Torque-Based Steering Assistance for Collision Avoidance During LaneChanges

Z. Ercan* & A. Carvalho & S. Lefevre & F.BorrelliDepartment of Mechanical EngineeringUniversity of California, Berkeley, USA*Department of Control and Automation Engineering*Istanbul Technical University, Istanbul, Turkey.

H. E. TsengFord Research Laboratories, Dearborn, MI,USA

M. GokasanDepartment of Control and Automation EngineeringIstanbul Technical University, Istanbul, Turkey.

ABSTRACT: Lane changing is a cognitively demanding task that drivers perform frequently while driving onhighways at high speed. This paper presents a predictive control framework which prevents collisions duringlane change maneuvers. Approaches based on steering angle control are commonly used in the literature. Theseapproaches may inadvertently take away the ultimate control authority of the driver. Also they involve a separatelower level steering control that requires subjective and complex steering feel calibration and steering torqueconstraints. The method proposed in this paper is based on a corrective torque formulation which incorporatesthe steering dynamics and driver response in the model. We tested the proposed controller in an experimentalvehicle with a human driver. The results demonstrate that the proposed method avoids collision threats withminimum intervention.

1 INTRODUCTION

Studies show that 4 to 10 percent of road accidents oc-cur during or around lane change maneuvers. In mostcases, the drivers do not initiate an avoidance maneu-ver because they fail to notice the presence of othervehicles in the surroundings (Fitch et al. 2009).

There is an emerging research interest on driversteering assistance systems (DSAS) which use envi-ronment perception sensors and electronically con-trolled actuators to manipulate the steering of the ve-hicle in order to improve safety. Some of the systemscould provide visual, haptic or auditory warnings toenhance the situational awareness of driver (Bartelset al. 2012). However, there are scenarios when anautonomous intervention could be necessary to pre-vent any means of collision. This paper focuses onthe design and implementation of a DSAS which usesa column-type electric power steering (EPS) actuatorto provide a torque intervention to the driver in orderto prevent hazardous situations.

The works in Gray et al. (2013) and Anderson et al.

(2010) focus on using steering angle as control in-put in design of semi-autonomous vehicle control. Al-though steering angle control provides robustness andsimplicity, it does not consider the human-machineinteractions arising from sharing the same physical in-terface and therefore the system does not let the driverintervene or take over smoothly. For example in thework of Gray et al. (2013), a lane departure avoid-ance system is proposed using a predictive controllerto optimize a corrective steering angle. In order torealize the torque intervention in a vehicle equippedwith EPS, the authors use a low-level controller tocalculate corrective torque reference. This controllerneeds to be tuned so that the magnitude and rate ofthe corrective torque feels comfortable to the driver.

On the other hand, steering torque input is usedto assist the driver in lane keeping applicationsin Minoiu-Enache et al. (2009) and Shimakage et al.(2002). The intervention is realized by a servo motormounted in the steering column. However, these stud-ies consider the driver torque as a disturbance and as-sume that driver does not apply any torque during the

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intervention.From a practical point of view, another approach to

realize such a safety intervention would be using thetorque overlay function of the steering actuator (i.e.EPS). The torque overlay function enables the su-perimposition of a corrective torque to the assistivetorque applied by the EPS. A lane keeping system us-ing torque overlay function is proposed in the workof Kim et al. (2016). However, driver torque is con-sidered as a disturbance in the control structure.

The work presented in Zafeiropoulos and Cairano(2013) formulates the human-machine interactions asconstraints on the driver’s steering feel and the pro-posed controller improves the stability of the vehicleduring aggressive maneuvers.

In this paper, we extend the work of Gray et al.(2013) to be directly implemented in a vehicleequipped with an EPS system. We propose a modelpredictive control (MPC) framework which optimizesa corrective torque as an input to the EPS actuator inorder to prevent collisions during lane change maneu-vers. We account for the human-machine interactionsby modeling the steering dynamics of the driver-in-the-loop system. The corrective torque is optimizedfor smoothness and utilizes minimum effort. The re-sulting controller is always active but intervenes onlyif the safety constraints are violated.

