Research Papers

Minimum Maneuver Time Calculation Using Convex Optimization

[+] Author and Article Information
Julian P. Timings

e-mail: julian.timings@gmail.com

David J. Cole

e-mail: david.cole@eng.cam.ac.uk
Driver-Vehicle Dynamics Group,
Department of Engineering,
University of Cambridge,
Trumpington Street,
Cambridge, CB2 1PZ, UK

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement, and Control. Manuscript received July 29, 2011; final manuscript received January 1, 2013; published online March 28, 2013. Assoc. Editor: Alexander Leonessa.

J. Dyn. Sys., Meas., Control 135(3), 031015 (Mar 28, 2013) (9 pages) Paper No: DS-11-1234; doi: 10.1115/1.4023400 History: Received July 29, 2011; Revised January 01, 2013

The problem of calculating the minimum lap or maneuver time of a nonlinear vehicle, which is linearized at each time step, is formulated as a convex optimization problem. The formulation provides an alternative to previously used quasi-steady-state analysis or nonlinear optimization. Key steps are: the use of model predictive control; expressing the minimum time problem as one of maximizing distance traveled along the track centerline; and linearizing the track and vehicle trajectories by expressing them as small displacements from a fixed reference. A consequence of linearizing the vehicle dynamics is that nonoptimal steering control action can be generated, but attention to the constraints and the cost function minimizes the effect. Optimal control actions and vehicle responses for a 90 deg bend are presented and compared to the nonconvex nonlinear programming solution.

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Milliken, W., and Milliken, D., 1995,Race Car Vehicle Dynamics, Society of Automotive Engineers, Warrendale, PA.
Blasco-Figueroa, J., 2000, “Minimum Time Manoeuvre Based in the GG-Speed Envelope,” M.S. thesis, School of Engineering, Cranfield University, Cranfield, UK.
Gadola, M., Vetturi, D., Cambiaghi, D., and Manzo, L., 1996, “A Tool for Lap Time Simulation,” Proceedings of the SAE Motorsport Engineering Conference and Exposition, Dearborn, MI.
Fujiuka, T., and Kimura, T., 1992, “Numerical Simulation of Minimum Time Cornering Behaviour,” JSAE Rev., 13, pp. 44–51.
Hendrikx, J., Meijlink, T., and Kriens, R., 1996, “Application of Optimal Control Theory to Inverse Simulation of Car Handling,” Veh. Syst. Dyn., 26, pp. 449–461. [CrossRef]
Casanova, D., 2000, “On Minimum Time Vehicle Manoeuvring: The Theoretical Optimal Lap,” Ph.D. thesis, School of Mechanical Engineering, Cranfield University, Cranfield, UK.
Kelly, D. P., 2008, “Lap Time Simulation With Transient Vehicle and Tyre Dynamics,” Ph.D. thesis, Cranfield University School of Engineering, Cranfield, UK.
Prokop, G., 2001, “Modeling Human Vehicle Driving by Model Predictive Online Optimization,” Veh. Syst. Dyn., 35, pp. 19–53. [CrossRef]
Gerdts, M., Karrenberg, S., Muller-Beßler, B., and Stock, G., 2009, “Generating Locally Optimal Trajectories for an Automatically Driven Car,” Optim. Eng., 10, pp. 439–463. [CrossRef]
Timings, J. P., and Cole, D. J., 2011, “Efficient Minimum Manoeuvre Time Optimisation of an Oversteering Vehicle at Constant Forward Speed,” Proceedings of the American Control Conference 2011, San Francisco, CA.
Timings, J. P., and Cole, D. J., 2012, “Vehicle Trajectory Linearisation to Enable Efficient Optimisation of the Constant Speed Racing Line,” Veh. Syst. Dyn., 50(6), pp. 883–901. [CrossRef]
Maciejowski, J., 2002, Predictive Control With Constraints, Prentice-Hall, London.
Boyd, S., and Vandenberghe, L., 2004, Convex Optimization, Cambridge University Press, New York.
Wright, S. J., 1997, “Applying New Optimization Algorithms to Model Predictive Control,” Chemical Process Control-V, CACHE, AIChE Symposium Series No. 316, Vol. 93, pp. 147–155.
Rao, C. V., Wright, S. J., and Rawlings, J. B., 1998, “Application of Interior-Point Methods to Model Predictive Control,” J. Optim. Theory Appl., 99, pp. 723–757. [CrossRef]
Pacejka, H., 2006, Tyre and Vehicle Dynamics, Butterworth-Heinemann, Oxford, UK.
Gordon, T., and Best, M., 2006, “On the Synthesis of Driver Inputs for the Simulation of Closed-Loop Handling Manoeuvres,” Int. J. Veh. Des., 40, pp. 52–76. [CrossRef]
Zhou, J., Lu, J., and Peng, H., 2009, “MPC-Based Supervisory Control and Constrained Optimal Tire Force Allocation for Enhanced Vehicle Handling and Stability,” Proceedings of the IAVSD 21st International Symposium on Dynamics of Vehicles on Roads and Tracks.
IBM Corporation, 2009, IBM ILOG CPLEX V12.1: User's Manual for CPLEX, http://www-01.ibm.com/software/integration/optimization/cplex-optimizer/
Falugi, E., Kerrigan, E., and van Wyk, E., 2010, “Imperial College London Optimal Control Software User Guide,” Department of Electrical and Electronic Engineering, Imperial College London, London, UK.


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Fig. 1

The model predictive control structure

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Fig. 2

5 DoF model with associated forces, torques, and dimensions in x-y plane

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Fig. 3

Predicted vehicle trajectory and corresponding reference trajectory parameters

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Fig. 4

Strategy used to evaluate true position of the vehicle relative to lines normal to the track centerline

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Fig. 5

Geometric definitions for derivation of intrinsic vehicle-track progression expression

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Fig. 6

Initial geometric definitions for derivation of displacement change expressions, due to a change in heading angle δφ, for two consecutive predicted vehicle paths

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Fig. 7

Initial geometric definitions for derivation of displacement change expressions, due to a change in vehicle speed δu, for two consecutive predicted vehicle paths

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Fig. 8

Approximation of y∧err(k+i) through the summation of the previous lateral displacement between the vehicle and track centerline, y∧err(k+i+1|k-1), and the change in lateral displacement between successive predicted vehicle path trajectories Δy∧err(k+i), which in turn is determined from δ∧x(k+i) and δ∧y(k+i)

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Fig. 9

Combined tire force characteristics for normal load Fzj=6000 N; peak force generated on a slip-circle of radius ≈0.12

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Fig. 10

A single time-varying plane constraint based on the predicted tire operating condition (α∧j(k+i+1|k-1),κ∧j(k+i+1|k-1),F∧zj(k+i+1|k-1)) used to constrain tire force within a maximum combined tire force surface

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Fig. 11

Illustration of how the linearization of a nonlinear term in the equation of motion may be exploited by the optimizer to increase the vehicle's longitudinal acceleration

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Fig. 12

Optimal path and speed; comparison between linear MPC and NLP controllers, for a 90 deg bend. Circles represent marked 1 s intervals.

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Fig. 13

Optimal path and speed; comparison of control, yaw rate and speed histories between linear MPC and NLP controllers, during 90 deg bend

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Fig. 14

Optimal path and speed; comparison of tire usage between linear MPC and NLP controllers, during 90 deg bend




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