Scenario Model Predictive Control for Lane Change Assistance andAutonomous Driving on Highways

Gianluca Cesari∗, Georg Schildbach†, Ashwin Carvalho†, Francesco Borrelli†

Abstract— This paper presents a novel design of controlalgorithms for lane change assistance and autonomous driv-ing on highways, based on recent results in Scenario ModelPredictive Control (SCMPC). The basic idea is to account forthe uncertainty in the traffic environment by a small numberof future scenarios, which is intuitive and computationallyefficient. These scenarios can be generated by any model-based or data-based approach. The paper discusses the SCMPCdesign procedure, which is simple and can be generalizedto other control challenges in automated driving, as well asthe controller’s robustness properties. Experimental resultsdemonstrate the effectiveness of the SCMPC algorithm andits performance in lane change situations on highways.

I. INTRODUCTIONA. Traffic Prediction Models

In the past decade, new sensor technology and the increas-ing capacity of computational hardware has lead to a risinginterest in Advanced Driver Assistance Systems (ADAS). Akey element for any ADAS feature is the recognition of trafficscenes based on measurements of the vehicle’s environmentvia onboard or external sensors. Traffic predictions comprisethe joint trajectory prediction for all relevant vehicles inthe estimated traffic scene, called target vehicles (TVs).Difficulties arise from complex vehicle dynamics, intricatetire-road interactions, and uncertainty in humans’ drivingactions.

Existing models for traffic predictions can be classifiedinto physics-based, maneuver-based and interaction-awaremodels; see [1] for a survey. Models can be further classifiedbased on the nature of the predictions. Deterministic modelspredict a single trajectory for each object in the trafficscene, and do not reflect any uncertainty associated with thefuture actions of human drivers [2]. Set-based predictionsover-approximate the future occupancy of target vehicles,given a prediction model and bounds on the drivers’ actions,allowing for a formal verification of safety [3], [4]. Stochasticmodels use standard distribution functions such as GaussianMixtures to model a driver’s behavior [5], [6]. Scenario-based models represent uncertainty by means of samples anddo not require an explicit probability distribution for theirpredictions [7], [8].

B. Model Predictive ControlModel Predictive Control (MPC) is a powerful approach

for the optimal control of multi-variable systems with con-straints on the inputs and states [9]. MPC has been used

∗ ETH Zurich, Zurich, Switzerland.(email:gianluca.cesari@me.com) †Hyundai Center of Excellence,University of California Berkeley, Berkeley (CA), United States.(email:schildbach|ashwinc|fborrelli@berkeley.edu)

successfully for vehicle control in a variety of ADAS appli-cations [10]. The key benefit of MPC is the easy integrationof predicted information, as well as constraints resulting fromtraffic predictions or road geometry.

The most basic approach, Deterministic MPC (DMPC),has successfully been adopted for path following in absenceof other traffic [11]. Stochastic MPC (SMPC) is basedon probabilistic traffic predictions and has been used forhighway driving [12]. In SMPC, chance constraints are usedto limit the risk of dangerous events.

However, a major drawback of SMPC is the requirementof a full stochastic uncertainty model and the associatedcomputational complexity with its forecasts, in particularfor nonlinear systems or non-Gaussian distributions. Thispaper examines the use of Scenario MPC (SCMPC) [13],[14], where traffic predictions are based on scenarios. Thisapproach is very intuitive and amenable to numerical pro-cessing. Moreover, the approach is very versatile, as thescenarios can be sampled from any stochastic model or dataset.

Recent studies have revealed a direct connection betweenthe number of scenarios and chance constraints on the systemstates [14], [15], using principles from scenario optimization[16], [17]. The number of scenarios required by SCMPC islow, typically less than a hundred for chance constraint prob-abilities of a few percent. The SCMPC does not guaranteerobust safety, but the goal is to explicitly bound the residualrisk and relate it to the number of generated scenarios.

C. Scenario-Based Model Predictive ControlIn this paper, two SCMPC algorithms for performing safe

lane change maneuvers on highways are introduced: onefor lane change assistance (LCA) and one for automatedhighway driving (AHD). A safe trajectory is considered to beone that respects all actuator, comfort, and safety constraints.

The main difference between the two proposed algorithmslies in the formulation of the state constraints and theoptimization problem for the controller. The LCA algorithmrequires that the lane change is initiated by a human driver,e.g., via the turn signal (using it for ADAS), or a higher-level path planner (using it for autonomous driving). TheAHD algorithm includes a constraint generation techniquewhich enables the controlled vehicle to plan and execute lanechange maneuvers based exclusively on the measured trafficconditions and the commanded reference cruise speed.

D. NotationThroughout the paper, τ and t are used to denote con-

tinuous and discrete time, respectively. The same variable

will be used for the continuous signal and its discrete timecounterpart. For example, v(τ) represents a velocity signalin continuous time, with τ ∈ [0,∞) in brackets, and vt,with t = 0, 1, 2, . . . as an index, represents the same signalsampled at the sampling time ts. A dot will be used toindicate a derivative with respect to (continuous) time, e.g.,acceleration a(τ) = v(τ). Bold letters are used to distinguishvectors and matrices from scalars.

II. EGO VEHICLE MODELA. Bicycle Model in Frenet Frame

The SCMPC uses a kinematic bicycle model [18, Sec. 2.2]in Frenet coordinates, aligned with the lane centerline [19];see Fig. 1. The model yields a good trade-off between modelcomplexity and prediction accuracy [11], [20].

The curvilinear abscissa s(τ) and the distance to the lanecenter line η(τ) describe the position of the center of gravity(CoG) of the ego vehicle (EV). The relative orientation φ(τ)is the difference between the inertial heading ψ(τ) and theorientation of the tangent to the lane centerline at the closestpoint. The model has two control inputs, the steering angleand the acceleration u(τ) , [γ(τ)(EV), a(τ)(EV)] and fivestates x(τ) , [s(τ), η(τ), φ(τ), v(τ), l(τ)], where l(τ)indicates the current lane of the vehicle. The EV being inits current lane of width wlane, the lateral distance η(τ)lies within the interval [−wlane/2, wlane/2]. The followingdiscrete-time equations of motion for the EV are obtainedby an explicit Euler method with sampling time ts:

˙st =vt

1− κtηtcos (φt + βt) (1a)

st+1 = st + ˙stts (1b)ηt = ηt + vt sin (φt + βt)ts (1c)

ηt+1 = ηt +

−wlane, if ηt ≥ wlane/2

wlane, if ηt < −wlane/2

0, otherwise(1d)

φt+1 = φt +

(vtlr

sinβt − κt ˙st

)ts (1e)

vt+1 = vt + atts (1f)

lt+1 = lt +

1, if ηt ≥ wlane/2

−1, if ηt < −wlane/2

0, otherwise(1g)

where βt = tan−1(

lrlf+lr

tan(γ(EV)t )

), and κ(s) represents

the lane curvature at s as sensed by a vision system. TheEV’s speed is denoted by v, and lf and lr are the lengthsfrom the CoG to the front and rear axles, respectively. Forcompact notation, (1) will be rewritten as

xt+1 = ft(xt,ut) . (2)

B. Input Constraints

Actuator limits on the steering angle, steering rate, andacceleration are captured by constraints on the control inputs:

umin ≤ ut ≤ umax , umin · ts ≤ ut+1−ut ≤ umax · ts . (3)

(t)(t)

(t)

(t)

O ●

(t)(t)

(t)

(t)

O ●

�(t)(t)

(t)

(t)

O ●v

�

lf

lr

CoG

Fig. 1: Vehicle position in a Frenet coordinate frame attachedto the lane center line.