2 SYSTEM DESCRIPTION

The system architecture is shown in Fig. 1. Thehuman driver and the controller can both applytorque , respectively τd and τc, to the steering col-umn. The steering wheel is the physical interface be-tween the driver and the controller. The “Vehicle &Environment” block represents the ego vehicle (EV)(i.e. where the controller action is implemented) andthe target vehicles (TVs) which are traveling in thevicinity of the EV. The “Sensors” block provides allthe measurements related to the EV, road geometry,and relative positions and velocities of the TVs. Theproposed controller is implemented in the “MPC”block which outputs a corrective torque input τc thatis exerted on the steering column by the EPS actuator.

The main contributions of this paper are:

• An implementation of predictive, driver-in-the-loop control framework for safe, assisted driv-ing. The control design uses a model of driver-in-the-loop steering dynamics.

• An approach for modeling driver-in-the-loopsteering dynamics.

• Experimental validation of the proposed frame-work with a human driver in a commercial vehi-cle without any modification to the steering ac-tuator.

Figure 1: System architecture

3 MODELING

In this section, we introduce the mathematical modelsused in the controller to describe the motion of the EVand simulated TVs.

3.1 Ego Vehicle Model

We use a bicycle model to describe the dynamics ofthe EV as illustrated in Fig. 2. The following set ofnonlinear differential equations describe the motionof the vehicle in a road-aligned coordinate frame (Ra-jamani 2011) :

x = ψy + ax (1a)

y = −ψx+2

m

(Fy,f + Fy,r

)(1b)

ψ =2

I

(lfFy,f − lrFy,r

)(1c)

eψ = ψ− κs (1d)

ey = x sin(eψ) + y cos(eψ) (1e)

s =1

1− κey

(x cos(eψ)− y sin(eψ)

). (1f)

The longitudinal and lateral velocity of the centerof gravity (CoG) are denoted as x and y, respectively,and ψ is the yaw rate. The longitudinal acceleration isdenoted as ax. ey and eψ denote the lateral position ofthe vehicle with respect to the lane center line and theorientation of the vehicle with respect to the tangentat the road center line, respectively. The road curva-ture κ is assumed to be known at each time step. Thelongitudinal position of the vehicle along the lane cen-ter line is denoted as s. The parameters of the vehiclemodel are the mass m, moment of inertia I and dis-

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tances from the CoG to front and rear axles, denotedas lf and lr, respectively.

x y

ψFy,r

Fx,r

Fx,fFy,f

Fc,r

Fl,r

Fc,f

Fl,fey

𝑋

𝑌

eψ

δf

αf

vr

vf

αr

s

Lane centerline

lr

lf

Figure 2: Notation used in the bicycle model.

Since we focus on the lateral control of the vehiclein this paper, we introduce the following assumption:

Assumption 1. We assume zero longitudinal acceler-ation in the EV model that is used in predictive con-troller (i.e. ax = 0 in (1a)).

We consider a front steerable vehicle model whereδ denotes the road steering angle of the front tires. Un-der normal driving conditions (i.e. small slip angles),the linear tire model provides a good approximationof the relationship between the lateral tire force andthe tire slip angle. In the linear regime, the front (rear)lateral tire force Fc,f (Fc,r) is proportional to the front(rear) tire slip angle αf (αr). That is,

Fc,i = −Cαiαi, i ∈ {f, r}, (2)

where Cα denotes the cornering stiffness of the tire.The slip angles αf and αr are approximated as,

αf =y + lf ψ

x− δ, αr =

y− lrψx

. (3)

The numerical values of the cornering stiffnessesof the front and rear tires are estimated experimen-tally. The lateral tire forces acting on the CoG in thevehicle body frame are represented by,

Fy,f = Fc,f cos δ, Fy,r = Fc,r. (4)

The nonlinear bicycle model described in (1)–(4) iscompactly written as,

ξ(t) = f(ξ(t), u(t), κ(t)), (5)

where the state vector is ξ = [x, y, ψ, eψ, ey, s]ᵀ, the

control input is u= δ and κ is the curvature parameter.

3.2 Driver-in-the-Loop Steering Dynamics Model

We represent the equivalent rotational dynamics ofthe steering system of a vehicle as

Jsθ+ bsθ = τd + τc − τr, (6)

where Js and bs represents the inertia and dampingof the steering system respectively. τd represents thedriver torque, τc is the corrective torque (i.e. input tothe steering actuator) and τr is the equivalent align-ment torque that is generated from road-tire interac-tion.

