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Minimum manoeuvre time of a nonlinear vehicle atconstant forward speed using convex optimisation

Julian P. Timings and David J. ColeDepartment of Engineering, University of Cambridge

Trumpington StreetCambridge, Cambs, CB2 1PZ, UK

Phone: +44 (0)1223 765201Fax: +44 (0)1223 332662

E-mail: david.cole@eng.cam.ac.uk

The aim of this research is to improve understanding of driver-vehicle dynamics in a rac-ing environment, through the development of computational tools that reflect the controlbehaviour of the human driver. The most recent attempts to model racing driver control be-haviour have resulted in computationally intensive nonlinear optimisation problems, whichdiscourages application to vehicle parametric study. To this extent, a computationally ef-ficient LTV MPC steering controller has been developed, able to push a nonlinear vehiclemodel to the limit of its handling capabilities. Additionally, a path optimisation algorithm,initially applied only to lateral control with constant vehicle forward speed, has been estab-lished which seeks to maximise vehicle progression as a function of time. The technique isdirectly applicable to the motorsport problem of minimising lap time.

Topics / Motorsport, Driver Behaviour and Driver Model, Driver-Vehicle System

1. INTRODUCTION

Research into the driver / vehicle interface hasbeen ongoing and remains a subject of much inter-est within the automotive field. In particular, acomprehensive understanding of this interaction isof considerable benefit to the motorsport industry.The research presented here focuses on enhancing ourknowledge of driver-vehicle dynamics during limit-handling manoeuvres.

The task faced by a racing driver is to find theoptimal balance of steer, brake and throttle controlsthat allows the car to traverse the circuit in the leastamount of time. Two requirements need to be con-sidered when searching for this optimum vehicle tra-jectory. The first requirement is to maximise thevelocity of the vehicle at all points throughout themanoeuvre, limited by the available tyre force andmaximum power produced by the engine. The sec-ond, potentially conflicting requirement, is to selecta path so that, given the track boundaries, the totaldistance travelled is minimised. A compromise be-tween these requirements is therefore sought whichresults in vehicle manoeuvre time being minimised.Previous research into this area can broadly be splitinto two categories of approach; path following suchas that used by Thommyppillai et al. [1] and nonlin-ear optimisation with some form of track boundaryconstraint. The former, relies on the concept that theoptimum racing line can be extracted from real worldtelemetry data and by accurately tracking this pre-

defined line while trying to maximise vehicle speedresults in an optimum lap. By eliminating path gen-eration from the problem Thommyppillai et al. areable to use Linear Optimal Preview Control Theory[2] to generate, o!-line, a set of control schemes opti-mised for various dynamic equilibrium vehicle states.These are then employed by the driver depending onthe vehicle operating condition. Such an approachhas the advantage of being more amenable to com-puter solution, however could actually restrict max-imising vehicle performance as it does not permit adriver to adjust his/her racing line depending on thevehicle’s set-up. In contrast to this approach, byforming the lap time minimisation problem as oneof nonlinear optimisation, it is possible to computeboth the optimum path and speed trajectories on-line. Such methods have successfully been developedand applied by a number of researchers [3, 4, 5, 6].However the fact that solving the required nonlinearoptimisation problem is extremely computational in-tensive and convergence to a solution is often notguaranteed, may explain the lack of published mate-rial detailing parametric studies.

Common to both types of approach are the inclu-sion of a number of human characteristics considered,to some extent, to represent the control behaviour ofthe human driver. These include: (i) preview infor-mation, designed to reflect the drivers ability see theroad/track ahead and make anticipatory control de-cisions; (ii) adaptive control, to represent a drivers

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acute ability to adapt rapidly to varying plant dy-namics or sudden alterations in operating conditionsand; (iii) an internal mental model of the vehicledynamics used to estimate vehicle conditions at fu-ture points in time [7]. The research presented hereuses these concepts and builds on the works of Keenand Cole [8] to develop a computationally-e"cientand robust technique for optimising the target patharound the lap of a racing circuit. In doing so wesuppose the driver model itself can be formed as atime-varying Model Predictive Control (MPC) con-troller. Focus here will the be on optimum steeringcontrol at constant forward speed.