C. Road Boundaries

The road boundaries impose the following bounds on thelateral position of the EV:

ηlane,min + ∆ηsafe ≤ ηt ≤ ηlane,max −∆ηsafe, (4)

where ∆ηsafe is a desired safety distance which also accountsfor the EV’s geometry and orientation. The upper bound isset when the vehicle travels on the leftmost lane of the road,while the lower bound when it travels on the rightmost lane.

III. TARGET VEHICLE PREDICTIONS

The SCMPC algorithm requires predictions for all relevantTVs in the EV’s environment, which are combined intotraffic scenarios. The TV predictions comprise two parts:(a) the maneuver prediction model that identifies the type ofmaneuver that a TV is likely to perform (Section III-A), and(b) the trajectory prediction model that generates possibletrajectories for the TV, based on the predicted maneuver(Sections III-B and III-C).

A. Maneuver Prediction Model

The learning-based approach of [21] is used to estimatethe distribution over a TV’s maneuvers. The human driveris modeled as a stochastic hybrid system, switching betweeni = 1, 2, 3 control strategies: lane keeping (LK), lane changeleft (LCL), and lane change right (LCR). A set of features zt,representing the current traffic scene at time t, is considered.The dependencies among these variables are described bya Hidden Markov Model (HMM), whose parameters arelearned from experimental data collected in a training phase.

The distribution over the control strategy it at time t, con-ditioned on the history of measurements z1:t = {z1, . . . , zt},is denoted P[it = i|z1:t] with i ∈ {LK,LCL,LCR}. It isrecursively computed by inference on the HMM; for detailssee [21]. Then, the TV’s trajectory is predicted separatelyfor each control strategy.

B. Lane Keeping (LK) Strategy

LK is the maneuver performed by a driver most of thetime. The resulting trajectory is approximated by a straightline with a constant orientation (relative to the lane cen-terline), traversed at a constant acceleration. The relativeorientation of the path is modeled by a random variableγ(TV ).

The resulting state predictions during LK are given by

sLK (t) = s(0) + xlong (t) cos(γ(TV)

), (5a)

ηLK (t) = η(0) + xlong (t) sin(γ(TV)

), (5b)

φLK (t) = γ(TV) , (5c)

vLK (t) = v(0) + a(TV) t . (5d)

Here, xlong = v(0)τ + 0.5 a(TV) τ2 is the vehicle’s longitudi-nal distance traveled along the path, and s(0), η(0), v(0)are the initial conditions for s, η, v at the beginning ofthe predictions. The values of γ(TV) and a(TV) are constantover the prediction horizon and drawn from the probabilitydistribution functions (PDFs) identified in Section III-D.

C. Lane Changing (LC) Strategies

LC trajectories are influenced by many factors, such as thedriver’s style and the current traffic situation. The cases fora lane change to the left (LCL) and to the right (LCR) areconsidered symmetric and thus handled analogously. In thispaper, a maneuver-based approach is used where the lanechange path is modeled as a sigmoid function:

f (s; a, b) =b

1 + exp (−as) . (6)

Here a = 4b tan (α), where α is the slope at the inflection

point of the sigmoid, and b is the ordinate value of thefunction for s → ∞. As the sigmoid function tends to 0for s→ −∞, a point must be selected which is consideredas the beginning of the path. We choose this to be the pointat which the sigmoid function has a value of v 2% of thefinal lateral position b. Hence, an auxiliary constant variablek = 4 is set so that the initial lateral offset c becomes

c(b) =b

1 + exp (k)= 0.018b . (7)

A constant variable d(a) = k/a shifts the origin of thelongitudinal axis to the lane change starting point, so thatthe lower horizontal asymptote approximates the current lanecenter line. The modified formulation of the sigmoid functionis

f (s; a, b) =b+ c(b)

1 + exp (−a (s− d(a)))− c(b) . (8)

The goal of the prediction model is to recognize whichtrajectory is being taken by a vehicle based on its currentand past states. Thus, it is necessary to determine the valuesof the parameters a and b to identify the path, as wellas the longitudinal position of the vehicle with respect tothe beginning of the maneuver at sLC(0). It is reasonableto expect a TV to change one lane at a time; hence b is

-40 -20 0 20 40Longitudinal position, s [m]

0.0

1.0

2.0

3.0

4.0

Lateral

position,η[m

]

a = 0.1

-40 -20 0 20 40Longitudinal position, s [m]

0.0

1.0

2.0

3.0

4.0

Lateral

position,η[m

]

a = 0.2

sigmoid2nd order approximation curve, begin2nd order approximation point, begin1st order approximation line1st order approx point2nd order approximation curve, end2nd order approximation point, end

Fig. 2: Approximation of different sigmoid functions.

assumed to be equal to the road’s lane width, subject tosome disturbance.

The first and second order derivatives of the vehicle’slateral position η(τ) with respect to its longitudinal positions(τ) are used to retrieve the values of a and sLC(0):

v(TV)lat ,

d η (τ)

d s (τ)=

dη(τ)dτ

ds(τ)dτ

and a(TV)lat ,

d2η (τ)

d s (τ)2 =

ddτ

dη(τ)ds(τ)

ds(τ)dτ

.

Since it is not possible to find an explicit expressionfor a and s in the first and second derivatives of thesigmoid function f(s), three Taylor series are considered toapproximate it; see Fig. 2 for an illustration.

A first-order Taylor approximation φ1 (s; a, b,ms) is con-sidered at the inflection point by setting the approximationpoint ms = 0. Its derivative

φ(1)1 (a; b) =

ddsφ1 (s; a, b,ms = 0) =

ba

4(9)

allows the extraction of an explicit form of a. A secondorder Taylor approximation φ2 (s; a, b,ms) has as first andsecond order derivatives φ(1)2 (s; a, b,ms) and φ(2)2 (a; b,ms),respectively, which do not allow us to find an explicit form ofa. Therefore, φ(2)2 is approximated with a second order Taylorseries Ψ2 (a;ma, b,ms), making it possible to determine thevalue of a. The approximation points can be considered astuning parameters, and they are chosen to be: ms1 = −10m, ms2 = 10 m and ma = 0.08.

