” Proc. Inst. Mech. Eng., Part D, 219(3), pp. 295–307. [CrossRef]
Schroer, R. T., Boggess, M. J., Bachmann, R. J., Quinn, R. D., and Ritzmann, R. E., 2004, “Comparing Cockroach and Whegs Robot Body Motions,” The Proceedings of the 2004 IEEE International Conference on Robotics and Automation, New Orleans, LA, IEEE.
Lee, J., Lamperski, A., Schmitt, J., and Cowan, N., 2006, “Task-Level Control of the Lateral Leg Spring Model of Cockroach Locomotion,” Fast Motions Biomech. Rob., 340, pp. 167–188. [CrossRef]
Nishigaki, H., and Kawashima, K., 1998, “Motion Control and Shape Optimization of a Suitlike Flexible Arm,” Struct. Optim.,15(3–4), pp. 163–171. [CrossRef]
Trease, B. P., 2008, “Topology Synthesis of Compliant Systems with Embedded Actuators and Sensors,” Ph.D. dissertation, University of Michigan, Ann Arbor, MI.
Carmichael, D. G., 1990, “Structural Optimization and System Dynamics,” Struct. Multidiscip. Optim., 2(2), pp. 105–108. [CrossRef]
Fares, M. E., Youssif, Y. G., and Hafiz, M. A., 2005, “Multiobjective Design and Control Optimization for Minimum Thermal Postbuckling Dynamic Response and Maximum Buckling Temperature of Composite Laminates,” Struct. Multidiscip. Optim., 30(2), pp. 89–100. [CrossRef]
Fathy, H. K., 2003, “Combined Plant and Control Optimization: Theory, Strategies and Applications,” Ph.D. dissertation, University of Michigan, Ann Arbor, MI.
Allison, J. T., and Nazari, S., 2010, “Combined Plant and Controller Design Using Decomposition-Based Design Optimization and the Minimum Principle,” The Proceedings of the 2010 ASME Design Engineering Technical Conferences, Montreal, Quebec, Canada.
Peters, D. L., Papalambros, P. Y., and Ulsoy, A. G., 2013, “Sequential Co-Design of an Artifact and Its Controller via Control Proxy Functions,” Mechatronics, 23(4), pp. 409–418. [CrossRef]
Papalambros, P. Y., and Wilde, D., 2000, Principles of Optimal Design: Modeling and Computation, 2nd ed., Cambridge University Press, Cambridge, UK.
Deshmukh, A., and Allison, J. T., 2013, “Design of Nonlinear Dynamic Systems Using Surrogate Models of Derivative Functions,” The Proceedings of the 2013 ASME Design Engineering Technical Conferences, Portland, OR.
Williams, P., and Trivailoa, P., 2005, “Optimal Parameter Estimation of Dynamical Systems Using Direct Transcription Methods,” Inverse Probl. Sci. Eng., 13(4), pp. 377–409. [CrossRef]
Reyer, J. A., and Papalambros, P. Y., 2000, “An Investigation into Modeling and Solution Strategies for Optimal Design and Control,” The Proceedings of the 2000 ASME Design Engineering Technical Conferences, Baltimore, MD.
Allison, J. T., 2013, “Co-Design of an Active Automotive Suspension Using Direct Transcription,” http://www.mathworks.us/matlabcentral/fileexchange/40504
Kasturi, P., and Dupont, P., 1998, “Constrained Optimal Control of Vibration Dampers,” J. Sound Vib., 215(3), pp. 499–509. [CrossRef]
Gobbi, M., 2001, “Analytical Description and Optimization of the Dynamic Behaviour of Passively Suspended Road Vehicles,” J. Sound Vib., 245(3), pp. 457–481. [CrossRef]
He, Y., and McPhee, J., 2005, “Multidisciplinary Design Optimization of Mechatronic Vehicle With Active Suspensions,” J. Sound Vib., 283(1), pp. 217–241. [CrossRef]
Allison, J. T., 2008, “Optimal Partitioning and Coordination Decisions in Decomposition-Based Design Optimization,” Ph.D. dissertation, University of Michigan, Ann Arbor, MI.
Fathy, H. K., Papalambros, P. Y., Ulsoy, A. G., and Hrovat, D., 2003, “Nested Plant/Controller Optimization With Application to Combined Passive/Active Automotive Suspensions,” The Proceedings of the 2003 American Control Conference, Denver, CO, IEEE.
Bourmistrova, A., Storey, I., and Subic, A., 2005, “Multiobjective Optimisation of Active and Semi-Active Suspension Systems With Application of Evolutionary Algorithm,” The Proceedings of the 2005 International Conference on Modelling and Simulation, Melbourne, Australia.
Alyaqout, S. F., Papalambros, P. Y., and Ulsoy, A. G., 2007, “Combined Design and Robust Control of a Vehicle Passive/Active Suspension,” The Proceedings of the 2007 European Control Conference, Kos, Greece.
Verros, G., Natsiavas, S., and Papadimitriou, C., 2005, “Design Optimization of Quarter-Car Models With Passive and Semi-Active Suspensions Under Random Road Excitation,” J. Vib. Control, 11(5), pp. 581–606. [CrossRef]
Shigley, J., Mischke, C., and Budynas, R., 2003, Mechanical Engineering Design, McGraw-Hill, New York.