We consider a steering system where the driver andthe steering actuator are mechanically coupled. Dur-ing a safety intervention, the driver feels the correc-tive torque through the steering wheel and could re-spond to the intervention in different ways. The drivercould either follow the corrective torque without re-sisting (i.e. compliant driver) or could oppose the in-tervention by generating a reaction torque (i.e. stiffdriver).

The predictive controller requires a model that re-lates the actuator input τc to output θ in order to pre-dict the future evolution of the steering dynamics. Us-ing this model, we can relate the actuator input to thestates of EV model presented in (1).

Hence, we propose the following assumptions toderive the “driver-in-the-loop steering model”.

Assumption 2. We define the driver torque as τd =τd,str + τd,react where τd,str and τd,react represent thesteering and the reaction torque of driver respectively.

Assumption 3. Driver applies a steering torque thatis equal to alignment torque (i.e. τd,str = τr) in theprediction horizon.

Assumption 4. The reaction torque of driver is de-fined as τd,react = Jdrθ + bdrθ + kdrθ where Jdr,bdr and kdr represent the dynamic properties of thedriver’s impedance which is coupled to the impedanceof steering system (Pick and Cole 2007).

Using the assumptions (2)-(4), we propose the“driver-in-the-loop steering model” from input τc tooutput θ as a second order single input-single out-put (SISO) system

d

dt

[θ

θ

]=

[0 1

−keqJeq− beqJeq

][θ

θ

]+

[01Jeq

]τc, (7)

where the parameters Jeq, beq, keq describe the equiv-alent inertia, damping and stiffness of the driver-in-the-loop steering system model.

Remark 1. The parameters in (7) are used as tun-able parameters to calibrate the torque interventionfeel. These parameters change with respect to re-sponse of driver and should be identified online. How-ever, in this paper we use constant values of the pa-rameters which are tuned during experiments that areperformed with a compliant driver.

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Table 1: IDM ParametersParameter Value unitsacceleration exponent µ 4 -max.acceleration a 1.5 m/s2

desired speed vdes 20 m/scomf.deceleration b 1.15 m/s2

min. gap to leading vehicle d0 5.0 mtime gap to leading vehicle T 1.5 s

We augment the vehicle dynamics model (5) insection 3.1 with the steering model in (7). The aug-mented nonlinear vehicle model can be compactlywritten as ζ(t) = f(ζ(t), u(t), κ) where the state vec-tor is ζ = [x, y, ψ, eψ, ey, s, θ, θ]

ᵀ, the control input isu = τc and κ is the curvature parameter.

We discretize the differential equations using a for-ward Euler approach with fixed sample time Ts to ob-tain ζk+1 = fd(ζk, uk, κ) where k denotes the currentdiscrete time step and t = kTs.

3.3 Simulated Target Vehicle Model

We use the Intelligent Driver Model (IDM) to sim-ulate virtual TVs that are traveling around EV. Thismodel represents the longitudinal dynamics of TVs ina car following scenario (Treiber et al. 2000). The ac-celeration of the TV is calculated by

v = a

[1−

(v

vdes

)µ−(d∗(v,∆vp)

d

)2]

(8a)

d∗(v,∆v) =

{d0 + Tv + v ∆vp

2√ab

if there is a PV0 otherwise

(8b)

The relative velocity is defined as ∆vp , vp − vwhere vp is the velocity of preceding vehicle (PV). Ifthere is a PV in front of simulated TV, the algorithmcalculates the acceleration to track the desired dis-tance d∗. If there is no PV (i.e. EV is not in front ofTV), the algorithm calculates the acceleration to trackthe desired speed vdes. All the relevant parameters aregiven in the Table 1.

4 MODEL PREDICTIVE CONTROLLERDESIGN

In this section, we present the design of the predictivecontroller which employs the models presented in theprevious section.

4.1 Safety Constraints

We define a “safety box” around each TV to representsafe distance margins between the EV and the TVs, asillustrated in Fig 3. The front and rear safety distancesare calculated by

dfront = d0 + tgap max(0,∆vt) (9a)

drear = d0 − tgap min(0,∆vt), (9b)

where d0 is a constant safety distance, ∆vt , vt − veis the relative velocity and tgap is the time headwaywhich is a design parameter that relates the size ofsafety box to relative velocity.