The next section starts by detailing the nonlin-ear vehicle model used for both the vehicle plant andinternal human model. We then go on to outline thestructure of the MPC controller and the formula-tion of the optimal control problem, including theminimum manoeuvre-time algorithm that results inan e"cient computational problem. The results ofa number of closed-loop driver simulations will nextbe presented before finishing with some conclusionsand future work.

2. VEHICLE MODEL

The model chosen to represent the lateral dynam-ics of the vehicle is the standard yaw/sideslip rep-resentation, often referred to as a simple car modelor bicycle model [9]. An illustration of the vehiclemodel is given in Fig. 1. The vehicle is assumedto have a steering system that acts in such a waythat a hand or steering wheel angle, !sw, results ina front road wheel angle, !, proportional to one overthe steering ratio, G. Assuming small steer angles,meaning the driving force, Fx, required to keep theforward speed constant remains small with respectto the tyre lateral force, the dynamic equations ofmotion for lateral velocity, v, and yaw rate, r, read:

Mt(v + ur) = !Fyf ! Fyr (1)Izz r = bFyr ! aFyf (2)

Fig. 1: Simple car model with associated forces anddimensions

The lateral tyre slip, ", is defined as the ratioof the lateral and longitudinal velocity of the tyre.Thus, for the front tyre it can be shown to be givenby

"f =v + ar

u! !sw

G(3)

and for the rear

"r =v ! br

u(4)

The lateral tyre force characteristics are assumed tovary with " according to the well known Magic For-mula expression:

Fyi = 2Di sin (Cyi arctan(Byi!i ! Eyi(Byi!i ! arctan(Byi!i))))(5)

In order to account for the tyre nonlinearities, thetyre dynamics can be approximated with a Linear-Time-Varying (LTV) model. By linearising the sys-tem at the current operating point, a local modelwhich accurately represents the tyre dynamics withinthe neighbourhood of that operating point can befound. Choosing front and rear lateral slip as set-point parameters allows the tyre characteristics tobe linearised using a Taylor Series expansion abouteach setpoint. Collecting only first order terms en-ables the lateral force of the tyres to be described bythe simple LTV expression:

Fyi(t) = Ciy(t)"i(t) + Diy(t) (6)where Ciy(t) and Diy(t) denote the time-varying lat-eral slip sti!ness and tyre force intercept at zero sliprespectively [8].

Utilising the results of Pick and Cole [10], it’s pro-posed that the vehicle model to be controlled com-prises of the vehicle dynamics coupled with the driver’sbandwidth-limiting Neuromuscular System (NMS) dy-namics. The NMS can be represented as a second or-der system acting on the steering wheel angle inputto the vehicle, given by

!sw + 2#n$n!sw + $2n!sw = $2

n!com (7)where !com is the steering wheel angle commandedfrom the driver’s brain and !sw now considered theoutput of the driver’s NMS. #n and $n denote thedamping ratio and natural frequency of the NMS re-spectively.

Coupling the NMS model and vehicle dynamicsallows the system to be represented using the discretetime state-space description:

x(k+1) = A(k)x(k) + B(k)#!com(k) + E(k) (8)

z(k) = C(k)x(k) (9)where x(k) is the vehicle system states, E(k) containsthe tyre force intercept terms from Equation 6 andthe steering input command has been split into thesum of the steering command from the previous timestep, now a system state, and the change in steeringcommand during the present step.

The chassis, tyre and NMS parameter set used inthe implementation of the vehicle system are givenin Table 1.

3. DRIVER MODEL

The driver model is built around a model pre-dictive or receding horizon controller, taking the ba-sic structure shown in Fig. 2. As opposed to the

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controller generating a fully optimal nonlinear con-trol law, which of great computational complexity,Maciejowski [11] demonstrates that by using a lin-ear internal model together with constraints imposedas linear inequalities, allows the constrained MPCproblem to be setup as a convex Quadratic Program-ming (QP) problem. The convexity of the objectivefunction guarantees termination of the optimisationproblem, avoiding problems with multiple local min-ima commonly associated with constrained nonlinearoptimisation. An additional advantage of posing theproblem in this way is that solution techniques forconvex QP problems are well established, with nu-merous e"cient solvers available.

