Cooperative look-ahead control for fuel-efficient and safe heavy-duty vehicle platooning
11 Cooperative look-ahead control for fuel-efficientand safe heavy-duty vehicle platooning
Valerio Turri, Bart Besselink, Karl H. Johansson
Abstract —The operation of groups of heavy-duty vehicles(HDVs) at a small inter-vehicular distance (known as platoon)allows to lower the overall aerodynamic drag and, therefore, toreduce fuel consumption and greenhouse gas emissions. However,due to the large mass and limited engine power of HDVs,slopes have a significant impact on the feasible and optimalspeed profiles that each vehicle can and should follow. Thereforemaintaining a short inter-vehicular distance as required byplatooning without coordination between vehicles can often resultin inefficient or even unfeasible trajectories. In this paper wepropose a two-layer control architecture for HDV platooningaimed to safely and fuel-efficiently coordinate the vehicles inthe platoon. Here, the layers are responsible for the inclusionof preview information on road topography and the real-timecontrol of the vehicles, respectively. Within this architecture,dynamic programming is used to compute the fuel-optimal speedprofile for the entire platoon and a distributed model predictivecontrol framework is developed for the real-time control of thevehicles. The effectiveness of the proposed controller is analyzedby means of simulations of several realistic scenarios that suggesta possible fuel saving of up to % for the follower vehiclescompared to the use of standard platoon controllers. I. I
NTRODUCTION
The transportation of goods has been fundamental to theworld economic development and the demand for freighttransportation, together with the global economy, is expectedto increase in the coming years. However, the transport sector,due to the burning of fuel, is responsible for a significantamount of greenhouse gas and CO emissions. In the EuropeanUnion, the transport sector amounts to roughly 29% of thetotal CO emissions and 11% of these emissions are directlyaccountable to road freight transportation [3], [17]. Globally,the CO emissions linked to the surface (road and rail) freighttransport sector are expected to increase up to 347% in thenext 40 years if no measure is taken [19]. In order to contrastthis increase and the related impact on the climate change,governments all over the world are agreeing in setting stringentlimitations on greenhouse gas emissions connected to roadfreight transportation [4], [16]. In order to cope with theselimitations, heavy-duty vehicle (HDV) manufactures are facingnumerous challenges. Furthermore, the expected increase ofthe oil price [19] and the need for maintaining competitivenessrequire them to design vehicles and technologies that areincreasingly fuel-efficient. The fuel cost for an HDV fleetowner, in fact, accounts roughly for the 35% of the totalcost of owning and operating a vehicle [2]. Therefore evena reduction of a few percent of the fuel consumption wouldlead to significant saving. The authors are with the ACCESS Linnaeus Centre and Department ofAutomatic Control, KTH Royal Institute of Technology, Stockholm, Sweden,email: [email protected], [email protected], [email protected].
An effective method to reduce fuel consumption and, con-sequently, greenhouse gas emissions, is HDV platooning. Byoperating groups of vehicles at small inter-vehicular distances,the overall aerodynamic drag can be reduced. As about onefourth of the HDV fuel consumption is related to aerodynamicdrag [20], platooning can have a large effect on the fuelconsumption. Indeed, experimental results in [5] and [13]have shown a reduction in fuel consumption up to 7%.However, in order to safely operate HDVs at the short inter-vehicular distances required for platooning, automation of thelongitudinal dynamics is necessary.In this work we present a novel control architecture forfuel-efficient and safe HDV platooning. Vehicle platooningis not a new control problem. The first works on vehicleplatooning appeared in the sixties, e.g., [18], [28], [30]. Themain focus of these early works was the theoretical study ofthe dynamics of a string of vehicles with local information,with a particular attention on the study of string stability,i.e., the attenuation of disturbances in position, speed andacceleration along the string of vehicles. The vehicle pla-tooning concept received the first application interest underthe Partners for Advanced Transportation Technology (PATH)project [32], where platooning (of passenger vehicles) has beeninvestigated as a means to increase highway throughput. Underthis project a control architecture based on vehicle-to-vehiclecommunication has been developed for the platoon formationand maintenance of fully autonomous vehicles [22]. Althoughthe environmental aspect was not the focus of the project,noteworthy results on fuel reduction due to HDV platooninghave been reported [15]. Since then, new projects and relatedpublications have appeared with focus on different aspects ofHDV platooning such as congestion, safety, fuel-efficiency anduser-acceptance [10], [11], [33].In the more recent COMPANION project [1], where thispaper finds its place, the fuel-efficiency of HDV platooningis the main focus. The project is not limited to pursue theefficiency of a single platoon, but rather to create a completefuel-efficient freight transportation system. This led to thedevelopment of a system architecture aimed to divide thiscomplex problem into hierarchical solvable subproblems [5],[27]. An adaptation of such architecture has three layers,namely the mission planner, the platoon controller and thelow-level vehicle controller, defined as follows: the missionplanner is responsible for the optimal routing of the HDVs andtheir synchronization in order to create and dissolve platoonsin optimized meeting points. This problem has been addressedin [26] where the authors propose a distributed framework forthe synchronization of single HDVs and platoons on the roadnetwork. The platoon controller of each platoon receives from a r X i v : . [ c s . S Y ] M a y the mission planner the optimal route and the average speedper link that the platoon should track. Therefore it controls thevehicles’ dynamics and it computes the inputs for the low-levelvehicle controllers of each HDV. In [7] a distributed controlframework over the platoon suitable for HDV dynamics ispresented. However the role of external factors, such as slopes,is not taken into account.Because of the large mass and the limited power of HDVs,altitude variations have a significant impact on their behavior.Even small slopes produce such large longitudinal forces onthe HDVs that they are often not able to keep constant speedduring uphill segments (because of limited engine power) andduring downhill segments without applying brakes (becauseof the significant inertia). Hence it is common that HDVshave to brake and therefore waste energy in order not toovercome the speed limit during downhill sections. This hasbeen addressed in [21] in order to design a control system thatoptimizes the fuel consumption of single HDVs driving overhilly roads. In this work the authors showed how, by usinglook-ahead control based on road topography information andspeed limits, it is possible to reduce the fuel consumptionof a single HDV up to . %. Slopes become more criticalin the case of HDVs driving in a platoon formation. In [8],the authors point out how the existing look-ahead strategiesfor single HDVs are not necessarily suitable for a platoonand that a dedicated approach is required. This is due tothe fact that the additional requirement of keeping a smallinter-vehicular distance between vehicles collides with thefact that HDVs experience significantly different longitudinalforces (e.g., gravity force depending on their mass and currentroad slope and air drag resistance depending on the distancefrom the previous vehicle). This appeared evident in theexperimental results of a three-vehicle platoon driving on ahighway presented in [5]. Even though in this work the HDVshave similar characteristics, the author highlighted how theuse of feedback controllers in particularly hilly sectors of thehighway could lead to an increase of the fuel consumptionof the follower vehicles compared to the case in which aredriving alone. These experimental data are further analyzed inSection II in order to obtain a good understanding of the role ofthe road gradient on HDV platooning. This analysis providesa motivation for the development of a novel cooperative look-ahead control for HDV platooning with the specific objectiveof fuel-efficiency, for which some early results have been pub-lished in [34]. Hence, this leads to the following contributionsof the current paper.First, a control architecture for the fuel-efficient and safecontrol of an HDV platoon is presented. The control architec-ture is divided into two layers, namely the platoon coordinatorand the vehicle controller layers. The platoon coordinatorcomputes the fuel-optimal speed profile for the entire platoonby taking into account information about the road ahead. Thisoptimal profile is communicated to the decentralized vehiclecontroller layer that safely tracks it and computes the real-timeinputs for each vehicle in the platoon.Second, two receding horizon strategies within this controlarchitecture are developed. The platoon coordinator relies on adynamic programming (DP) formulation [9] that exploits pre- view information on the road topography and speed limits tocompute a speed trajectory defined over space that is safe andfuel-optimal for the whole platoon. Here we emphasize thatthe platoon coordinator can handle heterogeneous platoons in asystematic way. The vehicle controller layer, instead, is solvedthough a distributed model predictive control formulation [14]that tracks the speed trajectory and a certain gap policy whileguarantying fuel-efficiency and safety. More precisely, it isproved that with this architecture no collision will occur withinthe platoon when up to one vehicle is controlled manually.The performance of the proposed control architecture isfinally evaluated through extensive simulations motivated byreal experimental scenarios and comparisons with existingapproaches for speed control and spacing policy are presented.The rest of the paper is organized as follows. In Section IIwe analyze the experimental results presented in [5]. InSection III we present the vehicle and platoon models usedin the controllers, whereas in Section IV we introduce thecontrol architecture. The platoon coordinator and the vehiclecontroller layers are discussed in Sections V and VI, whiletheir performance is studied in Sections VII, VIII and IX,by means of simulations. Finally, conclusions are stated inSection X. II. M OTIVATING EXPERIMENT