The longitudinal position of a vehicle performing a LCmaneuver can be expressed in terms of the progression ratio,i.e., the ratio of the longitudinal position s with respect tothe beginning of the maneuver and the length of the sigmoidfunction. A sigmoid length is defined as the value of s suchthat f(s) = 0.95b. The progression ratio function is derivedby (8) and it is defined as

Λ(a, s) =sa

2(k − log(0.95)). (10)

Therefore, the sigmoid’s parameters a and s, correspondingto the estimated values of a and s, can be determined usingthe following algorithm:

if a(TV)lat ≥ 0 thena← Ψ−12 (a

(TV)lat ;ma, b,ms1)

s← φ−(1)2 (vlat; a, b,ms1)

λ← Λ (a, s)if λ > λthreshold then

a← φ−(1)1 (vlat, b)

s← 0end if

elsea← Ψ−12 (a

(TV)lat ;ma, b,ms2)

s← φ−12 (vlat; a, b,ms2)λ← Λ (a, s)if λ < λthreshold then

a← φ−(1)1 (vlat, b)

s← 0end if

end ifIn the above algorithm, the first order approximation isadopted when the value of s calculated using the secondorder approximation is closer to the inflection point than athreshold given by the tuning variable λthreshold.

The path is assumed to be tracked at a speed affected bya constant, random acceleration, a(TV):

s(τ) =

∫ τ

0

vLC(τ) cos(φLC(τ))dτ , (11)

sLC(τ) = s(0)− s+ s(τ) , (12)

ηLC(τ) = η(0)− f(s) + f(s(τ); a, b) , (13)

φLC(τ) = tan−1(f (1)(s(τ); a, b)) , (14)

vLC(τ) = v(0) + a(TV) τ . (15)

Here s represents the maneuver progression and s(0), η(0)and v(0) are the measured values at prediction time t = 0.The LCR follows from a LCL maneuver by inverting thesigns of f(s(τ); a, b) and f (1)(s(τ); a, b). The values of band a(TV) are drawn from PDFs, as shown in the next section.

D. Traffic Model Identification

For the LK maneuver, the probability distributions of thevehicle orientation and the longitudinal acceleration are

γ(TV) ∼ N (µγ(TV) , σ2γ(TV)) , (16)

a(TV)LK ∼ N (µa(TV),lk, σ

2a(TV),lk) , (17)

where µγ(TV) , µa(TV),lk are the means and σ2γ(TV) , σ2

a(TV),lkthe

variances of the corresponding normal distributions.For LC maneuvers, the probability distributions of the lane

change offset and the longitudinal acceleration are given by

b ∼ N (µb, σ2b ) , (18)

a(TV)LC ∼ N (µa(TV),lc, σ

2a(TV),lc) , (19)

where µb, µa(TV),lc are the means and σ2b , σ2

a(TV),lcthe

variances of the corresponding normal distributions.

The LK and LC distribution parameters are identified froma large dataset provided by the Next Generation Simulation(NGSIM) program [22]. The set of recorded trajectories isrepresented in Fig. 3. Every gray line corresponds to thepath of a single vehicle. As no information about the lanewidths is available, lane positions are estimated according tothe assigned lane identifier. Changes in the lane identifier arerepresented by red dots in the Fig. 3.

Fig. 3: Raw traffic data and lane detection.

After pre-processing the data, the PDFs were identified asfollows.

1) Select a point on a given trajectory.2) Compute the predicted trajectories for many combina-

tions of (γ(TV), a(TV)LK ) or (b, a(TV)

LC ) depending on thetype of maneuver over a time horizon of 2 s.

3) Among all trajectories computed in Step 2, select theone with the lowest root-mean-square error (RMSE)with respect to the original trajectory.

4) Store the values of the parameters (γ(TV), a(TV)LK ) or

(b, a(TV)LC ) and the RMSE value corresponding to the

trajectory chosen in Step 3.Steps 2 and 3 are illustrated in Fig. 4. The outcome of thePDF identification and the corresponding numerical valuesare shown in Figs. 5 and 6. The average RMSE values forthe predictions were 0.15 m (v < 10m/s) and 0.17 m (v ≥10m/s) for the LK trajectories, and 0.21 m (v < 10m/s)and 0.24 m (v ≥ 10m/s) for the LC trajectories.

IV. SCENARIO GENERATION

Given the present traffic scene, SCMPC requires forecastsof traffic scenarios with associated uncertainty. Both thescenario generation based on TV predictions (Section III)and the controller design (see Sections V and VI) assumediscretized time steps t = 0, 1, 2, . . . , with a common sampletime ts. The SCMPC algorithm, however, can be based onany generic model-based or data-based method for scenariogeneration, if it fulfills the assumptions stated below. Thesescenarios can then be simplified or consolidated, as shownfor our algorithm.

Let N be the time horizon for the traffic predictions.Denote all uncertainty involved in the traffic predictions overthe horizon by the variable δt ∈ ∆t, where the support∆t has an arbitrary nature and dimensions. The followingtechnical assumption on the scenario generation is needed toderive the theoretical properties in Section VII.

Assumption 1 (Traffic Scenarios) (a) At every time stept, an arbitrary number K of samples of the uncertaintyδt ∈ ∆t over the horizon N (‘scenarios’) can be drawnfrom the traffic prediction model. (b) These sampled scenar-ios δ(1)t , ..., δ

(K)t are independent and identically distributed

(i.i.d.) with the real uncertainty outcome δt.

Assumption 1 requires only that sufficient scenarios areavailable and that they are representative of the actual uncer-tainty in the traffic scene. The availability of representativedata is a minimal requirement for any uncertainty prediction.

Note that the scenario-based uncertainty model can betime-varying and hence be conditioned on the latest infor-mation available at each time step, unlike many other MPCapproaches [23]. Neither the current joint probability distri-bution of δt nor the support ∆t need to be known explicitly.These are further advantages over similar approaches usingStochastic MPC [12] or Robust MPC [24].