Neuro-muscularmodel

Fig. 2: The predictive controller scheme

Here, the internal model approximates the non-linear vehicle dynamics using the LTV vehicle modeldetailed in §2. Use is also made of the variable-model-preview concept detailed by Keen and Cole [8].Many of driver controllers presented in the literature,[1, 12, 13], fix the internal/prediction model and con-trol input over the horizon and iterate these to esti-mate the future state and output trajectories, x(k+i)

and z(k+i) respectively. Typically in MPC, at eachtime step only the first element of the computed con-trol sequence is used for control of the vehicle plantand the remainder of the solution is discarded. How-ever, as presented in [8], by using variable-model-

Table 1: Parameters, symbols and values for vehicle,tyres and driver’s neuromuscular system.

Parameter Symbol Value

Vehicle mass Mt 1050 kgVehicle yaw inertia Izz 1500 kgm2

CoG to front axle distance a 0.92 mCoG to rear axle distance b 1.38 mSteer to road wheel angle ratio G 17Sti!ness factor (per tyre) Byi 17.5Shape factor (per tyre) Cyi 1.68Peak factor (per tyre) Dyi 3900 NCurvature factor (per tyre) Eyi 0.6NMS damping ratio "n 0.7NMS natural frequency #n 18.9 rads!1

preview an accurate estimation of the vehicle’s futurestate trajectories can be determined by using the dis-carded control commands from the previous cycle.Due to the time-varying nature of Equations 8 and9, the prediction results in a sequence of linearisedvehicle models that approximate the expected dy-namics of the vehicle at each future preview point.

3.1 Banded MPC formulationTo further improve computational e"ciency a dif-

ferent MPC formulation is adopted. The predictedstates, x(k+i), are not eliminated from the problemand are instead left as variables to be found by theQP solver. In doing so the additional equality con-straints are imposed:

x(k+i+1) = A(k+i)x(k+i) +B(k+i)#!com(k+i) + E(k+i)

(10)for i = 1 . . . Np where Np is the number of previewpoints. The QP problem can now be written in itsgeneral form

min!

J(%) =12%T$% + &T % ($ = $T " 0) (11)

subject to%% = ' (12)

and&% # $ (13)

where we chose the variables to be ordered such that

% =

!

"""""""""#

#!com(k)

x(k+1)

#!com(k+1)

x(k+2)...

#!com(k+Np!1)

x(k+Np)

$

%%%%%%%%%&

(14)

On the face of it introducing additional variablesinto the problem is likely to slow the computationsdown. However doing so results in the remaining ma-trices of Equations 11-13 having a banded structurewhere all their non-zero entries lie close to the prin-cipal diagonal, see [11] for details. Rao et al. [14]demonstrate how the bandedness can be exploitedto speed up the factorization involved when solvingQP problems. Rao et al. go on to explain how thebanded structure is particularly advantageous whenusing long horizons (as is the case here) since compu-tational time increases approximately linearly withNp as opposed to cubically as is the case when usinga densely structured MPC scheme.

3.2 Path optimisationAn alternative, but not equivalent, way to min-

imise achievable manoeuvre times is to try and max-imise the distance travelled in a fixed amount of time[5, 15]. With time as the independent variable, thetask here is to express the incremental distance dsr,travelled along a reference line, corresponding to theincremental distance ds = V T $ uT travelled by the

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vehicle in time T . Maximising dsr therefore max-imises the progression of the vehicle along the track.For the present case the reference line is consideredto be the centreline of the track with all distanceevaluations made with respect to lines normal to thecentreline. Fig. 3 sets out the intrinsic geometric def-initions used to derive the quadratic objective func-tion J which maximises vehicle progression along thetrack centreline.

predicted vehicle trajectory

track centreline

Fig. 3: Geometric definitions for derivation of intrin-sic vehicle-track progression expression

From Fig. 3 it is possible to derive the follow-ing expression for the extra distance travelled alongthe track centreline reference in terms of the head-ing angle and lateral displacement errors between thepredicted vehicle and track reference path.

