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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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. 5

Geometric definitions for derivation of intrinsic vehicle-track progression expression

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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. 3

Predicted vehicle trajectory and corresponding reference trajectory parameters

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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. 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. 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. 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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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. 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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