Research Papers

A Mathematical Model of Driver Steering Control Including Neuromuscular Dynamics

[+] Author and Article Information
Andrew J. Pick

Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK

David J. Cole

Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UKdjc13@eng.cam.ac.uk

J. Dyn. Sys., Meas., Control 130(3), 031004 (Apr 09, 2008) (9 pages) doi:10.1115/1.2837452 History: Received December 11, 2006; Revised September 10, 2007; Published April 09, 2008

A mathematical driver model is introduced in order to explain the driver steering behavior observed during successive double lane-change maneuvers. The model consists of a linear quadratic regulator path-following controller coupled to a neuromuscular system (NMS). The NMS generates the steering wheel angle demanded by the path-following controller. The model demonstrates that reflex action and muscle cocontraction improve the steer angle control and thus increase the path-following accuracy. Muscle cocontraction does not have the destabilizing effect of reflex action, but there is an energy cost. A cost function is used to calculate optimum values of cocontraction that are similar to those observed in the experiments. The observed reduction in cocontraction with experience of the vehicle is explained by the driver learning to predict the steering torque feedback. The observed robustness of the path-following control to unexpected changes in steering torque feedback arises from the reflex action and cocontraction stiffness of the NMS. The findings contribute to the understanding of driver-vehicle dynamic interaction. Further work is planned to improve the model; the aim is to enable the optimum design of steering feedback early in the vehicle development process.

Copyright © 2008 by American Society of Mechanical Engineers
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Figure 1

Linear yaw∕sideslip vehicle model

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Figure 2

Results of driving simulator measurements by Cheong (12) showing the average vehicle path during lane-change maneuvers performed by eight test subjects immediately after changes in the steering system. In (a), the steering stiffness Kst is switched between values of 3Nm∕rad, 10Nm∕rad, and 20Nm∕rad, in the order given in the figure; the steering angle ratio is fixed nrsw=16. In (b), the steering ratio nrsw is switched between values of 1, 16, and 50; the torque needed to maintain a given road wheel steer angle is fixed.

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Figure 3

Vehicle and previewed road path showing global position of vehicle and road at time kT where ypi is the road path lateral displacement and ψpi is the road path heading angle

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Figure 4

Mean path taken by test subjects through lane change and performance of LQR controller when linked directly to the vehicle. Error bars show ±1 standard deviation from the mean path taken by test subjects.

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Figure 5

Linear driver model with path-following control and NMS dynamics

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Figure 6

Cost C as a function of reflex stiffness, showing the optimum reflex stiffness for each of the three cars. Evaluated with Br=1Nms∕rad, Ka=5Nm∕rad, Gd=0, and qTα=0.01.

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Figure 7

Path-following performance with three values of cocontraction stiffness. Evaluated with Kr=5Nm∕rad, Br=1Nms∕rad, and Gd=0 for Car 1.

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Figure 8

Cost C, Eq. 11, against active stiffness Ka generated through cocontraction, showing optimum active stiffness in each of the three test cars. Evaluated with Br=1Nms∕rad, Kr=5Nm∕rad, and Gd=0.

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Figure 9

Comparison of cocontraction torque predicted by the simulation and mean cocontraction torque measured and averaged for all test subjects (data taken from Ref. 11). The model prediction is taken from Fig. 8 with qTα=0.001 and using c=1.8rad−1.

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Figure 10

Bode plot showing the true vehicle and steering system dynamics, for Car 1, G(s), and three approximations Ĝ(s)

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Figure 11

Cost C, Eq. 11, as a function of active stiffness, showing effect of reference model accuracy on optimum active stiffness. Evaluated with Br=1Nms∕rad and Kr=5Nm∕rad for Car 1.

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Figure 12

Active stiffness during ten successive maneuvers in Car 1, calculated from measured cocontraction shown in Fig. 5.15 of Ref. 2 using Eq. 12

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Figure 13

Robustness of driver model: (a) vary steering torque feedback and (b) vary steering angle gain. Evaluated with Br=1Nms∕rad, Kr=5Nm∕rad, and Ka=10Nm∕rad, reference Model II.




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