A. Target Areas

Conditioned on the measured state of the TVs, scenariosfor a TV j are obtained based on its estimated controlstrategy ij ∈ {LCL,LCR,LK}, as discussed in SectionIII-A. Then, for each strategy, a scenario number Kj,LCL,Kj,LCR, or Kj,LK is identified:

Kj,LCL = round(P[ij = LCL

]·K), (20)

Kj,LCR = round(P[ij = LCR

]·K), (21)

Kj,LK = K −KLCL −KLCR. (22)

Finally, a predicted trajectory is obtained for each scenarioby feeding the sampled parameters into the corresponding

0 5 10 15 20

∆s [m]

0

0.5

1

1.5

2

∆η[m

]

0 5 10 15 20 25 30 35 40

∆s [m]

-1

-0.5

0

0.5

1

∆η[m

]

Fig. 4: Lane change (up) and lane keeping (bottom) trajecto-ries fitting: comparison between the real trajectory (red) andsome calculated trajectories (blue) given the initial conditions(black cross). The best fitting trajectory has thicker circles.

Fig. 5: Lane keeping parameters PDFs fitting.

Fig. 6: Lane changing parameters PDFs fitting.

prediction model (Sections III-B, III-C). The predicted posi-tions are based on (5) for a LK and (11) for a LC maneuver,and they are estimated at the discrete sample times.

The range occupied by a TV is called target area andmodeled by a rectangle. For each scenario of TV j andat each time step t, the vertices of the correspondingtarget area are associated with the set ρ(k)j,t , where k =1, . . . ,K is the scenario index. For each type of maneuvermj ∈ {LCL,LCR,LK}, the sets ρ

(k)j,t are combined into

the convex hull Rmj

j,t containing all the corresponding ρ(k)j,t ,

e.g., RLCLj,t contains Kj,LCL subsets ρ

(k)j,t associated to the

LCL maneuver. Each convex hull includes a number ofcoordinates of vertices of the corresponding target areasand it is characterized by a centroid (Cx, Cy), or geometriccenter, which is used to reference the position of the wholeset. An example of scenario-based predictions of a vehicleperforming a lane change is shown in Fig. 7. The rectangularshapes representing the target areas are included inside theirregular polygons corresponding to the convex hulls Rmj

j,t

considered within the prediction horizon. To reduce thecomputational complexity, the convex hull is substituted bythe smallest rectangular box aligned with the lane whichcontains the convex hull. This exemplification was chosen at

Fig. 7: Scenario-based predictions of a LCL maneuver. Thevehicle area (blue box while in the initial lane, red box whenmoving to the other lane) is drawn for each scenario at allpredicted time steps. The green convex hull represents thetarget area. ∆s and ∆η are the TV’s longitudinal and lateraldisplacement with respect to its initial position.

the cost of introducing a larger uncertainty in TV prediction.

B. Safety Distances

Safety distances are included in the control algorithm byexpanding the area occupied by each TV. First, each TVis embedded in a bigger safety box. Then, for each TVj, the vertices of the associated safety box substitute thevertices in the sets ρ

(k)j , k = 1, . . . ,K. As a result, the

corresponding convex hulls Rmj , m ∈ {LCL, LCR, LK}become enlarged as well. For simplicity of notation, sincethe safety constraints are generated at each horizon step, theindex t is dropped in the sequel.

Define E andRmj as the closed sets of the points inside therectangle representing the EV’s shape and the points insidethe convex hull Rmj , respectively. Hence, the signed distancebetween the two sets is given by

ξmj ,

min

a∈E, b∈Rmj

d(a, b) if@ b ∈ E , a ∈ Rmj ,

− maxa∈E, b∈Rm

j

{d(a, b)| a ∈ Rmj , b ∈ E}if ∃ b ∈ E , a ∈ Rmj .

(23)

where d(a, b) is the Euclidean distance. When ξmj > 0, itmeans that the EV is keeping the required safety distancefrom the scenario-based prediction of TV j.

The EV must strictly respect the safety distances under allsampled scenarios. Because of the finite sampling, there isa residual probability ε that the safety distances may indeedbe violated. This probability will be bounded in Section VII.

V. LANE CHANGE ASSISTANCE

The objective of SCMPC for lane change assistance is todetect and perform safe lane change trajectories on highways.A safe trajectory is considered to be one that respects allactuator, comfort, and safety constraints, as described inSection II. In order to simplify the optimization problem,the safety constraints to TVs, as described in Section IV-B,are simplified to comprise only longitudinal safety distances.

Assumption 2 (Safety Distances) (a) If the EV stays on itscurrent lane, it is only responsible to maintain a longitudinalsafety distance dEV to its front vehicle. (b) In a lane changesituation, the EV needs to maintain a safety distance dEV

to the TVs in its front on the current lane and target lane,as well as a safety distance dTV to the TV behind it onthe target lane. (c) In closed-loop, the safety distances mayoccasionally be violated (e.g., if the TVs perform emergencybraking maneuvers).

Lateral constraints are also in place, confining the EV to itscurrent lane or, in a lane change situation, to its current laneand target lane. Assumption 2 will translate into the chanceconstraints on the state of the EV [14], [25]. In a particularscenario δ(k)t , the safety distances translate directly into stateconstraints on the longitudinal and lateral position of the EV.

If a lane change is desired, the algorithm requires it tobe triggered by a human driver, e.g., via the turn signal orby a higher-level path planner, e.g., for autonomous driving.Hence the initiation of a lane change is given as an externalsignal to the SCMPC controller.

As illustrated in Fig. 8, the current lane (cl) and target lane(tl) are considered separately under each scenario. The safetydistances on lane l ∈ {cl, tl} under scenario k ∈ {1, . . . ,K}are based on the trajectories of all TVs, as given by δ(k)t .

TVs behind the EV are required to maintain a safetydistance dTV, depending on the corresponding TV’s speed.TVs in the front require the EV to maintain a safetydistance of dEV that varies with the EV’s speed. Hence thelongitudinal position of the EV (s coordinate) is constrainedby a safety interval [s

l,(k)min,t, s

l,(k)max,t], as shown in Fig. 8. The

lateral position of the EV (η coordinate) is constrained bythe boundaries of the current lane (l = cl) and/or the targetlane (l = tl), [ηlmin,t, η

lmax,t].

s

η

cl

tldTV dEV

dEVdEV

︸ ︷︷ ︸

safety interval on current lane (cl)

safety interval on target lane (tl)︷ ︸︸ ︷

Fig. 8: Construction of safety intervals [sl,(k)min,t, s

l,(k)max,t] at time

step t for a particular scenario k. The EV is shown in darkgray and three TVs in light gray. The hatched regions on thecurrent lane l = cl and the target lane l = tl are unsafe.

A. Lane Change Timing

The SCMPC algorithm has a prediction horizon of N timesteps, i.e., a total preview time of Nts. It consists of an outerloop and an inner loop. The outer loop iterates to find theearliest time to initiate a lane change tlc within the interval[0, Nts]. The inner loop attempts to compute a safe trajectorygiven the lane change initiation time tlc; i.e., observing thelane change constraints under all K scenarios.