In this section we analyze the experimental results presentedin [5] in order to reach a good understanding of the impact ofthe road gradient on HDV platooning and motivate the need forthe design of a look-ahead control framework for fuel-efficientHDV platooning.In this experiment a platoon of three similar HDVs (samepowertrain and mass of . , . , . tons, respectively) isdriven over a km highway stretch between the Swedishcities of Mariefred and Eskilstuna. The topography for thisroad is displayed in Figure 1, where the red color highlightsthe uphill and downhill sections where the slope is too largefor a nominal HDV (whose parameters are displayed in Table Iin Section VII) to maintain a constant speed of m/swithout braking or exceeding the engine power limit. Forthe considered road, the steep sections represent % of thetotal length. Overall, the follower vehicles, by platooning,manage to save . % and . %, respectively. However in [5]it is shown that the fuel-efficiency drops significantly in theroad sector where the slope is more varying. In this studywe therefore analyze the behavior of the first two vehicleswhile driving over the particularly hilly stretch highlightedin Figure 1 as Sector A for which an increase of the fuelconsumption of the second HDV of % compared to the caseof driving alone has been reported. The behavior is shown inFigure 2. The first vehicle tracks a reference speed of . m/s using cruise control and it switches to braking mode onlywhen the speed limit of . m/s is reached, while the secondvehicle tracks a headway gap (a distance proportional to itsspeed) from the first vehicle and it switches to braking modeonly when the headway gap reaches a certain threshold (referto [5] for a complete characterization of the controllers). In theanalyzed sector three critical segments highlighted in Figure 2 are identified where the use of feedback controls shows itslimitations. • Segment 1 : due to the steep downhill the first HDV isnot able to maintain the reference speed and thereforeit accelerates while coasting. The second vehicle, whiletrying to track the headway gap policy, follows thesame behavior. However, due to the reduced experiencedair resistance, during the downhill the second vehicleaccelerates more than the first one and, when the criticalheadway gap is reached, it brakes. In this case the co-ordination between the accelerations of the two vehicleswould have the potential to avoid the undesired braking. • Segment 2 : the headway gap deviates significantly fromthe reference one, due to a large relative speed at thebeginning of the uphill segment and a change of gearduring the segment. The second vehicle, in order toreduce the headway gap error, significantly increases therelative speed. Once the critical headway gap is reached,it brakes strongly. In this case, the prediction of thevehicles behaviors would have allowed the second vehicleto reduce the relative speed before reaching the referenceheadway gap and, therefore, to avoid the undesired brak-ing. • Segment 3 : here the second vehicle shows a more criticalbehavior compared to the first downhill. In fact, duringdownhills, the vehicles’ actuators work close to saturation(small fueling and small braking) which is not suitable forfeedback controller. Therefore in Segment 3 the controlstate of the second vehicle continues to switch betweenfueling and braking modes. In this case, the use ofa receding horizon framework would have allowed topredict correctly the vehicle behavior depending on theslope and, by taking into account the actuators’ saturation,therefore, to obtain a smoother behavior of the vehicle.The analysis of these experimental results provides a strongmotivation for the development of a cooperative look-aheadcontrol strategy for HDV platooning based on a recedinghorizon framework where the road gradient and the vehiclesahead can be explicitly taken into account.III. M
ODELING
HDVs are complex systems with a large number of inter-acting dynamics. Due to their heavy load, the braking andpowertrain systems of an HDV have to generate and transferextremely high torques. In this section we first present the ve-hicle system architecture upon which the proposed controller isdesigned. Second, we introduce the model of the longitudinaldynamics of a single vehicle and a platoon with a particularfocus on the components that play a significant role for thefuel consumption. Finally, we present the fuel model used toestimate the fuel consumption.
A. Vehicle system architecture
The functioning of an HDV is guaranteed by a large numberof system units that communicate with each other throughthe controller area network (CAN) bus. A simplified controlarchitecture of an HDV is shown in Figure 3, where only the a l t i t ud e [ m ] not steepsteep25 26 27 28 29 30 31 − −
10 space [km] b r a k e [ m / s ] b t o r q u e [ % ] T T d i s t a n ce [ m ] d d r,2 s p ee d [ m / s ] v v Figure 2. Experiment results presented in [5] relative to the first two vehiclesof a three vehicle platoon driving along the Sector A highlighted in Figure 1.The first plot shows the road topography, whereas the second plot showsthe speed of the two vehicles; the third plot shows the real and reference(according to a headway gap policy) between the vehicles; finally the forth andfifth plots shows respectively the normalized engine torque for both vehiclesand the normalized braking force for the second vehicle (the braking actionof the first vehicle is not available). For additional details, see [5].
WSU GPS RadarData ProcessingHigh-level ControllerBMS EMS GMS a ∗ , a ∗ br a ∗ , a ∗ br a ∗ , a ∗ br a ∗ , a ∗ br C AN bu s Figure 3. Simplified system vehicle architecture. system units that are of interest for our work are displayed[5]. A more detailed description of a complete vehicle systemarchitecture is given in [24]. The communication with theoutside world relies on the wireless sensor unit (WSU). Thisunit shares real-time information with the other vehicles withinthe platoon. The global positioning system (GPS) computesthe absolute position of the vehicle, while the radar measuresthe distance and relative speed between the current vehicleand the preceding one. The real-time state information ofthe platoon coming from the WSU, the GPS and the radarare fused by the data processing unit and sent to the high-level controller. The high-level controller computes the desired a l t i t ud e [ m ] not steepsteep Figure 1. Road topography for the km highway stretch between the Swedish cities of Mariefred and Eskilstuna. The red color highlights the uphill anddownhill sections where the slope is too large for the considered HDVs to maintain a constant speed of m/s without braking or exceeding the engine powerlimit. acceleration and a boolean variable defining which low levelcontroller should track it. In particular, the brake managementsystem (BMS) controls the braking actuators, while the engineand gear management systems (EMS and GMS) control theengine, the gearbox and the clutch to provide the requestedacceleration. B. Vehicle and platoon model
In this subsection we derive the model of the longitudinaldynamics of a single vehicle and the platoon that is then usedin the controller formulation. Using Newton’s second law, thelongitudinal dynamics of a single vehicle can be expressed by: m i ˙ v i = F e ,i + F b ,i + F g ,i ( α ( s i )) + F r ,i + F d ,i ( v i , d i ) , ˙ s i = v i , (1)where v i and s i are the states of vehicle i , respectively, thespeed and the longitudinal position, m i is its mass and F e ,i , F b ,i , F g ,i , F r ,i and F d ,i are the forces acting on the vehicle.We collect v i and s i in the state vector x i = [ v i s i ] T ). Morespecifically, F e ,i and F b ,i are the control inputs and repre-sent the forces generated by the powertrain and the brakingsystem, respectively. The engine force F e ,i is characterized inSection III-C, while the braking force F b ,i is assumed to belimited by the road friction and therefore bounded by − m i η i gµ ≤ F b ,i ≤ , (2)where µ and η i denote the (positive) road friction coeffi-cient and the braking system efficiency, respectively. Next, F g ,i ( α ( s i )) is the force due to the gravity, modeled as F g ,i ( α ( s i )) = − m i g sin( α ( s i )) , (3)where g is the gravity acceleration and α ( s i ) the road slope atposition s i . The rolling resistance is represented by F r ,i andis modeled as F r ,i = − c r m i g, (4)where c r is the rolling coefficient. Finally F d ,i ( v i , d i ) is theaerodynamic drag, modeled as F d ,i ( v i , d i ) = − ρA v C D ( d i ) v i , (5)where ρ is the air density, A v is the cross-sectional area ofthe vehicle and C D is the air drag coefficient. In order to takeinto account the influence of the inter-vehicular distance on theaerodynamic force that plays an essential role in platooning, . . . n o r m a li ze dd r ag c o e ffi c i e n t [ % ] experimental dataregression curve Figure 4. Experimental data [23] and regression curve of the normalized dragcoefficient experienced by an HDV as function of the distance to the previousvehicle. the drag coefficient C D is defined as a function of the distanceto the previous vehicle d i . This dependence is modeled by C D ( d i ) = C D, (cid:18) − C D, C D, + d i (cid:19) , (6)where the parameters C D, and C D, have been obtained byregressing the experimental data presented in [23]. The effectof the short inter-vehicular distance on the leading vehicle isneglected since it is smaller than one on the follower vehicles.The experimental data and the regression curve are displayedin Figure 4. Remark 1.