dsr ! ds = yerrd(r(err

2

2+ yerrd(r ! rrd(r

(err2

2

$ yerrd(r ! rrd(r(err

2

2(15)

provided (i) "err2

2 % 1 and (ii) yerr % rr. In or-der to maximise the distance dsr ! ds, it is conve-nient to introduce a constant pseudo distance, xp

[15], as a target distance in which the solver tries toget dsr ! ds to track, thus maximising dsr. In linewith Equation 11 the following candidate quadraticobjective function is proposed, taking into accountpossible penalisations of steering e!ort:

J(k) =Np'

i=1

(q1

)yerr(k+i)d(r(k+i) ! xp

*2

! q2rr(k+i)d(r(k+i)(err(k+i)

2

2

+

+Np!1'

i=0

&#!com(k+i)&2R (16)

which replicates Equation 15 providedxp ' yerr(k+i)d(r(k+i). To ensure equal contribu-tions from the path optimisation terms when min-imising the objective function, the weights q1 and q2

are defined asq2 = q12xp (17)

To be able to use Equation 16 as an objectivefunction within the framework of the steering con-troller developed thus far, it is necessary that theterms yerr and (err can be linearly reproduced fromthe system states, as state-space outputs. As men-tioned above, it is possible to use the discarded con-trol commands from the previous optimisation toform an estimate of the vehicle’s future state trajec-tories. By using this to describe a predicted coordi-nate path of the vehicle over the next horizon, futuretrack centreline and boundary information can thenbe gathered relative to this path. From this, (err isdefined simply as

(err(k+i) = ((k+i) ! (r(k+i) (18)

for i = 1 . . . Np and where ( is the predicted directionof the vehicle absolute velocity. (err can therefore beproduced straightforwardly from the predicted vehi-cle states and track reference information. However,the perpendicular displacement, yerr, between thevehicle and the track centreline requires further con-sideration.

3.3 Linearising lateral displacement errorsSince we make use of the predicted state trajec-

tories from the previous optimisation, in order toevaluate the lateral displacement error during thefollowing optimisation, yerr(k+i), it is necessary tofirst derive an expression which describes the lateraldisplacement error between two successive predictedvehicle paths. With the vehicle forward speed con-stant, this depends only on the di!erences in headingangles. Fig. 4 illustrates the connection between thetwo paths when an intrinsic coordinate descriptionis used. An approximation of the change in lateraldisplacements are found using

!y(k+i) = !y(k+i!1) cos (((k+i+1|k!1) ! ((k+i|k!1))

+ ds(((k+i) ! ((k+i+1|k!1)) (19)

for i = 1 . . . Np. By including !y as a system state,provided there are small variations in state trajec-tories between consecutive optimisations, allows yerr

to be calculated and optimised during the next timestep using

yerr(k+i) = yerr(k+i|k!1) + !y(k+i) (20)

for i = 1 . . . Np and where yerr(k+i|k!1) are the lat-eral errors calculated by apply the previous optimumcontrol sequence.

3.4 Track boundary and tyre slip constraintsHaving derived a linearised lateral displacement

error between successive predicted vehicle paths, it

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Fig. 4: Initial geometric definitions for derivation ofintrinsic lateral displacement error expression for twoconsecutive predicted vehicle paths

is relatively straightforward to include a constraintwithin the optimisation problem, in the form of Equa-tion 13, which ensures the vehicle stays within thetrack boundaries. The boundary constraints are there-fore defined as

yerr(k+i) # wr/2 ! w

!yerr(k+i) # wr/2 ! w (21)

where wr is the width of the track/circuit and w isthe half-track-width of the vehicle.

In addition to boundary constraints, the controllermakes use of stabilising constraints on tyre lateralslip "f and "r. These restrict the tyres from oper-ating beyond their saturation limit where the it isunlikely optimum performance will be found. As de-tailed in [16] this instability arises, in part, due tothe method used by the controller to linearise thetyre model. Nevertheless in practice a skilled driveris likely to maintain the tyres operating within thepositive slope region of its force curves.

