Research Papers

Optimal Control of a Formula One Car on a Three-Dimensional Track—Part 2: Optimal Control

[+] Author and Article Information
D. J. N. Limebeer

Department of Engineering Science,
University of Oxford,
Parks Road,
Oxford OX1 3PJ, UK
e-mail: david.limebeer@eng.ox.ac.uk

G. Perantoni

Department of Engineering Science,
University of Oxford,
Parks Road,
Oxford OX1 3PJ, UK
e-mail: giacomo.perantoni@gmail.com

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received December 13, 2013; final manuscript received December 16, 2014; published online January 27, 2015. Assoc. Editor: Shankar Coimbatore Subramanian.

J. Dyn. Sys., Meas., Control 137(5), 051019 (May 01, 2015) (13 pages) Paper No: DS-13-1507; doi: 10.1115/1.4029466 History: Received December 13, 2013; Revised December 16, 2014; Online January 27, 2015

The optimal control of a Formula One car on a three-dimensional (3D) track is studied. The track is described by its geodesic and normal curvatures, and its relative torsion. These curvature parameters are obtained from noisy measurement data using the optimal estimation technique described in Part 1. The optimal control calculations presented are based on the aforementioned track model and a vehicle model that is responsive to the geometric features of a 3D track. For vehicle modeling purposes, the track is treated as a plane tangent to a nearby point on the track's spine. This tangent plane moves under the car and is orthogonal to the principal normal vector m at the nearby spine point. Results are presented that compare two-dimensional (2D) and 3D minimum-lap-time results, with the two compared. The Barcelona Formula One track studied in Part 1 is used again as an illustrative example.

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Fig. 1

Side view of a Formula One car showing its mass center Mc, its geometric center Gc, and its center of pressure Cp

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Fig. 2

Plan view of a Formula One car with some of its basic geometric parameters

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Fig. 3

Differential-geometric description of a track segment T. The independent variable s represents the elapsed centerline distance traveled in the direction indicated. The track half-width is N, with ξ the car's yaw angle relative to the spine's tangent direction. The inertial reference frame given by nx, ny, and nz.

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Fig. 4

Tire force system. The car's yaw angle is ξ with respect to the Darboux frame, which is defined in terms of the vectors t and n.

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Fig. 5

Car aerodynamic maps. The drag coefficient CD is the solid (blue) curve; the down-force coefficient CL is the dotted–dashed (red) curve. The aerodynamic center of pressure is given by the dashed (magenta) curve in meters from the front axle. The “+” symbols represent measured data points.

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Fig. 6

Radau integration error. (Black) dotted–dashed line in the error bound (57) with f(2N–1)(η) = 1, with the (red) line given by log(EN) = 24 − 7.5 N.

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Fig. 7

3D view of the “Circuit de Catalunya” with all its dimensions given in meters

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Fig. 8

Approximate distances to mid corner on the Circuit de Catalunya track (in meters from the start–finish line  SF)

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Fig. 9

Elevation and geodesic curvature of the track. Distances are given in meters, while the curvature is given in radians per meter. Positive curvature is associated with right-hand bends.

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Fig. 10

Product of the geodesic curvature and the track camber angle. Positive values are indicative of positive cambering, while negative values are associated with adverse cambering.

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Fig. 11

Forces acting on a cornering car at speed u, turn radius r and road camber angle β. The car is traveling into the page and turning right.

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Fig. 12

Speeds and time difference for the 2D and 3D track. The (red) curve gives the speed of the car on the flat track, while the (blue) curve is the speed on the 3D track. The solid (black) curve is the time difference; negative values indicate that the car is faster on the 2D track.

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Fig. 16

Speed difference dashed (red) and the centripetal force in the car's z-direction solid (black). The speed difference is positive when the car is faster on the 3D track. The centripetal force provides additional down force when it is positive.

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Fig. 13

Engine and braking power as a function of distance from the start/finish line. The (red) curve illustrates the power delivered to the rear wheels, the (blue) shows the power delivered to the front wheels, while the (black) dashed plot is the sum of the two.

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Fig. 14

Racing line on a flat (red) and a 3D (blue) track. The left-hand diagram shows turns ② and ③, while the right-hand diagram is for turn ④.

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Fig. 15

Speed difference dashed (red) and the gravitational force component in the car's x-direction solid (black). The speed difference is positive when the car is faster on the 3D track. The car is traveling downhill when the gravitational force component is positive.



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