In this work we have chosen to model the air dragcoefficient on the basis of the experimental data presented in[23] relative to the second vehicle of a two buses platoondriving at km/h. In the literature reports on air dragcoefficient or fuel consumption measures based on both realexperiments [6], [13], [25] and fluid dynamics simulator [29]are presented. They show a reduction of the air drag coefficientfor short inter-vehicular distances. How the reduction relatesto the inter-vehicular distance varies. This variability has beenattributed to the weather condition (e.g, temperature, humidityor wind) and the shape of the vehicles. The model of a platoon of N v vehicles is given by thecombination of equations (1) for i = 1 , ..., N v and the distancedefinition: d i = (cid:40) ∞ , if i = 1 ,s i − − s i − l i − , if i ≥ , , (7)where l i denotes the length of vehicle i .
800 1 ,
000 1 ,
200 1 ,
400 1 ,
600 1 ,
800 2 ,
000 2 , , , , engine speed [rpm] e n g i n e t o r q u e [ N m ] optimal BSFC Figure 5. BSFC map for a hp engine regenerated from [31]. The plotshows the BSFC expressed in g/kWh as function of the engine speed andtorque. The dotted lines represent equal power curves, while the blue thickline represents the collection of the fuel-optimal operation points for variousgenerated powers.
C. Fuel model
The powertrain is a complex system composed by engine,clutch, gearbox and final gear that allows to transform thefuel’s energy into longitudinal force. In this subsection wederive a simple model of the powertrain that captures the in-trinsic relation between consumed fuel and generated tractionforce. In the model derivation we ignore transmission energylosses and the rotational inertia of the powertrain componentsbecause they are assumed to be negligible when compared tothe vehicle mass.Engine performance is typically described by the brake spe-cific fuel consumption (BSFC), that defines the ratio betweenconsumed fuel and generated energy for various operationpoints (i.e., engine speed and generated torque). In Figure 5we show the BSFC map for an HDV engine of hp [31],where the dotted lines represent the collection of operationpoints with equal generated power. This map can be easilyconverted in one that defines the fuel flow δ i as functionof the engine speed ω i and the generated engine power P i ,i.e., δ i = φ i ( ω i , P i ) . By assumption, the engine power P i ,passing through the clutch, the gearbox and the final gear iscompletely transferred to the wheels. The rotational speed,instead, changes between the transmission components andis finally transformed into longitudinal speed by the wheels.Ultimately, under the assumption of no longitudinal slip, thevehicle speed v i can be defined as v i = k i g i ω i , where k i is aconstant gain and g i is the gear ratio of the gearbox. Thereforethe fuel flow can be expressed as a function of the speed v i ,the traction force F e ,i and the gear ratio g i as δ i = φ i (cid:18) v i k i g i , F e ,i v i (cid:19) . (8)In order to be efficiently used in the control design, the fuel o p t i m a l f u e l fl o w [ g/ s original datalinear regression0 50 100 150 200 250 3001 , ,
500 power [kW] o p t i m a l e n g i n e s p ee d [ r p m Figure 6. The plots show the optimal fuel flow and engine speed as function ofthe generated power. In the first plot we also display the fuel model expressedin (10) obtained by the regression of the raw data. model is further simplified by removing the dependence ofthe fuel flow δ i on the gear ratio g i through the introductionof an additional assumption: the gear ratio can be changedcontinuously on a unlimited span and the gear managementsystem chooses the most efficient gear ratio. Hence, weredefine the fuel model as δ i = min ω i φ i ( ω i , F e ,i v i ) = φ opt ,i ( F e ,i v i ) . (9)The resulting curve φ opt ,i ( · ) is depicted in Figure 6 and islinearly regressed in order to obtain the fuel model used inthe controller design, defined by δ i = p ,i F e ,i v i + p ,i . (10)From this analysis we can also obtain the bounds on thegenerated power that are independent from the engine speedand the gear ratio: P min ,i ≤ F e ,i v i ≤ P max ,i . (11)In Figure 6 the two fuel models in (9) and (10), and thecorrespondent optimal engine speed are displayed. We notethat the approximation error is negligible.IV. P LATOON CONTROLLER ARCHITECTURE
The system architecture for the look-ahead HDV platooncontroller is shown in Figure 7. The mission planner suggestsroutes and platoon opportunities. The platoon coordinatorsupervises the platoon behavior exploiting road information.The vehicle controllers execute the speed profiles for theindividual vehicles.The platoon coordinator layer exploits available informationon the topography of the planned route to find a fuel-optimalspeed profile for the entire platoon, while satisfying theaverage speed requirement provided by the mission planner.Hereby, in order to capture the dynamics of the road topog-raphy, it considers a horizon of several kilometers and takesthe constraints of all vehicles in the platoon into account. Asa result, it can be guaranteed that every vehicle in the platoonis able to track the required speed profile. A single speedtrajectory is computed by the platoon coordinator, representingthe speed of the platoon. However, when this speed profile is platoon coordinatormission planner roaddatabase α s , v smax , v smin ¯ v vehicle controller a ∗ ( t ) x ( t ) v s , ∗ ( · ) x ( t ) vehicle controller a ∗ ( t ) x ( t ) v s , ∗ ( · ) , τ x ( t ) vehicle controller a ∗ ( t ) x ( t ) v s , ∗ ( · ) , τ x ( t ) x ( t ) x ( t ) d ( t ) d ( t ) Figure 7. System architecture for look-ahead HDV platooning. specified as a function of space (i.e., position on the road)and the inter-vehicle spacing is chosen according to a puretime delay, every individual vehicle in the platoon can trackthis single speed profile. It is remarked that this layer cantypically operate in a receding horizon fashion, providing anupdated speed profile roughly every 10 seconds or when therecalculation is needed due to a strong deviation from theoriginal speed profile. Finally, as this layer is not safety-criticaland not related to a specific vehicle, it can be implemented inany of the platooning vehicles or even in an off-board roadunit. In Section V, we present a DP approach to formulate andsolve the stated problem.The vehicle controller is responsible for the real-time con-trol of each vehicle in the platoon and is aimed at tracking thedesired speed profile as resulting from the platoon coordinator.It also exploits the communication between vehicles of theassumed trajectories to ensure the proper spacing strategy.This layer guarantees the safety of platooning operations tofor instance avoid collisions between trucks. Because of thesafety critical aspect, this layer is implemented in a distributedfashion in each vehicle of the platoon. More precisely, eachvehicle controller runs in the block named high-level controllerin the system vehicle architecture shown in Figure 3. InSection VI a distributed model predictive control approach forthis problem is discussed.Figure 8 shows how the optimization problems in the pla-toon coordinator and the vehicle controllers interact, and theirmathematical structure. Note how the platoon coordinator, inorder to have a good prediction of the consumed fuel overthe horizon, uses an accurate non-linear model of the vehicle,while the vehicle controller layer, in order to enable fastcomputation necessary for the real-time control of the vehicle,uses a linear vehicle model.V. P
LATOON COORDINATOR
The platoon coordinator is the higher layer of the platooncontrol architecture. It takes as inputs the average speed re- Platoon coordinator a min N v (cid:88) i =1 fuel i subj. to Non-linear HDVs modelConstraints on state and inputConstraint on avarage speedSame speed profile for all HDVsVehicle i controller a min Deviation from reference profilesubj. to Linear HDV modelConstraints on state and inputSafety constraintSoft constraint on brakingreference speed profile,gap policy
Figure 8. Optimal control problems solved in the platoon coordination andvehicle controllers. quirement ¯ v from the mission planner and the current vehiclesstate x i ( t ) from their vehicle controllers. By exploiting theavailable information on the planned route (i.e., slope data α s and speed limits v smax ), it generates a unique feasible andfuel-optimal speed profile v s , ∗ ( · ) defined over space for allthe vehicles within the platoon (i.e., v s , ∗ i ( z ) = v s , ∗ ( z ) for i = j, ..., N v , where z is the space variable). Furthermore,according to safety criteria, it specifies the time gaps τ i ,defined as the time delay between two consecutive vehiclespassing through the same point, i.e., s i ( t ) = s i − ( t − τ i ) . (12)Note that this spacing policy is consistent with the requirementthat all vehicles have to follow the same speed profile overspace. This can be easily shown by computing the timederivative of the left hand side, ds i ( t ) dt = v i ( t ) = v s i ( s i ( t )) , (13)and the right hand side of (12), ds i − ( t − τ i ) dt = v i − ( t − τ i )= v s i − ( s i − ( t − τ i )) = v s i − ( s i ( t )) , (14)where v s i ( s ) denote the speed of vehicle i at space s . In fact,by combining the time gap definition (12) with (13) and (14),we obtain v s i ( s ) = v s i − ( s ) .The coordinator layer is implemented using a DP framework[9] that runs in closed-loop. The parameters that characterizethe DP problem are the discretization space ∆ s DP , the horizonlength H DP and the refresh frequency f DP . We also define thehorizon space length as S DP = H DP ∆ s DP . In the coming subsections we introduce all the componentsof the DP formulation, i.e., the vehicle model, the constraintson the input and states and finally the cost function.