Lateralposition[m

]

Longitudinal position [m]30 35 40 45 50 55 60 65 70 75 80

−2

0

2

4

6

Scenario 1 : Target Lane is not available.

Scenario 2 : Target Lane is available.

Safety Box

Tv

Ev

Target Lane

Ego Lane

Front safety distanceRearsafety distance

Figure 3: The safety box concept and calculation of safety con-straints

Using the model in (8), we predict the position ofTV and calculate safety boxes for each prediction infinite horizon. The collision avoidance constraints areformulated as upper and lower bounds on the lateralposition ey of the EV in the prediction horizon (i.e.for i = 1 . . .N + 1) using Eq. (10)

eymin,i= 0.5(wc −wlane) (10a)

eymax,i=

{0.5(wlane −wc) if se,i ∈ St,i1.5wlane − 0.5wc otherwise

(10b)

where wlane and wc denote the width of lane and theEV. The subscript i denotes the prediction index, se,iis the longitudinal position of EV at prediction index iand St,i = [st,i−drear, st,i +dfront] denotes the size ofsafety box at prediction index i.

Finally, we can express the time-varying safetyconstraints in prediction horizon as,

eymin,i≤ ey,i ≤ eymax,i

for i = 1 . . .N + 1. (11)

Remark 2. The size of the safety box is associated tothe conservativeness of the intervention by the con-troller as the safety constraints are calculated basedon safety box.

4.2 Input Constraints

We define magnitude and rate constraints on the ac-tuator input (i.e. τc) in order to have a smooth andadequate level of torque intervention:

τmin ≤ τc ≤ τmax (12a)

∆τmin ≤ ∆τc ≤ ∆τmax. (12b)

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Table 2: Process Model ParametersParameter Value Parameter Valuem 1830 kg lf 1.1520 mI 3477 kg m2 lr 1.6930 mCαf

40703 N / rad Cαr64495 N / rad

wc 1.86 m Jeq 0.5 kg m2

beq 2.5 Nm s/rad keq 6.25 Nm / rad

4.3 Optimal Control Problem

We use the framework of nonlinear model predic-tive control (NMPC) for designing the DSAS con-troller. In NMPC, a constrained finite horizon opti-mal control (CFTOC) problem is solved at each timestep. The first optimal input is applied to the vehicleand the process is repeated at the next time step usingthe new state measurement as the initial condition.

In the CFTOC problem (13), the optimization vari-ables are the corrective torque input (u = τc) and theslack variable ε. The following numerical optimiza-tion problem is solved in real-time,

minuk,ε

N−1∑i=0

(||ui|k||2Q + ||∆ui|k||2R

)+Mε (13a)

s.t. ζi+1|k = fd(ζi|k, ui|k, κi|k) (13b)

τmin ≤ ui|k ≤ τmax (13c)

∆τmin ≤ ∆ui|k ≤ ∆τmax (13d)

eymin,i+1− ε ≤ eyi+1

≤ eymax,i+1+ ε (13e)

ε ≥ 0, i = 0 . . .N − 1

ζ0|k = ζk , u−1|k = uk−1, (13f)

where N is the prediction horizon. ζi|k denotes theith state prediction at time step k obtained by ap-plying the optimal control input sequence uk ={u0|k, u1|k, . . . , uN−1|k} to the discrete-time dynamics(13b) starting from the measured state at the currenttime step (13f). We impose the safety constraints assoft constraints (13e) by introducing the slack vari-able ε in order to keep the optimization problem fea-sible in case of constraint violations due to model mis-match. The actuator input constraints are formulatedin (13c) and (13d).

5 EXPERIMENTAL RESULTS

The control design presented in Sections (3)-(4) hasbeen validated experimentally on a test vehicle. Thesteering actuator is a production level EPS. The sen-sors, embedded computer and actuators communicatethrough a CAN bus. During the experiments, we usedthe parameters given in Table 2 and Table 3.

Table 3: MPC ParametersParameter Value Parameter ValueTs 0.1 s Tmpc 0.2 sτmin −5 Nm ∆τmin −5 Nm / sτmax 5 Nm ∆τmax 5 Nm / seymin −0.67 m eymax 4.07 mwlane 3.4 m Q,R (10,10)M 10000 N 10

The objective of the test is to evaluate the system’sability to prevent high risk lane changes. We simu-lated a TV around the test vehicle (next called EV). Attime t = 2s, driver of the EV simulated an inattentivedriver behavior by initiating a lane change maneuverwhile there is a fast TV in the target lane. The mea-sured lateral position of the EV along with the open-loop (OL) predictions from the controller are illus-trated in Fig. 4. The results show that the controllerprevents the vehicle from departing the lane.