Assumption 3 (Lane Change Constraints) (a) When inte-grated into the current (or target) lane, the EV must observe

the safety interval on the current (or target) lane. (b) Duringa LC maneuver, the EV must respect also the safety intervalon the target lane. (c) During a LC maneuver, the EV mustbe able to safely return to the current lane, so it must observethe safety intervals on both the current and target lanes.

The EV may not be able to move into the safety intervalon the target lane immediately, as in Fig. 8. Then it canfirst adjust its position and speed on the current lane beforeperforming the lane change. It is assumed that an earlier lanechange time is always preferred. The outer loop performs asimple search over a set of candidate lane change times

Tlc ⊂{

0, ts, . . . , N · ts}. (24)

For each candidate lane change time tlc ∈ Tlc, the innerloop computes the control inputs (steering, acceleration) andevaluates the feasibility of a lane change within tlc; seeSection V-C. Note that, by Assumption 3, if a lane changeis feasible within tlc, then it is also feasible within anycandidate lane change time greater than tlc. Hence a simplebisection algorithm can be used for the outer loop.

B. State ConstraintsLet [s

l,(k)min,t, s

l,(k)max,t] denote the safety interval on lane

l ∈ {cl, tl} at time t ∈ {1, . . . , N} under scenario k ∈{1, . . . ,K}:

slmin,t , maxk∈{1,...,K}

sl,(k)min,t , (25a)

slmax,t , mink∈{1,...,K}

sl,(k)max,t . (25b)

The longitudinal limits smin,t, smax,t and the lateral limitsηmin,t, ηmax,t are then defined as the limits of the currentlane while t ≤ tlc and that of both lanes when t > tlc. Theresulting state constraints are

xmin,t ≤ xt ≤ xmax,t , (26)

where xmin,t , [smin,t ηmin,t −∞ 0 lt]T and xmax,t ,

[smax,t ηmin,t +∞ vmax lt]T, and vmax denotes the speed limit.

Remark 4 (Constraint Consolidation) Given the sampledvalues of all scenarios δ(1)t , . . . , δ

(K)t , the safety intervals for

each lane and hence the state constraints (26) can be readilycomputed for each time step t.

C. Multi-Stage Scenario ProgramFor the inner loop of SCMPC, the lane change time tlc is

fixed. The control inputs ut, . . . ,ut+N−1 over the predictionhorizon are then computed by solving the following Multi-Stage Scenario Program (MSSP):

minuT ,xT+1

∑i∈T

(xi+1 − xref,i+1)TQ(xi+1 − xref,i+1)

+ uTiRui , (27a)

s.t. xi+1 = fi(xi,ui

)∀ i ∈ T , (27b)

umin(xi) ≤ ui ≤ umax(xi) ∀ i ∈ T , (27c)umin ≤ ui+1 − ui ≤ umax ∀ i ∈ T , (27d)xmin,i+1 ≤ xi+1 ≤ xmax,i+1 ∀ i ∈ T . (27e)

Here T , {t, . . . , t+N−1}, and Q ∈ R5×5, R ∈ R2×2 arepositive definite weighting matrices for tuning. The referencetrajectory xref,t+1, . . . ,xref,t+N for the EV is constructedaccording to the reference speed and the lane change time tlc.Since the SCMPC chooses the final trajectory in a safe anddynamically feasible manner, the exact shape of the referenceis secondary. Here it is chosen as a simple step change inthe lateral coordinate.

The MSSP is a nonlinear program, which can be solved bystandard algorithms [26], [27]. The objective function (27a)is a convex quadratic function that penalizes the deviationof the state from the reference as well as the input usage.The dynamic constraints (27b) are nonlinear. The inputconstraints (27c,d) can be linearized by substituting the initialstate xi ≡ xt. The state constraints (27e) are implementedas soft constraints, to ensure feasibility of the program.

VI. AUTONOMOUS HIGHWAY DRIVING

A. State Constraints

In the case of autonomous driving, the state constraint gen-eration is based on classifying the TVs into four categories.The categories depend on the longitudinal distance betweenthe TV and the EV, and the lane in which the TV is driving.

Let ∆s and ∆η be the longitudinal and lateral distancein the lane reference frame of a TV convex hull R centroid(Cs, Cη) and the EV. It follows that

∆s = Cs − s and ∆η = Cη − η.

A TV convex hull R is defined to be on the same laneas its centroid. Thus, as shown in Fig. 9, a scenario-basedprediction of a TV, denoted by R, belongs to category T1, T2,T3 or T4 depending on: (i) the relative longitudinal distance∆s between the EV and R and (ii) the lane on which R lays.Since each category may include more than a TV convexhull R, only that which is characterized by the smallestvalue of ξ is considered in the process of generating theconstraint. We formulate the obstacle avoidance constraintsas linear constraints on the state of the ego vehicle (similar to[28]). The constraint generation algorithm makes use of thelongitudinal and lateral extremity values of the TV convexhulls. In particular, the lateral maximum distances from thehull’s centroid are denoted σleft and σright while the frontand the rear distances are denoted σfront and σrear. Given aconvex hull R in the lane reference frame with n vertices(si, ηi), i ∈ {1, . . . , n} including the safety boxes of a TV’sscenario-based predictions, define

σright(left) , min(max)i∈{1,...,n}

{ηi − Cη

∣∣∣(si, ηi) ∈ R}, (28)

σfront(rear) , max(min)i∈{1,...,n}

{si − Cs

∣∣∣(si, ηi) ∈ R}. (29)

The algorithm considers for every category three rangesof |∆s| delimited by parameters d1 and d2 chosen so thatd1 > d2 > 0. Therefore, three distinct rules are applied togenerate the constraint depending on |∆sj |.

Ego

T1

T2T3

T4Lane 1

Lane 2

∆sT < 0 ∆sT ≥ 0

Ego T1

Ego & T1 on same lane

j"sT j > d1

Ego

T1

Ego & T1 on di,erent lanes

Ego T1d2 < j"sT j 5 d1

Ego

T1

Ego T1j"sT j 5 d2

Ego

T1

Fig. 9: Target vehicles notation (top) and T1 constraintgeneration (bottom). In all figures the vehicles are assumedto drive towards the right direction.

• |∆sj | > d1: a linear constraint is set perpendicular tothe lane direction, at the TV’s closest vertex to the EV.Hence, {

s < Cs − σrear, for T1,T2,s > Cs + σfront, for T3,T4.

(30)

• d2 < |∆sj | ≤ d1: a linear constraint is built connectinga vertex of the EV to one of the TV are in order toobtain the steepest positive slope (for T1 and T3) ornegative slope (for T2 and T4). In case of T1 and T3,the slope never decreases below 0; while for T2 and T4,it never increases above 0.