A. Platoon model
The platoon coordinator layer uses a discretized version ofthe vehicle model (1), where the discretization is carried outin the space domain using the implicit Euler approximation.The discretized vehicle model is: v s i ( z k ) v s i ( z k ) − v s i ( z k − )∆ s DP = F se ,i ( z k ) + F sb ,i ( z k ) (15a) − m i g [sin( α ( z k )) + c r ] − ρA v C D ( d s i ( z k ))( v s i ( z k )) ,v s i ( z k ) t s i ( z k ) − t s i ( z k − )∆ s DP =1 , (15b)where z k is the discretized space variable, v s i ( z k ) , F se ,i ( z k ) , F sb ,i ( z k ) and d s i ( z k ) are the speed, the engine and brakingforces and the distance to the previous vehicle expressed asfunction of space, respectively.The advantage of using the space discretization is that, byrelaxing the average speed requirement, there is no constraintdepending on time. The relaxation is done by removing theaverage speed constraint and introducing instead travel timeover the horizon in the cost function, hereby using an appro-priate weighting. This allows to ignore the time dynamics andtherefore reduce significantly the computational complexity.A drawback of the space discretization is that the distancedefinition (7) cannot be expressed in the space domain. In-stead, the following approximated expression, as function ofthe current vehicle speed v s i ( z k ) , is used: d s i ( z k ) = v s i ( z k ) τ i − l i − . (16)In the DP formulation we refer to (15a) as v s i ( z k − ) = f s v,i ( v s i ( z k ) , u s i ( z k )) , where u s i ( z k ) is theinput vector defined as u s i ( z k ) = [ F se ,i ( z k ) , F sb ,i ( z k )] T . B. Model constraints
The platoon model is constrained by introducing bounds onthe input and the speed.
1) Input constraints:
According to (2) and (11), the engineand braking forces are bounded by the following constraints: P min ,i v s i ( z k ) ≤ F se ,i ( z k ) ≤ P max ,i v s i ( z k ) , − m i η i gµ ≤ F sb ,i ( z k ) ≤ . (17)In the DP formulation, we refer to these constraints as u s i ( z k ) ∈ U s i ( z k ) .
2) State constraints:
In order to take into account the roadspeed limits, the speed is bounded by v min ( z k ) ≤ v s i ( z k ) ≤ v max ( z k ) . (18)We refer to this constraint as v s i ( z k ) ∈ V s ( z k ) .Moreover, in order to require all the vehicle to follow thesame speed profile, the constraint v s i ( z k ) = v s ( z k ) , i = 1 , ..., N v . (19) is introduced. The practical effect of this constraint is to reducethe search space of the dynamic programming algorithm to onedimension rather then the number of vehicles in the platoon,enabling fast computation. C. Cost function
The objective of the platoon coordinator layer is to definethe optimal speed profile that minimizes the fuel consumptionof the whole platoon, while maintaining a certain averagespeed. This is done by defining the cost function as theweighted sum of two terms: a first term J f ( v s ( z j ) , u s i ( z j )) for j = k, ..., k + H DP − and i = 1 , ..., N v representing the fuelamount consumed by the platoon and a second term J t ( v s ( z j )) for j = k, ..., k + H DP − representing the travel time overthe horizon, i.e., J DP ( v s ( z j )) , u s ( z j )) = J f ( v s ( z j ) , u s i ( z j ))+ βJ t ( v s ( z j )) , (20)where β represents a trade-off weight . The term J f ( v s ( z j ) , u s i ( z j )) is computed by using the fuel model(10), taking also into account a final term representing thekinematic energy of the platoon at the end of the horizon: J f ( v s ( z j ) , u s i ( z j ))= N v (cid:88) i =1 k + H DP − (cid:88) j = k ∆ s DP (cid:18) p ,i F se ,i ( z j ) + p ,i v s ( z j ) (cid:19) − N v (cid:88) i =1 p ,i m i ( v s ( z h + H DP − )) . The term J t ( v s ( z j )) is obtained by using the time model (15b): J t ( v s ( z j )) = k + H DP − (cid:88) j = k ∆ s DP v s ( z j ) . D. Dynamic programming formulation
We now have all the elements to formulate the DP problemsolved in the platoon coordinator: min u s ( z j ) J DP ( v s ( z j ) , u s ( z j )) (21a)subj. to v s i ( z j − ) = f s v,i ( v s i ( z j ) , u s i ( z j )) , (21b) u s i ( z j ) ∈ U s i ( z j ) , (21c) v s i ( z j ) = v s ( z j ) ∈ V s ( z j ) , (21d) z k = s ( t ) , (21e) v s ( z k ) = v ( t ) , (21f)for j = k, ..., k + H DP − , where the equations (21e) and (21f)represent the initial conditions of the DP formulation. Instead of the constraint on the average speed of Figure 8, the parameter β is tuned to give the desired average time VI. V
EHICLE CONTROLLER
This section focuses on the distributed model predictivecontrollers running in the vehicle controller layer.Each vehicle controller runs locally. Vehicle i receives theoptimal speed profile v s , ∗ ( · ) and the time gap τ i from theplatoon coordinator and state information from the precedingvehicle. By tracking the optimal speed profile and gap policyrequirement, and satisfying a safety constraint, it generates theoptimal state and input trajectories, respectively x ∗ i ( ·| t ) and a ∗ i ( ·| t ) , and the desired acceleration a ∗ i ( t ) for the vehicle low-level controllers. The parameters that characterize the MPCformulation are the discretization time ∆ t MPC , the horizonsteps number H MPC , the refresh frequency f MPC and the lengthof the horizon defined as T MPC = H MPC ∆ t MPC .In the coming subsections we introduce all the componentsof the MPC formulation, i.e., the vehicle model, the constraintson the input and state, the safety constraint and finally the costfunction.
A. Vehicle model
In the MPC formulation the vehicle is described by x i ( t j +1 | t k ) = Ax i ( t j | t k ) + Ba i ( t j | t k ) , (22)where A (cid:44) (cid:20) t MPC (cid:21) , B (cid:44) (cid:20) ∆ t MPC (cid:21) . The variables x i ( t j | t k ) = [ v i ( t j | t k ) s i ( t j | t k )] T and a i ( t j | t k ) denote the predicted state (speed and position) and controlinput (acceleration) trajectories of vehicle i associated to theupdate time t k , respectively. We also introduce three additionaltrajectories associated to each update time t k that will be usedlater in the MPC formulation: • the optimal state trajectory x ∗ i ( t j | t k ) , • the state reference trajectory ¯ x i ( t j | t k ) , • the assumed state trajectory ˆ x i ( t j | t k ) ,for j = k, ..., k + H MPC − and the corresponding in-put control trajectories defined likewise. While the pre-dicted and optimal trajectories are function of the opti-mization variable, the reference and assumed trajectoriesare pre-computed. More precisely the reference trajectories ¯ x i ( t j | t k ) = [¯ v i ( t j | t k ) ¯ s i ( t j | t k )] T and ¯ a i ( t j | t k ) are computedfrom the reference trajectory v s , ∗ ( · ) and the current position s ( t k ) of the vehicle. In particular, ¯ s i ( t j | t k ) is defined recur-sively as ¯ s i ( t j | t k ) = (cid:40) s i ( t j ) , j = k, ¯ s i ( t j − | t k ) + ∆ t MPC ¯ v s , ∗ (¯ s i ( t j − | t k )) , j > k, while ¯ v i ( t j | t k ) is defined as ¯ v i ( t j | t k ) = ¯ v s , ∗ ( s i ( t j | t k )) . The control input reference trajectory ¯ a i ( t j | t k ) is defined asfinite differences of ¯ v i ( t j | t k ) , i.e., ¯ a i ( t j | t k ) = (¯ v i ( t j +1 | t k ) − ¯ v i ( t j | t k ))∆ t MPC . The assumed state and control input trajectories are computedfrom the optimal and real trajectories of the vehicle as ˆ x i ( t j | t k ) = (cid:40) x i ( t j ) , j < k,x ∗ i ( t j | t k − ) , k ≤ j < k + H MPC , (23)and ˆ a i ( t j | t k ) likewise. As mentioned at the beginning of thissection, each vehicle communicates the assumed trajectory ˆ x i ( t j | t k ) to the follower vehicle. In this case, the use ofthe optimal trajectory computed the previous step reflects theassumption of a maximum communication delay of ∆ t MPC . B. Input and model constraints