MeasurementOL Predictions

Latera

lPosition[m

]

Time [sec]0 2 4 6 8 10 12

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Figure 4: The lateral position and the open-loop predictions ofthe EV during the experiment.

The predictive controller assessed the risk of col-lision by predicting the positions of the EV and TVand intervened only if it is safety critical. The cor-rective torque is illustrated in Fig. 5. The correctivetorque is zero when the predicted driver actions resultin a safe trajectory. However, if the predicted trajec-tory violates the safety constraints, an intervention istriggered to guide the driver back to the safe region ofthe road.

At time t = 3.2s, the predictive controller startedapplying a clockwise (i.e. negative in magnitude) cor-rective torque to steer the vehicle to right. In Fig. 6, weillustrate a snapshot of the experiment at t = 3.8s. Itis observed that without intervention, the controllerpredicted the vehicle would leave the lane thereforeviolating the safety constraints. By applying the cor-rective torque, the controller updated its predictionsto satisfy the safety constraints.

Starting from time t = 3.8s, driver started tosteer in clockwise direction as a result of correctivetorque. The steering wheel angle measurements andthe open-loop predictions are given in Fig. 7. Thecorrective torque decayed to zero when driver steeredback to safety portion of the road. Because of the rateconstraints, the corrective torque is applied graduallywhich provides a smooth transition instead of rapidchanges in corrective torque dynamics.

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AppliedOL Sequence

CorrectiveTorq

ue[N

m]

Time [sec]0 2 4 6 8 10 12

−4

−3

−2

−1

0

Figure 5: The corrective torque sequence during the experiment.

TV OL Traj.EV OL Traj.EV CL Traj.

Lateralposition[m

]

Longitudinal position [m]

Time : 3.8 s, Torque : -3.432 Nm, Rel.Dist : 12.352 m, Rel.Vel : -9.4314 m/s

50 60 70 80 90 100 110

− 2

0

2

4

6

TV(k) TV(k+N|k)

EV(k) EV(k+N|k) with intervention

EV(k+N|k) withoutintervention

Figure 6: The snapshot at time t = 3.8s. This figure shows theopen-loop (OL) predictions of EV and TV along with the ac-tual (closed-loop (CL)) trajectory of EV.

MeasurementOL Prediction

SteeringW

heelAngle

[rad]

Time [sec]0 2 4 6 8 10 12

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

Figure 7: The steering wheel angle of EV during the experiment.

6 CONCLUSIONS

We propose a predictive, human-in-the-loop con-trol framework for preventing collisions during lanechange maneuvers. We focus on modeling the steer-ing dynamics such that we could use the native inputof the EPS actuator (i.e. torque) as a direct optimiza-tion variable in predictive controller. This has the fol-lowing advantages:

• Compared to an optimal steering angle approach,a low-level controller is no longer needed to cal-culate the torque input from a desired correctivesteering angle.

• The actuator limits can be systematically han-dled in the predictive controller.

As both the driver and controller act on the steeringwheel, it is important to account for the reaction ofthe driver. Our approach enables this by modeling thedriver-in-the-loop steering model which integrates thedriver torque dynamics into steering system model.

We implement the presented control framework ona test vehicle and validate the performance with a hu-man driver. The experimental results demonstrate theeffectiveness of the proposed controller in preventinghigh risk lane change maneuvers.

As the next step of our research, we will work onexperimental tests with more drivers, using differentparameters in the driver-in-the-loop steering model toobserve their affects on the steering feel of the driversduring intervention.

REFERENCES

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Shimakage, M., S. Satoh, K. Uenuma, & H. Mouri (2002).Design of lane-keeping control with steering torque in-put. JSAE Review 23(3), pp. 317 – 323.

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Zafeiropoulos, S. & S. D. Cairano (2013, June). Vehicleyaw dynamics control by torque-based assist systemsenforcing driver’s steering feel constraints. In 2013American Control Conference, pp. 6746–6751.

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