• |∆sj | ≤ d2: if the EV and TV are on the same lane, alinear constraint is set perpendicular to the lane directionat the TV’s closest vertex to the EV as in (30). Instead,if the EV and TV are on different lanes, a constraintparallel to the direction of march is applied to the TV’sclosest vertex to the EV. That is{

η < Cη + σleft, for T1,T4,η > Cη − σright, for T2,T3.

(31)

Fig. 9 shows an example of the rules used to generatethe constraints for the T1 case. In this figure, the constraintis drawn as a red line, while the safety box is the graysurface centered on the corresponding TV. The constraintsfor the other cases (T2, T3 and T4) can be deducted bysymmetrically applying the given rules.

Finally, since the safety constraints on the state are linear,they can be expressed at each time step t in the formAs,txt ≤ bs,t where the subscript s underline the fact thatthese scenario-based constraints are generated as a result ofa stochastic process.

B. Multi-Stage Scenario Program

The optimal control input u∗t , for t ∈ T , {t, . . . , t+N−1}, is computed at every time step by solving the followingMulti-Stage Scenario Program (MSSP):

minui

(∑i∈T

(xi − xref,i)TQ(xi − xref,i)

+ uTiRui + ∆uT

iRR∆ui

)+ (xN − xref,N )TQN (xt+N − xref,t+N ), (32a)

s.t. xi+1 = fi(xi,ui), ∀ i ∈ T, (32b)

∆ui =(ui − ui−1

)/ts, ∀ i ∈ T, (32c)

Ad,i+1xi+1 ≤ bd,i+1, ∀ i ∈ T, (32d)As,i+1xi+1 ≤ bs,i+1, ∀ i ∈ T (32e)umin ≤ ui ≤ umax, ∀ i ∈ T, (32f)∆umin ≤∆ui ≤∆umax, ∀ i ∈ T. (32g)

Here Q ∈ R5×5, R ∈ R2×2, Rr ∈ R2×2 and QN ∈ R5×5

are positive definite weighting matrices, which can be usedfor tuning. The matrices Q and QN penalize the deviationof the state from the reference trajectory, while R and RR

respectively weight the magnitude and the rate of the inputs.The system prediction (32b) is initialized with the state xt ofthe EV, measured at time t. The polyhedron defined by thedeterministic constraint is described by Ad,i+1 and bd,i+1,while the scenario-based constraints are expressed by As,i+1

and bs,i+1.

VII. BOUND ON SAFETY DISTANCE VIOLATIONS

So far, no indication has been made for the selectionof the sample size K. In this section, the framework ofscenario-based optimization [15]–[17] is used to establisha connection between K and the satisfaction of chanceconstraints, in the sense defined below [25].

Definition 5 (Chance Constraints) (a) A chance constraintis defined by the frequency of safety distance violations Vtbetween EV and TVs over time t = 0, 1, . . . . In particular, theSCMPC controller must guarantee that the safety distanceof each EV and TV is not violated in more than a fractionε ∈ (0, 1) of all time steps t = 0, 1, . . . .

Here ε ∈ (0, 1) is a tuning parameter that represents theconservatism of the driving style. A small ε corresponds to alow-risk driving style, a large ε makes the controller performmore aggressive maneuvers. The following results are basedon the SCMPC theory in [14], to which the reader is referredfor further details.

Let P[Vt] denote the probability of a safety constraintviolation at time t. It is hard to bound P[Vt] directly, becausethe MSSPs in this paper are not convex, as required by thetheory presented in [14]. However, for each case it can beshown that the support constraints contributed to the MSSPthrough the traffic scenarios can be upper bounded, eventhough the optimization problem is not convex. The argumentin both cases is essentially the same: for the case of lane

change assistance, all but one constraints are completelyredundant; and for the case of autonomous highway driving,all but two constraints are redundant. The following technicalassumption is required for the theoretical analysis of thealgorithms presented in Sections V and VI.

Assumption 6 (Feasibility) (a) There always exists a fea-sible solution to the MSSPs (27) and (32). (b) For the lanechange assistance algorithm, the outer loop iterations arereplaced by a fixed lane change time tlc; e.g., the highestvalue in Tlc.

A. Lane Change Assistance

Let the lane l ∈ {cl, tl} and time step t ∈ {0, 1, . . . } befixed and consider the front safety distance i ∈ {1, . . . , N}steps into the future (the back safety distance works analo-gously). The violation of the front safety distance can occuronly if slmax,t+i has been chosen higher than

slmax,t+i

(δt), (33)

which is the unknown true longitudinal position for the EVunder the unknown true scenario δt:

P[Vt]≤ P

[slmax,t+i

(δt)< slmax,t+i

]. (34)

Therefore, to upper bound P[Vt]

by ε, it suffices to upperbound the right-hand side of (34). This can be accomplishedby a basic sampling lemma [29].

Theorem 7 (Violation Bound) Let Assumptions 1, 2, 3, 6hold and (27b) be an exact model of the controlled vehicle.Then

P[slmax,t+i

(δt)< slmax,t+i

]≤ 1

K + 1, (35)

and hence the sample size K ≥ 1ε − 1 ensures satisfaction

of the chance constraints according to Definition 5.

Proof: All scenarios δ(1)t , . . . , δ(K)t are independent and

identically distributed with δt (Assumption 1), so each oneis equally likely to generate the tightest bound for the safetydistance [29], and hence (35). State constraint satisfactionfollows by Assumption 6 along the lines of [14].

B. Autonomous Highway Driving

The argument now follows a similar pattern to the previouscase. By construction, it holds true that if the safety distanceis violated, i.e., the signed distance ξmj,k is smaller than 0,then the scenario-based constraints are violated, i.e.,

As,t(δt)xt > bs,t+1(δt) . (36)

Note that the converse is not true. However, it follows that

P[Vt]

= P[ξmj,t ≤ 0, ∀j, m = LCL, LCR, LK

]≤

P[As,t(δt)xt > bs,t+1(δt)

]. (37)

Hence, in order to upper bound the safety distance violations,it suffices to upper bound the right-hand side of (37).

At a fixed time t, the front and lateral safety distancesare considered. While performing lane keeping, the EV is

only responsible for maintaining a safety distance to thefront TV (T1 or T2) and the lateral TV (the most restrictivebetween T1 and T4, or T2 and T3) . During a LC maneuver,the EV must also respect the safety distance required bythe oncoming vehicles (T3 or T4). In both cases, by thegeometric constructions described in Section VI-A, there canbe at most two support constraints.