In order to take into account the bounds on the brakingforce (2) and the engine power (11), as done in the platooncoordinator layer, the control input a i is bounded by thefollowing non-linear constraint: − η i µg + F ext ( x i , ˆ s i − ) m i ≤ a i ≤ P i, max m i v i + F ext ( x i , ˆ s i − ) m i , (24)where F ext ( x i , ˆ s i − ) denotes the summation of the externalforces acting on the vehicle and is defined as F ext ( x i , ˆ s i − ) = − m i g (sin( α ( s i )) + c r ) − ρA v C D (ˆ s i − − s i − l i ) v i . (25)The control input is additionally bounded by a soft constraintin order to allow braking only if necessary, i.e., when thesafety constraint (see section VI-C) is activated or the brakingis required by the platoon coordinator. This is formulated asfollows: a i + (cid:15) i ≥ min( a c ,i , ¯ a i ) , (cid:15) i ≥ , (26)where (cid:15) i is the softening variable and a c ,i is the coastingacceleration (i.e., no braking and fuel injection) and is definedas: a c ,i = P i, min m i v i + F ext ( x i , ˆ s i − ) m i . (27)In the MPC formulation we refer to the constraint (24),(25)as a i ( t j | t k ) ∈ A i ( x i ( t j | t k )) and to the soft constraint (26),(27)as a i ( t j | t k ) + (cid:15) i ( t j | t k ) ∈ A e ,i ( x i ( t j | t k )) .The speed is bounded according to the constraint (18) as v min ( s i ( t j | t k )) ≤ v i ( t j | t k ) ≤ v max ( s i ( t j | t k )) . In the MPC formulation, we refer to this constraint as v i ( t j | t k ) ∈ V ( s i ( t j | t k )) . C. Safety constraint
The platoon is intended to operate on standard highwayswhere other vehicles are present. The designed controllertherefore should be able to cope with cases where the platoonbehavior deviates from the predicted one because of internaldisturbances (e.g., gear shift) or external events (e.g., relatedto the traffic situation or a vehicle cutting into the platoon). Inthis section we focus on the safety problem, leaving to furtherwork the study of how such events should be handled (i.e.,autonomously or switching to manual driving).The platoon is considered safe if, whatever a vehicle in theplatoon does, there exists an input for all the follower vehicles such that collision can be avoided. The safety of the platoonis guaranteed by ensuring that the state of each vehicle lieswithin a safety set and it is firstly studied by considering twoadjacent vehicles and later extended to the entire platoon. Inhere we consider the following vehicle continuous dynamics: ˙˜ x i = (cid:20) ˙˜ v i ˙˜ s i (cid:21) = f (˜ x i , ˜ a i ) = (cid:20) ˜ a i ˜ v i (cid:21) , (28)where ˜ v i , ˜ s i and ˜ a i are the speed, position and accelerationof vehicle i , respectively.Let us now focus on the dynamics of two adjacent vehiclesdescribed by (cid:20) ˙˜ x i − ˙˜ x i (cid:21) = F (˜ x i − , ˜ x i , ˜ a i − , ˜ a i ) = (cid:20) f (˜ x i − , ˜ a i − ) f (˜ x i , ˜ a i ) (cid:21) , (29)where the acceleration of the current vehicle ˜ a i is thecontrol input, while the acceleration of the previous ve-hicle ˜ a i − is the exogenous input that can be regardedas a disturbance. We also introduce the admissible set ˜ X = { [˜ x T i − ˜ x T i ] T : ˜ v i − ≥ , ˜ v i ≥ , ˜ s i − − ˜ s i ≥ l i − } as theset of all admissible states, where l i denotes the length ofvehicle i . In order to obtain a closed form of the safety set, thefollowing conservative approximations of the the exogenousand control inputs are introduced: ˜ a i − ∈ A p (˜ x i − ) = (cid:40) [ a min ,i , a max ,i ] , if ˜ v i − > , [0 , a max ,i ] , if ˜ v i − = 0 , (30a) ˜ a i ∈ A f (˜ x i ) = (cid:40) [ a min ,i , a max ,i ] , if ˜ v i > , [0 , a max ,i ] , if ˜ v i = 0 , (30b)where a min ,i , a min ,i , a max ,i and a max ,i are lower and upperbounds on the minimum and maximum possible accelerationsof vehicle i , respectively. Such bounds are computed underreasonable assumptions on the vehicles and road properties,i.e., the vehicles’ speed is limited ( ≤ ˜ v i ≤ v max ), theadmissible vehicles’ weight is bounded ( m i ∈ M ) and the roadslope α is bounded ( | α | ≤ α max ). For example, the bounds a min ,i and a min ,i can be computed as follows: a min ,i = min ≤ v ≤ v max ,m ∈ M, | α |≤ α max ,d ≥ a min ,i ( v, m, α, d ) ,a min ,i = max ≤ v ≤ v max ,m ∈ M, | α |≤ α max ,d ≥ a min ,i ( v, m, α, d ) , where a min ,i ( v, m, α, d ) = − µη i g − g sin( α ) − c r − ρA v C D ( d ) v and − µη i g represents the maximum braking capacity ofvehicle i . Note that, due to the definition of the bounds andbecause of the dominance of the − µη i g term in the definitionof a min ,i , the following inequalities hold: a min ,i ≤ a min ,i ≤ , (32a) a max ,i ≤ a max ,i . (32b)In order to guarantee the safety of the subsystem (29), weshould guarantee that the state [˜ x T i − ˜ x T i ] T always lies in asafety set S included in ˜ X , for any admissible trajectory of d i = ˜ s i − − ˜ s i − l i − [m] ˜ v i [ m / s ] ˜ v i − = m / s ˜ v i − = m / s ˜ v i − = m / s ˜ v i − = m / s Figure 9. Contour plot of the safety set boundary ∂ S . The variable ˜ d i denotesthe distance between the two adjacent vehicles. the previous vehicle. We now define the safety set S ⊆ ˜ X ,displayed in Figure 9, as S = { [˜ x T i − ˜ x T i ] T : g j (˜ x i − , ˜ x i ) ≥ , j = 1 , ..., } , (33)where g (˜ x i − , ˜ x i ) = ˜ s i − − ˜ s i − l i − − ˜ v i − a min ,i − + ˜ v i a min ,i ,g (˜ x i − , ˜ x i ) = ˜ s i − − ˜ s i − l i − ,g (˜ x i − , ˜ x i ) = ˜ v i − ,g (˜ x i − , ˜ x i ) = ˜ v i (34)and we state the following result: Lemma 1.
Given the dynamic system (29) and the constraints (30a) and (30b) on the exogenous and control inputs respec-tively, there exists a control law ˜ a i = φ ([˜ x T i − ˜ x T i ] T ) ∈ A f (˜ x i ) such that for all [˜ x T i − ( t ) ˜ x T i ( t )] T ∈ S and ˜ a i − ∈ A p (˜ x i − ) ,the condition [˜ x T i − ( t ) ˜ x T i ( t )] T ∈ S holds for all t ≥ t . Inother words, S is a robust controlled invariant set [12].Proof. By using Nagumo’s theorem for robust controlledinvariant sets [12], the lemma can be proved by showing thatfor all [˜ x T i − ˜ x T i ] T ∈ ∂ S (defined as the boundary of S ) thereexists an ˜ a i ∈ A f such that, for all ˜ a i − ∈ A p , the relation ∇ g j (˜ x i − , ˜ x i ) T F (˜ x i , ˜ x i − , ˜ a i − , ˜ a i ) ≥ (35)holds for all j such that g j (˜ x i − , ˜ x i ) = 0 . Because of thestructure of the problem, the control input ˜ a i is chosen as ˜ a i = (cid:40) a min ,i , if ˜ v i > , , if ˜ v i = 0 , (36)for any [˜ x T i − ˜ x T i ] T ∈ ∂ S and ˜ a i − ∈ A p (˜ x i − ) . We organizethe proof by considering the [˜ x T i − ˜ x T i ] T ∈ ∂ ˜ S defined by theactivation of each g j (˜ x i − , ˜ x i ) ≥ : • for [˜ x T i − ˜ x T i ] T such that g (˜ x i − , ˜ x i ) = 0 , and g j (˜ x i − , ˜ x i ) ≥ , for j = 2 , , , ∇ g (˜ x i − , ˜ x i ) T F (˜ x i − , ˜ x i , ˜ a i − , ˜ a i )= (cid:18) − ˜ a i − a min ,i − (cid:19) ˜ v i − − (cid:18) − ˜ a i a min ,i (cid:19) ˜ v i , = (cid:18) − ˜ a i − a min ,i − (cid:19) ˜ v i − ≥ , where the equality and inequality hold because of how ˜ a i is defined by (36) and g (˜ x i − , ˜ x i ) ≥ . • for [˜ x T i − ˜ x T i ] T such that g (˜ x i − , ˜ x i ) = 0 , and g j (˜ x i − , ˜ x i ) ≥ , for j = 1 , , , ∇ g (˜ x i − , ˜ x i ) T F (˜ x i − , ˜ x i , ˜ a i − , ˜ a i ) = ˜ v i − − ˜ v i ≥ , where the inequality holds by noticing that the combi-nation of g (˜ x i − , ˜ x i ) ≥ , g (˜ x i − , ˜ x i ) = 0 and therelation (32a) gives ˜ v i − ≥ ( a min ,i /a min ,i )˜ v i . • for [˜ x T i − ˜ x T i ] T such that g (˜ x i − , ˜ x i ) = 0 , and g j (˜ x i − , ˜ x i ) ≥ , for j = 1 , , , ∇ g (˜ x i − , ˜ x i ) T F (˜ x i − , ˜ x i , ˜ a i − , ˜ a i ) = ˜ a i − ≥ , where the inequality holds because of (30a). The samecan be verified in a similar way for [˜ x T i − ˜ x T i ] T such that g (˜ x i − , ˜ x i ) = 0 and g j (˜ x i − , ˜ x i ) ≥ for j = 1 , , .The choice of the safety set guarantees that the followervehicle can react to the emergency braking maneuver ofits predecessor, such that both vehicles come to a standstillwithout colliding. We now extend the result in Lemma 1 tothe safety of the whole platoon. More precisely, we proofthat whatever a vehicle does, there exists an input for all thefollower vehicles, such that collision can be avoided. This isformalized by the following theorem: Theorem 1.