Azarm, S., 1982, “An Interactive Design Procedure for Optimization of Helical Compression Springs,” University of Michigan, Technical Report No. UM-MEAM-82-7.
Stoicescu, A., 2009, “On the Optimal Design of Helical Springs of an Automobile Suspension,” UPB Sci. Bull., Series D71(1), pp. 81–94. Available at: http://scientificbulletin.upb.ro/rev_docs_arhiva/full1052.pdf.
Sayers, M. W., and Karamihas, S. M., 1998, The Little Book of Profiling, University of Michigan Transportation Research Institute, Ann Arbor, MI.
Dixon, J. C., 2007, The Shock Absorber Handbook, 2nd ed., John Wiley & Sons Ltd., Chichester, UK.
Lion, A., and Loose, S., 2002, “A Thermomechanically Coupled Model for Automotive Shock Absorbers: Theory, Experiments and Vehicle Simulations on Test Tracks,” Vehicle System Dynamics, 37(4), pp. 241–261. [CrossRef]
Bose, A. G., 1990, “Wheel Assembly Suspending,” US Patent No. 4,960,290, October.
Jones, W. D., 2005, “Easy Ride: Bose Corp. Uses Speaker Technology to Give Cars Adaptive Suspension,” IEEE Spectrum42(5), pp. 12–14. [CrossRef]
Allison, J. T., 2013, “Engineering System Co-Design with Limited Plant Redesign,” Eng. Optim., 46(2), pp. 200–217. [CrossRef]
Inman, D. J., 2014, Engineering Vibrations, 4th ed. Pearson Education, Inc., Upper Saddle River, NJ.
Wang, F. C., 2001, “Design and Synthesis of Active and Passive Vehicle Suspensions,” Ph.D. dissertation, University of Cambridge, Cambridge, UK.
Thite, A. N., 2012, “Development of a Refined Quarter Car Model for the Analysis of Discomfort Due to Vibration,” Adv. Acoust. Vib., 2012, p. 863061. [CrossRef]
Allison, J. T., Khetan, A., and Lohan, D. J., 2013, “Managing Variable-Dimension Structural Optimization Problems Using Generative Algorithms,” The Proceedings of the 10th World Congress on Structural and Multidisciplinary Optimization (WCSMO), Orlando, FL.
Guo, T., and Allison, J. T., 2013, “On the Use of MPCCs in Combined Topological and Parametric Design of Genetic Regulatory Circuits,” The Proceedings of the 10th World Congress on Structural and Multidisciplinary Optimization (WCSMO), Orlando, FL.
Clune, J., and Lipson, H., 2011, “Evolving 3D Objects with a Generative Encoding Inspired by Developmental Biology,” The Proceedings of the 2011 European Conference on Artificial Life, Paris, France.
Campbell, M. I., Rai, R., and Kurtoglu, T., 2012, “A Stochastic Tree-Search Algorithm for Generative Grammars,” ASME J. Comput. Inform. Sci. Eng., 12(3), p. 031006. [CrossRef]


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Fig. 1

Illustration of continuity defect ζ1 between time segments 1 and 2 using the multiple shooting method

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Fig. 2

Conceptual solution trajectories through design and state subspaces

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Fig. 3

Analysis structure of (a) DT and (b) DT for co-design

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Fig. 4

DT Jacobian structure: (a) dense and (b) sparse plant constraints

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Fig. 5

Quarter-car vehicle suspension model

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Fig. 6

Helical compression spring with squared ground ends

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Fig. 7

Typical nonlinear damping curve

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Fig. 8

Sectional view of a single-tube telescopic damper

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Fig. 9

Sectional view of the piston compression valve

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Fig. 10

Damper heat transfer model

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Fig. 11

Structure of the plant constraint Jacobian

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Fig. 12

System responses for sequential optimization. (a) Sprung mass response to ramp input. (b) Sprung mass response to rough road input. (c) Control force (ramp input). (d) Control force (rough road input)

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Fig. 13

System responses for simultaneous optimization. (a) Sprung mass response to ramp input. (b) Sprung mass response to rough road input. (c) Control force (ramp input). (d) Control force (rough road input)

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Fig. 14

Pareto sets illustrating the tradeoff between maximum control force and system objective function value (both sequential and simultaneous solution approaches)




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