4. PATH OPTIMISATION SIMULATIONRESULTS

To demonstrate the performance of the path opti-misation algorthim and evaluate the various approx-imations made during its derivation in order for thecost function to remain quadratic in linearly definedsystem outputs, a simulation was setup using a dou-ble 90 deg “s-bend” track of finite width. Since thevehicle will have to negotiate a coupled left and rightbend, the results give a good indication of how wellthe algorithm is working, in particular in the transi-tion phase between the two corners where large ori-entation angle errors occur. Results in which thevehicle is driven at an undemanding forward speedof 20 ms!1 are presented in Fig. 5. The simulationand control parameters used during the simulationsare set out in Table 2.

Table 2: Simulation and constraint parameters usedduring the path optimisation simulations

Parameter Symbol Value

Discrete time step T 2 msPreview horizon Np 300Maximise distance travelled weight q1 1"$com e!ort weight R 100Track/road width wr 10 mVehicle half track w 0 mPseudo distance xp 100 m

The results of this simulation show how whenthe vehicle is driven well within its dynamic lim-its, the steering controller minimises the overall dis-tance travelled by hugging the inside lines of the twocorners with a smooth, approximately straight-line,transition between the two. Furthermore, evaluationof (err and yerr reveals both stay well within thelimits set out in order for Equation 15 to hold.

0 50 100 150 200 250 300 350

0

20

40

60

80

100

120

y glo

bal(m

)

xglobal (m)

vehicle pathtrack centrelinetrack boundaries

finish

start

Fig. 5: Path optimised simulation results for s-bendmanoeuvre at low speed with marked 1 s intervals

To further test the controller a repeat manoeu-vre was performed, this time with the vehicle speedincreased to 30 ms!1 so that the driver/steering con-troller would have to make full use of the circuitwidth and vehicle operating limits. The vehicle pathand parameter time histories are shown in Figs. 6and 7 respectively. As can be seen from Fig. 6,shortly after the start of the simulation the controllerbeings to steer the vehicle towards the outside of thetrack in anticipation of the approaching corner se-quence. The controller stops short of driving the ve-hicle to the very edge of the circuit as the future pre-dicted trajectories have enabled the controller to es-tablish it can navigate the upcoming bend(s) with anearly turn-in and still operate within the tyre forceslimits. The setup parameters chosen mean the ve-hicle is front axle limited, the time histories of Fig.7 demonstrate how the controller ensures the vehi-cle continually operates at the front side-slip anglecorresponding to max lateral tyre force.

5. CONCLUSIONS

The research presented here contributes a com-putationally e"cient and robust technique for opti-

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0 50 100 150 200 250 300 350

0

20

40

60

80

100

120y g

lobal(m

)

xglobal (m)

vehicle pathtrack centrelinetrack boundaries

finish

start

Fig. 6: Path optimised simulation results for s-bendmanoeuvre at high speed with marked 1 s intervals

0 2 4 6 8 10 12 14!20

0

20

!(m

s!2)

0 2 4 6 8 10 12 14!0.1

0

0.1

"f

0 2 4 6 8 10 12 14!0.05

0

0.05

"r

0 2 4 6 8 10 12 14!2

0

2

# sw

(rad)

t (s)

(rad)

(rad)

Fig. 7: Path optimised time histories for s-bend ma-noeuvre at high speed

mising the vehicle path around a racing circuit. Thishas been achieved through the linearisation of boththe vehicle dynamics and positioning of the vehiclerelative to the circuit boundaries, consequently en-abling the optimisation to be formed as a convexQuadratic Programming problem. Further compu-tational gains have been achieved through the use abanded structured MPC scheme.

Simulations have shown how the technique cansuccessfully optimise the path a nonlinear vehicletravelling at a demanding constant forward speed.Constraining tyre slip within the positive slope re-gion of the force characteristics has permitted thetyres to continually operate at their lateral force limit,allowing the total vehicle distance travelled duringthe simulated manoeuvre to be minimised. Futurework includes application of the minimum time op-timisation algorithm to optimal combined path andspeed profile control with the inclusion of a morecomplex vehicle model.

ACKNOWLEDGMENTS

The authors would like to acknowledge the financialsupport of the UK Engineering and Physical ScienceResearch Council and Renault F1 Team.

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