Consider a vehicle with index i < N v andall its follower vehicles i ∈ I = { i + 1 , ..., N v } sat-isfying the dynamics in (28) . Then, there exists a controllaw ˜ a i = φ (˜ x i , ˜ x i − ) ∈ A f (˜ x i ) , i ∈ I such that for all [˜ x T i − ( t ) ˜ x T i ( t )] T ∈ S and ˜ a i ∈ A p (˜ x i ) , the condition [˜ x T i − ( t ) ˜ x T i ( t )] T ∈ S holds for all t ≥ t and all i ∈ I .Proof. The application of Lemma 1 for i = i + 1 provesthe existence of an input ˜ a i ∈ A f (˜ x i ) that ensures that [˜ x T i − ( t ) ˜ x T i ( t )] T ∈ S for all t ≥ t . Then, by noting that A f (˜ x i ) ⊆ A p (˜ x i ) according to (32), it follows that ˜ a i ∈A p (˜ x i ) .The theorem is then proven by induction over thevehicle index, hereby repetitively applying Lemma 1.This result is adapted to the MPC formulation in orderto guarantee the safety of the platoon. More precisely, eachvehicle, knowing the assumed state trajectory of the vehicleahead, can compute the safety set for its own predicted state.By taking into account that, according to the definition of theassumed state in (23), the real state of the previous vehicle isknow with a one step delay, the safety set S translates to thefollowing safety constraint on each follower vehicle state: s i ( t j +1 | t k ) − v i ( t j +1 | t k )2 a min ,i ≤ ˆ s i − ( t j − | t k ) − ˆ v i − ( t j − | t k )2 a min ,i − l i − , (40)for i = 2 , ..., N v . Note that for safety purpose only the safetyconstraint for j = k is necessary. In fact it guarantees that,if at the update time t k the current state of each followervehicle is safe, then it is going to be safe also at the updatetime t k +1 . However, the safety constraint for j > k gives optimal trajectories that are safe over the whole horizon andtherefore produces a smoother and more fuel-efficient behaviorof the platoon. In the MPC formulation, we refer to the safetyconstraint (40) as f safe ( x i ( t j +1 | t k )) ≥ . D. Cost function
The objective of the vehicle controller layer is to follow theoptimal trajectory and the gap policy requirement providedby the platoon coordinator layer. This can be formulated byintroducing the following cost function: J MPC i ( x i ( · , t k ) ,a i ( · , t k ) , (cid:15) i ( · , t k ))= k + H MPC − (cid:88) j = k || x i ( t j | t k ) − ˆ x i − ( t j − T i | t k ) || ζ i Q + || x i ( t j | t k ) − ¯ x i ( t j | t k ) || (1 − ζ i ) Q + || a i ( t j | t k ) − ¯ a i ( t j | t k ) || R + || (cid:15) i ( t j | t k ) || P , where ζ i = (cid:40) , if i = 1 , ¯ ζ, if i = 2 , ..., N v (41)and T i represents the discretized version of the time gap τ i (i.e., T i = (cid:98) τ i / ∆ t MPC (cid:99) ). The parameters Q , R and ¯ ζ ∈ [0 , can be chosen in order to have a good trade-off between ref-erence trajectory, gap policy tracking and actuators excitation.The weight P related to the softening-variable of the constraint(26) is chosen relatively large such that only the activationof the safety constraint f safe ( x i ( t j +1 | t k )) ≥ can require asignificant braking force. E. Model predictive control formulation
We now have all the elements to formulate the MPCproblem: min a i ( · ,t k ) ,(cid:15) i ( · ,t k ) J MPC i ( x i ( · , t k ) , a i ( · , t k ) , (cid:15) i ( · , t k )) (42a)subj. to x i ( t j +1 | t k ) = Ax i ( t j | t k ) + Ba i ( t j | t k ) , (42b) a i ( t j | t k ) ∈ A i ( x i ( t j | t k )) , (42c) a i ( t j | t k ) + (cid:15) i ( t j | t k ) ∈ A e ,i ( x i ( t j | t k )) , (42d) v i ( t j | t k ) ∈ V ( s i ( t j | t k )) , (42e) f safe ( x i ( t j +1 | t k )) ≥ , if i ≥ , (42f) (cid:15) i ( t j | t k ) ≥ , (42g) x i ( t k | t k ) = x i ( t ) , (42h)where j = k, ..., k + H MPC − and (42h) represents the initialcondition of the MPC problem. For implementation purposethe state-dependent constraint set in (42c), (42d) and (42e)will be replaced respectively by A i (ˆ x i ( t j | t k )) , A e ,i (ˆ x i ( t j | t k )) and V (ˆ s i ( t j | t k )) . Taking into account that the safety constraint(42f) is quadratic and convex, the MPC problem can berecasted into a quadratic constraint quadratic programming(QCQP) problem for which efficient solvers exist.The output of the vehicle controller is the desired acceler-ation a ∗ i ( t k ) (defined as a ∗ i ( t k ) = a ∗ i ( t k | t k ) , where a ∗ i ( ·| t k ) is the optimal input trajectory resulting from the MPC) and aboolean variable a br ,i defined as a br ,i = (cid:40) , if a ∗ i ( t k ) < a ∗ c ,i ( t k | t k ) , , if a ∗ i ( t k ) ≥ a ∗ c ,i ( t k | t k ) , (43)that states if the desired acceleration should be tracked by theBMS or the EMS.VII. P ERFORMANCE ANALYSIS OF THE PLATOONCOORDINATOR
In this section we analyze the performance of the platooncoordinator (as presented in Section V and shown in Figures7 and 8) by focusing on fuel-efficiency. We compare itsperformance with other standard controller setups. To makethe analysis independent from the low-level tracking strategy,we assume in this section that the HDVs can follow exactlythe speed trajectories and spacing policies defined by the high-level controllers.