Theorem 8 (Violation Bound) Let Assumptions 1, 2, 6hold and (32b) be an exact model of the controlled vehicle.Then

P[As,t(δt)xt > bs,t+1(δt)

]≤ 2

K + 1, (38)

and hence the sample size K ≥ 2ε − 1 ensures satisfaction

of the chance constraints according to Definition 5.

VIII. EXPERIMENTAL RESULTS

In this section, we demonstrate the ability of the SCMPCalgorithm for autonomous highway driving, as describedin Section VI, to operate in real-time on our experimentalvehicle in the presence of virtual TVs. An Oxford TechnicalSolutions (OxTS) RT 2002 sensing system, comprising of aglobal positioning system (GPS) and an inertial measurementunit (IMU), combined with a GPS base station are used forlocalizing the vehicle. Computations related to the SCMPCalgorithm are performed on a Speedgoat real-time mobiletarget machine (2.16 GHz Intel Core 2 Duo processor, 1GB RAM) running a Simulink Real-Time application. Thesensors, actuators and computing platform on the vehiclecommunicate via a CAN bus.

The MSSP (32) is solved in real-time using the nonlinearoptimization package NPSOL [30]. A prediction horizon ofN = 10 is used which results in 2 × N = 20 optimizationvariables. There are NTV number of state constraints per timestep of the horizon yielding a total of N · NTV constraints.The scenario-based traffic prediction and controller run at asampling time of 0.2 s which is the maximum computationtime of the proposed approach on our platform.

The traffic scene for the experiments consists of one TVinitially in the left lane with the EV in the right lane. Forsafety reasons, the reference speed for the EV is 8 m/s. Avirtual (simulated) TV is used which tracks a reference speedof 4 m/s. At t = 0 s, the vehicles are at rest and the TV is100 m ahead of the EV. The TV begins a lane change to cutin front of the EV at approximately t = 4 s.

Snapshots of the traffic scene during the experiment at fourtime instants are shown in Fig. 10. The solid blue and redrectangles depict the EV and TV, respectively, at the giventime instants. The shaded blue rectangles show the predictedopen-loop trajectory of the EV as computed by the SCMPCalgorithm. The shaded red rectangles are the convex hulls ofthe TV areas over the prediction horizon. The solid blue andred lines are the closed-loop trajectories of the EV and TV,respectively.

At t = 1 s (Fig. 10a), the EV and TV are proceedingin the same direction on different lanes. At t = 11 s (Fig.10b), the TV is in the process of changing lane. It is seen

s [m]

η[m

]

100 150 200−4

−2

0

2

4

6

8

(a) t = 1 s

s [m]

η[m

]

150 200 250−4

−2

0

2

4

6

8

(b) t = 11 s

s [m]

η[m

]

220 240 260 280 300−4

−2

0

2

4

6

8

(c) t = 22 s

s [m]

η[m

]

240 260 280 300 320−4

−2

0

2

4

6

8

(d) t = 25 s

Fig. 10: Snapshots of the traffic scene during the experiment.The EV (blue) and a TV (red) are represented with theircorresponding open-loop predictions (light blue and red) andclosed-loop trajectories (solid lines).

that the actual trajectory of the TV lies within the scenariobased predictions of the TV’s future motion. Also, the speeddifference of the two vehicles is making the EV approachthe TV ahead. However, the EV is still far away from theTV, hence, it plans to stay in its lane. At t = 22 s (Fig. 10c),the TV is classified as a T1 vehicle and the relative distancebetween the TV and the EV is between the thresholds d1 andd2 (see Section VI-A). The corresponding safety constraintcauses the EV to plan a collision avoidance maneuver andchange lanes to the left. Fig. 10d shows the traffic scene att = 25 s while the EV is changing lanes, gaining the abilityto overtake the slower TV on the left.

IX. CONCLUSION

This paper has presented algorithms to perform safe lanechanges on highways. It has been shown how scenario-basedtraffic prediction models can be combined with the latestdevelopments in scenario MPC. A maneuver-based model torapidly predict lane keeping and lane changing trajectoriesof target vehicles has been developed and validated usingreal traffic data from a large dataset. The real-time capa-bility of the proposed approach has been demonstrated viaexperiments on our test vehicle.

Future research will focus on extensive field tests onhighways in real traffic. The sensors on our experimentalvehicle will be used to detect surrounding target vehicles.Furthermore, it will involve a more detailed study andmodeling of traffic scenarios, based on actual traffic data, toreduce the uncertainty affecting the probabilistic distributionsfrom which scenarios are sampled.

ACKNOWLEDGMENTS

This material is based upon work partially supported bythe National Science Foundation under Grant No. 1239323.Any opinions, findings, and conclusions or recommendationsexpressed in this material are those of the authors and donot necessarily reflect the views of the National ScienceFoundation. The work was also supported by the HyundaiCenter of Excellence at UC Berkeley. Gianluca Cesari wasfunded by the Master Thesis Grant issued by the Zeno KarlSchindler Foundation.

REFERENCES

[1] S. Lefevre, D. Vasquez, and C. Laugier, “A survey on motion predic-tion and risk assessment for intelligent vehicles,” Robomech Journal,vol. 1(1), pp. 1–14, 2014.

[2] J. Jansson, “Collision avoidance theory with application to automo-tive collision mitigation,” Ph.D. Dissertation, Linkoping University,Linkoping, Sweden, 2005.

[3] M. Althoff, D. Heß, and F. Gambert, “Road occupancy predictionof traffic participants,” Proceedings of the 16th International IEEEConference on Intelligent Transportation Systems, pp. 99–105, 2013.

[4] M. Althoff and J. M. Dolan, “Online verification of automatedroad vehicles using reachability analysis,” In IEEE Transactions onRobotics, vol. 30, pp. 903–918, 2014.

[5] C. Laugier, I. E. Paromtchik, M. Perrollatz, M. Yong, J.-D. Yoder,C. Tay, K. Mekhnacha, and A. Negre, “Probabilistic analysis ofdynamic scenes and collision risks assessment to improve drivingsafety,” IEEE Intelligent Transportation Systems Magazine, vol. 2(4),pp. 4–19, 2011.

[6] J. Wiest, M. Hoffken, U. Kreßel, and K. Dietmayer, “Probabilistictrajectory prediction using gaussian mixture models,” in IntelligentVehicles Symposium, Alcala de Henares, Spain, 2012.

[7] A. Broadhurst, S. Baker, and T. Kanade, “Monte carlo road safetyreasoning,” in Intelligent Vehicles Symposium, Las Vegas (NV), UnitedStates, 2005.

[8] A. Eidehall and L. Petersson, “Statistical threat assessment for generalroad scenes using monte carlo sampling,” IEEE Transactions onIntelligent Transportation Systems, vol. 9(1), pp. 137–147, 2008.

