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Research Papers

# Optimal Control of a Formula One Car on a Three-Dimensional Track—Part 1: Track Modeling and Identification

[+] Author and Article Information
Giacomo Perantoni

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

David J. N. Limebeer

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

If a × b = c, then R(a × b) = Rc. If $a'=Ra,b'=Rb$ and $c'=Rc$, then $a'×b'=c'$, since the relative orientations and lengths of these vectors have not changed. Consequently (Ra) × (Rb) = Rc = R(a × b).

Throughout this paper we will assume that we are dealing with constant-speed curves and so derivatives with respect to time t and arc length s are equivalent; $(ds/dt)=1$. We will use a dot to denote derivatives with respect to either variable.

Interpolation can be used to enforce this condition if the number of measurement points for the two boundaries is different.

Contrary to accepted convention we use χ to represent the state vector and υ to denote the control vector. This notation is used to avoid confusion with the first translational component x of a position vector, and with the longitudinal velocity u of a vehicle as is used in Part 2 [30].

The forward Euler scheme for finite differences can be used for x0, whereas xf has to be defined using a backward finite-differentiation rule.

The World Geodetic System is a standard for use in cartography and geodesy. It comprises a standard coordinate frame for the Earth, a datum or reference ellipsoid for altitude data and a gravitational equipotential surface (the geoid) that defines nominal sea level. The latest revision is WGS 84 that was last revised in 2004.

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 August 7, 2014; published online January 27, 2015. Assoc. Editor: Shankar Coimbatore Subramanian.

J. Dyn. Sys., Meas., Control 137(5), 051018 (May 01, 2015) (11 pages) Paper No: DS-13-1506; doi: 10.1115/1.4028253 History: Received December 13, 2013; Revised August 07, 2014; Online January 27, 2015

## Abstract

The identification of three-dimensional (3D) race track models from noisy measured GPS data is treated as a problem in the differential geometry of curves and surfaces. Curvilinear coordinates are adopted to facilitate the use of the track model in the solution of vehicular optimal control problems. Our proposal is to model race tracks using a generalized Frenet–Serret apparatus, so that the track is specified in terms of three displacement-dependent curvatures and two edge variables. The optimal smoothing of the curvature and edge variables is achieved using numerical optimal control techniques. Track closure is enforced through the boundary conditions associated with the optimal control problem. The Barcelona formula one track is used as an illustrative example.

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## Figures

Fig. 1

A ribbon R generated by a spine curve C. The spine is described in terms of its torsion and curvature, while the camber of the ribbon comes from the twist angle ν(s).

Fig. 8

Distances (in meters) to the right-hand (solid black) and left-hand (dashed red) boundaries of the track as measured from the track's spine

Fig. 2

Three-dimensional ribbon for the extended Lemniscate of Bernoulli. The solid and dashed lines are the optimized boundaries and centerline, respectively. The dots represent the noisy simulated boundary-point measurements on which the solution is based. The ribbon gray scale represents the z-coordinate of the optimized centerline and is calibrated according to the color bar given.

Fig. 3

Parameters for the extended lemniscate ribbon. The theoretically correct values are the red dashed curves, while the optimal estimates are the solid curves. The color convention is the same as that given in Table 1. The top plot shows the relative torsion Ωx, the second the normal curvature Ωy, and the third the geodesic curvature Ωz. The distances between the ribbon boundaries and its spine are shown in the bottom diagram. The theoretical boundaries come from Eq. (40).

Fig. 4

Sums of the left-hand and right-hand boundary errors for different weight selections. The color convention is the same as that used in Table 1.

Fig. 6

Magnified views of corner 10 (left-hand side), and corners 14 and 15 (right-hand side). The measured boundary points are shown as black dots, with the dashed red curve the spine curve estimate.

Fig. 7

Estimated curvature of the Circuit de Catalunya. The plots show, respectively, the relative torsion Ωx, the normal curvature Ωy, and the geodesic curvature Ωz (in radians per meter). The optimal model-based estimates are shown as the solid black curves, while the finite-difference approximations are the red dashed curves.

Fig. 5

Ribbon-based model of the Circuit de Catalunya. The solid lines are the optimized track boundaries. The track spine's z-coordinate is shown using a gray scale.

Fig. 9

Error estimates for the track boundaries (as calculated in the ribbon tangent plane in meters). The solid black curve corresponds to the right-hand side boundary, while the dashed red curve corresponds to the left-hand boundary.

## Errata

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