A. Experiment setup
The comparison is done by using as benchmark the scenariointroduced in Section II. We therefore consider a platoon oftwo HDVs driving over the km road stretch shown inFigure 1 and investigate the controller performance for bothhomogeneous and heterogeneous platoons. The performancemetrics chosen to compare the different control configurationsare the energy and the fuel consumed by the HDVs. Insome comparisons the consumed energy is preferred over theconsumed fuel because it can be directly related to the energiesdissipated by the various forces (i.e., gravity, rolling, drag andbraking forces).The control configurations considered in the comparisonsinclude three control strategies and three gap policies. In detail,the following control strategies are considered: • cruise control (CC): the first vehicle keeps the constantreference speed v CC on low-grade slopes. If the uphillslope is too large to maintain constant speed, the en-gine generate the maximum power P max until the speedreaches v CC again. If the downhill slope is too largeto maintain constant speed without braking, the enginecoasts (i.e., does not inject any fuel, generating thereforethe minimum power P min ) until the speed reaches v CC again. However, if the HDV reaches the speed limit v max ,the brakes are activated in order not to overcome it; • look-ahead control (LAC): the first vehicle exploits theslope information of the road ahead in order to minimizeits own fuel consumption. • cooperative look-ahead control (CLAC): the first vehiclefollows the speed profile generated by the platoon coor-dinator proposed in this paper.The following gap policies are considered: • space gap (SG): the second vehicle keeps a constantdistance d SG from the first vehicle; • headway gap (HG): the second vehicle keeps a constantheadway time τ HG from the first vehicle, i.e., it keeps adistance proportional to its speed ( d HG ( t ) = τ HG v i ( t ) ); • time gap (TG): the second vehicle keeps a constant timegap τ TG from the first vehicle according to (12).In order to be able to maintain exactly the desired gappolicies as previously assumed, the second vehicle is allowedto overcome the theoretical maximum engine power P max ,i and to brake if necessary. In addition, in order to obtain a faircomparison it is ensured, by tuning the trade-off parameter β of the LAC and CLAC formulations (see (41)), that thedifferent control strategies have the same average speed ¯ v andthe parameters d SG , τ HG and τ TG are chosen such that the vehi-cles in the different gap policies have the same distance whendriving at constant speed ¯ v (i.e., d SG = ¯ vτ HG = ¯ vτ TG − l ).Finally in order to remove the influence of the residualkinematic energy, the initial and final speeds are constrainedto be the same in all the controller configurations. B. Fuel-efficiency analysis for different control strategies
In this section we present the results of the platoon behaviorfor the three different control strategies, while keeping a TGpolicy ( τ TG = 1 . s). In the first part, as in the motivationalexample of Section II, we focus on the homogeneous platoonscenario, while in the second part we consider two heteroge-neous platoons (i.e., where the second vehicle is respectivelyheavier and lighter than the leading one). Table IV
EHICLE ’ S PARAMTERS
Parameter Valuemass ( m i ) tlength ( l i ) mroll coefficient ( c r ) × − vehicle cross-sectional area ( A v ) m maximum engine power ( P max ,i ) kWminimum engine power ( P min ,i ) − kW We now consider a platoon of two identical vehicles,whose parameters values are displayed in Table I. We startthe comparison by analyzing the comprehensive bar diagramdisplayed in Figure 10 representing the energy consumed byeach vehicle of the platoon for the three control strategies (thecorresponding fuel consumption is displayed in the centralcolumn of Table II). This energy is normalized respect tothe energy consumed by a single vehicle driving alone usingCC. The consumed energy is additionally split into variouscomponents representing the energy dissipated by each force,namely the gravity, roll, drag and braking force. We can firstnotice how the second vehicle, for all the control strategies,consumes less energy compared to the first one, due to thesignificant reduction of the energy associated to the drag force.Second, comparing the three control strategies, we can observehow the use of the LAC allows both vehicles to save energy,respectively . % and . % compared to the use of the CC.Instead, by switching from the LAC to the CLAC, the firstvehicle consumes . % more energy, while the second onesaves . % of energy; therefore the platoon, given by theunion of the two vehicles, saves . % of energy. This resultis in line with our expectation since the LAC optimizes thefuel consumption of the first vehicle, while the CLAC targets N o r m a li ze d e n e r g y c o n s u m t i o n [ % ] E g E r E d E b CC LAC CLAC
Figure 10. Comparison of the energy consumed by each vehicle of a platoon( m = m = 40 t), for the three control strategies, namely CC, LAC andCLAC, while keeping a TG policy, driving along the km road displayedin Figure 1. Each bar represents the consumed energy normalized respect tothe the energy consumed by a single vehicle driving alone using CC. Theconsumed energy is split into various components representing the energydissipated by each force, namely the gravity ( E g ), roll ( E r ), drag ( E d ) andbraking ( E b ) force. the reduction of the fuel consumption of the entire platoon.Consequently, the saving of the CLAC strategy with respectto the LAC strategy are expected to increase for platoons ofmore vehicles. Going into the details of the various consumedenergy components, first we notice that the gravity and rollenergy components are the same for both vehicles for allthe considered control strategies. This is due to the fact thatthe gravity energy depends only on the difference of altitudebetween the initial and final points, while the roll energy onlydepends on the driven distance that is the same by experimentdesign specification. The drag energy, instead, is significantlydifferent for the two vehicles because of its dependence onthe distance to the preceding vehicle, while it is approximatelythe same for the different control strategies. What significantlychanges between the different control strategies is the energydissipated by braking.In order to understand the role of the control strategies inthe braking usage in Figure 11 we show part of the simulationresults corresponding to the road highlighted as segment B inFigure 1. In this study we have chosen to focus on a downhillsection because this is where the braking action is taking place.The comparison of the platoon behaviors follows: • CC: during the downhill, starting from speed v CC , thefirst HDV accelerates while coasting due to the large roadgrade. In the meantime the second vehicle has to brakeslightly in order to maintain the time gap and compensatethe reduced drag force compared to the first vehicle. At . km, in order not to overcome the speed limit, bothvehicles need to brake significantly; • LAC: by exploiting the topography information of theroad ahead, the first vehicle reduces its speed before thedownhill by anticipating the coasting phase such that thespeed limit is reached only when the slope grade is smallenough to stop accelerating while coasting and thereforeit avoids braking. The second vehicle, as in the CC case, − p o w e r [ k W ] CCLACCLAC405060 a l t i t ud e [ m ] not steepsteep202224 s p ee d , [ m / s ] CCLACCLAC35 36 37 38 39 40 41 − p o w e r [ k W ] CCLACCLAC
Figure 11. Comparison of the behavior of an homogeneous platoon (i.e., m = m = 40 t) for three different control strategies, namely CC, LACand CLAC, while keeping a TG policy, driving along the Sector B displayedin Figure 1. The first plot shows the road altitude, where the red color isused to highlight the sections too steep to keep a constant speed of m/s while respecting the power limit and avoiding braking; the second plotshows the speed profiles for the three control strategies followed by bothvehicles (because of the SG policy); finally the third and forth plots showthe summation between the generated power by the engine and the brakingsystems for the two vehicles and three control strategies; the black linesin such plots define the theoretical minimum and maximum engine power,respectively P min ,i and P max ,i (hence if the power crosses the lower powerlimit P min ,i , the respective vehicle is braking). has to brake slightly while the first vehicle is coasting butit also avoids the significant braking phase at the end ofthe downhill; • CLAC: since in this case the optimization is done consid-ering the fuel consumption of both vehicles, with respectto the LAC case the first vehicle starts to loose speedearlier before the downhill. This allows it to fuel sightlyduring the downhill, allowing the second vehicle to coastmeanwhile and, as in the LAC case, to reach the speedlimit only when the slope grade is small enough to stopaccelerating while coasting. In this case both HDVs donot need to brake.Note that, in the case of longer downhill segments, the lowerspeed bound does not allow the vehicle to decrease the speedenough before the downhill in order not to hit the upper speedlimit during the downhill. This is why in some sections of the km benchmark road, in the LAC case, the first vehicle and,in the CLAC case, both vehicles still need to brake.So far we have considered the case of an homogeneousplatoon. What we want to investigate now is the the role ofthe different control strategies in the case of heterogeneousplatoons. To answer this question, in Table II we have re-ported the normalized fuel consumption for the cases of twoheterogeneous platoons and the same homogeneous platoonpreviously considered. More in detail, the HDVs have thesame powertrain, but their masses vary between , and t. Analyzing the table we can notice how in the case of a heavier second vehicle the CLAC allows to save . % of fuelcompared to the CC, while, in the case of an lighter secondvehicle, it allows to save . %. However if we only analyze thelast row we can note how, with the use of the CLAC, the orderof the vehicles does not significantly change the normalizedfuel consumption.Concluding, the proposed controller (CLAC) has a signifi-cant impact on the reduction of the energy and fuel consump-tion. In detail, the majority of the fuel saving is related to thereduction of energy dissipated by braking during the downhillsections. The impact of such a controller grows in the case ofheavier follower vehicle. Table IINormalized fuel consumption of the vehicles in the platoon for differentcontrol strategies and scenarios (vehicle weights). The fuel is normalizedrespect to the fuel consumed by the respective HDV driving alone usingCC. For the acronyms explanation refers to Section VII-A [%]. mass t t t t t tCC . . . . . . LAC . . . . . . CLAC . . . . . . C. Fuel-efficiency analysis for different gap policies
In the previous analysis we have always considered a TGpolicy. The aim of this section is to compare the platoonperformance for different gap policies, namely the SG, HG andTG policies, while keeping the same control strategy (in theanalysis we have considered CC). Note that in order to be ableto follow the required gap policy the second vehicle is allowedto exceed the maximum engine power. In this section we onlyfocus on the homogeneous platoon, since the results for anheterogeneous platoon are qualitatively the same. In Figure 12we show the comprehensive bar diagram representing thenormalized energy consumed by each vehicle of the platoonfor the three gap policies, while using CC as control strategy.Since the first vehicle uses the same control strategy, theenergy consumption defers only for the second vehicle. It isinteresting to notice that, similarly to the comparisons done inthe previous section, the main difference between the energyconsumption of the second vehicles is related to the energydissipated by braking. More in detail the HG policy allowsthe second vehicles to save % over the SG policy, while theTG policy allows to save an additional . % of energy. Inorder to understand the role of the gap policy on the brakingenergy, we show the platoon behavior driving over a synthetichill composed by an uphill section with constant slope grade,a flat section and a downhill section with constant slope grade.The platoon behavior for such a hill is shown in Figure 13.Analyzing the second vehicle behavior for each gap policy,the following can be observed: • TG policy: as argued in Section V, the time gap allowsthe vehicles to follow the same speed profile over space.That means that the generated forces and therefore the generated powers (because of the equal speed result)are equivalent except for a reduction of the air dragcomponent in the second vehicle. Therefore the powergenerated by the second vehicle, as can be observed inFigure 13, is approximately a biased equivalent of thatone generated by the first vehicle. • SG policy: the space gap, instead, requires the vehiclesto follow the same speed profile over time. An interest-ing consequence can be observed, for example, at thebeginning of the uphill section shown in Figure 13; assoon as the first vehicle enters the uphill section anddecelerates because of limited engine power, the secondvehicle, which is still in the flat section, has to brake inorder to respect the space gap requirement. In general,excluding the offset given by the drag power, every timethe slope increases (in Figure 13, entering the uphill andleaving the downhill sections), the second vehicle has togenerate less power than the first vehicle, while everytime the slope decreases (in Figure 13, leaving the uphilland entering the downhill sections) the second vehiclehas to generate more power than the first vehicle. Asa consequence, the second vehicle has respectively tobrake and to exceed the power limit in order to followthe required SG policy. • HG policy: the headway gap can be considered as a trade-off between a time gap and a space gap. In fact, forexample, as soon as the first vehicle enters the uphillsection and starts to decelerate, the distance between thetwo vehicles is allowed to decrease, but this decrease isnot as fast as in the case of the time gap.The results obtained by the analysis of the platoon behaviorin the case of the synthetic hill are valid also in the case ofthe original scenario. In conclusion, the time gap allows tosave more energy compared to the space and headway gaps.In addition the time gap allows all the vehicles to follow thesame speed trajectory in space and therefore it scales well withthe number of vehicles in the platoon. The complete resultsfor the normalized fuel consumption are reported in Table III.