[9] J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theoryand Design. Madison (WI): Nob Hill Publishing, 2009.

[10] A. Carvalho, S. Lefevre, G. Schildbach, J. Kong, and F. Borrelli,“Automated driving: The role of forecasts and uncertainty: A controlperspective,” European Journal of Control, vol. 24, pp. 14 – 32, 2015.

[11] J. Kong, M. Pfeiffer, G. Schildbach, and F. Borrelli, “Autonomousdriving using model predictive control and a kinematic bicycle vehiclemodel,” in Intelligent Vehicles Symposium, Seoul, Korea, 2015.

[12] A. Carvalho, Y. Gao, S. Lefevre, and F. Borrelli, “Stochastic predictivecontrol of autonomous vehicles in uncertain environments,” in 12thInternational Symposium on Advanced Vehicle Control, Tokyo, Japan,2014.

[13] L. Blackmore, M. Ono, A. Bektassov, and B. C. Williams, “A proba-bilistic particle-control approximation of chance-constrained stochasticpredictive control,” IEEE Transactions on Robotics, vol. 26(3), pp.502–516, 2010.

[14] G. Schildbach, L. Fagiano, C. Frei, and M. Morari, “The scenarioapproach for stochastic model predictive control with bounds onclosed-loop constraint violations,” Automatica, vol. 50(12), pp. 3009–3018, 2014.

[15] G. Schildbach, L. Fagiano, and M. Morari, “Randomized solutions toconvex programs with multiple chance constraints,” SIAM Journal onOptimization, vol. 23(4), pp. 2479–2501, 2013.

[16] M. C. Campi and S. Garatti, “The exact feasibility of randomized so-lutions of uncertain convex programs,” SIAM Journal on Optimization,vol. 19, pp. 1211–1230, 2008.

[17] ——, “A sampling and discarding approach to chance-constrainedoptimization: Feasibility and optimality,” Journal of OptimizationTheory and Applications, vol. 148, pp. 257–280, 2011.

[18] R. Rajamani, Ed., Vehicle Dynamics and Control, 2nd ed. New York:Springer, 2012.

[19] M. Werling, J. Ziegler, S. Kammel, and S. Thrun, “Optimal trajectorygeneration for dynamic street scenarios in a Frenet frame,” in IEEEConference on Robotics and Automation, Anchorage (AK), UnitedStates, 2010.

[20] C. M. Kang, S.-H. Lee, and C. C. Chung, “Comparative evaluationof dynamic and kinematic vehicle models,” in 53rd Conference onDecision and Control, Los Angeles (CA), United States, 2014.

[21] S. Lefevre, Y. Gao, D. Vasquez, E. Tseng, R. Bajcsy, and F. Borrelli,“Lane keeping assistance with learning-based driver model and modelpredictive control,” in 12th International Symposium on AdvancedVehicle Control, 2014.

[22] U.S. Department of Transportation, “Next generation simulation pro-gram, U.S Highway 101 dataset.” http://ngsim.fhwa.dot.gov, Jan. 2007.

[23] G. Schildbach, “Scenario-based optimization for multi-stage stochasticdecision problems,” Ph.D. Dissertation, Eidgenossische TechnischeHochschule Zurich, Zurich, Switzerland, 2014.

[24] Y. Gao, A. Gray, H. E. Tseng, and F. Borrelli, “A tube-based robustnonlinear predictive control approach to semiautonomous groundvehicles,” Vehicle System Dynamics, vol. 52(6), pp. 802–823, 2014.

Gianluca Cesari received his ‘Laurea’ degreein Automation Engineering (2012) from the Uni-versity of Bologna, Italy, and B.Eng. degree inElectrical Engineering (2013) from the Tongji Uni-versity, Shanghai, China. In 2016, he completedhis M.S. in Robotics, Systems and Control atETH Zurich, Switzerland. During his education,he has worked more than one year in the fieldof industrial automation. Now, his area of interestand specialization is optimization-based predictivecontrol applied to autonomous mobile robots.

Georg Schildbach Georg Schildbach obtainedhis Masters degrees in Applied Mechanics (Dipl.-Ing.) and Industrial Engineering (Dipl. Wirtsch.-Ing.) from the Technical University of Darmstadtin 2008. After working in the financial industryfor two years, he joined the Automatic ControlLaboratory at ETH Zurich, where obtained hisPh.D. degree in the field of control and opti-mization in 2014. He held the position of anAssociate Director at the Hyundai of Excellenceat UC Berkeley until 2016, and now works for

Elektronische Fahrwerksysteme GmbH in the area of vehicle control andautomated driving.

[25] G. Schildbach and F. Borrelli, “Scenario model predictive control forlane change assistance on highways,” in IEEE Intelligent VehiclesSymposium, Seoul, South Korea, 2015, pp. 611–616.

[26] D. Luenberger and Y. Ye, Linear and Nonlinear Programming, 3rd ed.Berlin et al.: Springer, 2008.

[27] J. Nocedal and S. Wright, Numerical Optimization, 2nd ed. NewYork: Springer, 2006.

[28] J. Nilsson, Y. Gao, A. Carvalho, and F. Borrelli, “Manoeuvre gener-ation and control for automated highway driving,” in The 19th worldcongress of the international federation of automatic control (IFAC),2014.

[29] G. C. Calafiore and M. C. Campi, “Uncertain convex programs:Randomized solutions and confidence levels,” Mathematical Program-ming, Series A, vol. 102-1, pp. 25–46, 2005.

[30] P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright, “User’sguide for NPSOL: Version 4.0,” 1986.

Ashwin Carvalho received the B.Tech. degree inMechanical Engineering from the Indian Instituteof Technology Bombay in 2011. He is currentlypursuing the Ph.D. degree in Mechanical Engi-neering (Controls) at the University of California,Berkeley, with minors in Artificial Intelligenceand Optimization. His research interests lie instochastic model predictive control, environmentmodeling and real-time optimization for controlunder uncertainty.

Francesco Borrelli received the ‘Laurea’ degreein computer science engineering in 1998 fromthe University of Naples ‘Federico II’, Italy. In2002 he received the PhD from the AutomaticControl Laboratory at ETH-Zurich, Switzerland.He is currently a Professor at the Departmentof Mechanical Engineering of the University ofCalifornia at Berkeley, USA.

He is the author of more than one hundredpublications in the field of predictive control. Heis author of the book Constrained Optimal Control

of Linear and Hybrid Systems published by Springer Verlag, the winner ofthe 2009 NSF CAREER Award and the winner of the 2012 IEEE ControlSystem Technology Award.

In 2008 he was appointed the chair of the IEEE technical committeeon automotive control. His research interests include constrained optimalcontrol, model predictive control and its application to advanced automotivecontrol and energy efficient building operation.