Table IIINormalized fuel consumption of the vehicles in the platoon for differentcontrol strategies and gap policies. The fuel is normalized respect to the fuelconsumed by the respective HDV driving alone using CC. For the acronymsexplanation refers to Section VII-A [%].SG HG TG CC . . . . . . LAC . . . . . . CLAC . . . . . . VIII. P
ERFORMANCE ANALYSIS OF THE VEHICLECONTROLLER
In this section we analyze the performance of the vehiclecontroller layer (as presented in Section VI and shown inFigures 7 and 8) by focusing on the safety aspect. The analysisis based on the simulation result displayed in Figure 14 and N o r m a li ze d e n e r g y c o n s u m t i o n [ % ] E g E r E d E b SG HG TG
Figure 12. Comparison of the energy consumed by each vehicle of anhomogeneous platoon (i.e., m = m = 40 t) for three different gap policies,namely space (SG), headway (HG) and time (TG) gap policies, while usingCC as control strategy, driving along the km road displayed in Figure 1.For the plots explanation refer to the caption of Figure 10. s p ee d [ m / s ] SPHGTG05 a l t i t ud e [ m ] not steepsteep0 0 . . . . − p o w e r [ k W ] SPHGTG − p o w e r [ k W ] SPHGTG
Figure 13. Comparison of the behavior of a homogeneous platoon (i.e., m = m = 40 t) for three different gap policies, namely space (SG),headway (HG) and time (TG) gap policies, while using CC as control strategy,driving over a synthetic hill. For the plots explanation refer to the caption ofFigure 11; note that the second plot shows only the speed trajectories of thesecond vehicle (the speed trajectory of the first vehicle coincides with thatone of the second vehicle in the case of TG policy). Figure 15, where the leading HDV of a three vehicles platoondriving on a flat road brakes repeatedly with different brakingprofiles. Here we assume that the leading vehicle in the brakingphases is manually driven and, therefore, the control systemdoes not know a priori the braking profile. The consideredvehicles are identical with the parameters as defined in Table I.
A. Safety analysis
Here we focus on the safety analysis of the distributedvehicle controller layer and, in particular, we analyze the roleof the safety constraint in various situations. In Figure 14, the −
20 time [s] p o w e r [ M W ] P P P s p ee d [ m / s ] v v v d i s t a n ce [ m ] d d d s,2 d s,3 Figure 14. Behavior of a three identical vehicles platoon driving on a flatroad. The leading HDV brakes three times at , and s, with a brakingdeceleration of respectively , and m/s for . s. The first plot showsthe speed of the three vehicles: the second plot shows the distance betweenthe vehicles and the respective safety distance computed using an adaptationof inequality (40); the third plot shows the summation between the generatedpower by the engine and the braking systems of the vehicles. leading vehicle is braking with deceleration of , and m/s for . s at respectively , and s. In the second plot ofthis figure, the effective distances and that ones that wouldactivate the safety constraint (we will refer to it as the safetydistance) are shown. First we can notice how, in line with ourexpectation, the second and third vehicles are braking (seethe third plot) only when the effective distance touches thesafety distance. In fact here we recall that, according to howthe vehicle controller is designed (see Section VI-E), only theactivation of the safety constraint or a braking request from theplatoon coordinator can lead to a significant braking action.Consequently, during the first braking of m/s , both followervehicles do not brake, despite the deviation of their statesfrom the reference trajectories. During the second braking of m/s , instead, the safety constraint of the second vehicle areactivated and therefore it requires a braking action. Finally,during the third braking of m/s , the safety constraints ofboth follower vehicles activate and therefore they both brake.Note that the safety constraint is designed such that fuel-efficiency has priority on driver comfort. In fact, in this case,in order to be fuel-efficient, the braking action is required onlywhen the platoon is in a safety critical situation. However, apriori knowledge of the braking profile of the first vehicle (e.g.having a model of the driver or handling the braking actionautonomously) would have allowed to have a smoother andless intense braking action.In Figure 15, we consider a more challenging scenario inwhich the first vehicle brakes with higher intensity, simulatingan emergency situation. More precisely it brakes at s witha deceleration of m/s for s and at s with the samedeceleration until it arrives to full-stop. We can notice how,also in this scenario the safety constraint in each vehiclecontroller layer activates the braking action and guarantees −
50 time [s] p o w e r [ M W ] P P P d i s t a n ce [ m ] d d d s,2 d s,3 s p ee d [ m / s ] v v v Figure 15. Behavior of a three identical vehicles platoon driving on a flatroad. The leading HDV brakes a first time at s for s with a deceleration of m/s and a second time at s with a deceleration of m/s until arrivingto full-stop. For the plot explanation refer to the caption of Figure 14. no collision between the vehicles.IX. P ERFORMANCE ANALYSIS OF THE INTEGRATEDSYSTEM
In this section we analyze the simulation results displayedin Figure 16 of the platoon under the control of the integratedcontrol architecture (i.e., platoon coordinator and vehicle con-troller). More precisely in this analysis we consider a platoonof three identical vehicles (whose parameters are defined inTable I) driving along the Sector A highlighted in Figure 1.This is the same sector for which the experimental results in[5] are displayed in Figure 2 and analyzed in Section II.At first glance, as expected from the platoon coordinatorformulation, we can notice how all the vehicles follow thesame speed and distance profiles in the space domain. Addi-tionally, in order to follow such profiles, we can observe in thelast plot how the second and third vehicle, thanks to the airdrag reduction, need to generate less power than the leadingvehicle. We now continue the analysis by focusing on the threesegments highlighted in Figure 16: • Segment 1: due to the steep downhill, all vehicles are notable to maintain the constant speed without braking and,therefore, accelerate. However the platoon coordinatorrequires the leading vehicle to fuel slightly such that thefollower vehicles can coast. In this case, the coordinationrole of the platoon coordinator allows to avoid brakingaction to all vehicles. • Segment 2:
Since no gear shift is simulated the vehiclesare able to maintain the time gap requirement during theuphill. • Segment 3: due to the longer downhill compare to thefirst one, the platoon exhibits a different behavior. First,the platoon coordinator requires all vehicles to decreasethe speed to the minimum allowed (in this simulation itis set to m/s ) in order to hit the speed limit as late as a l t i t ud e [ m ] not steepsteep25 26 27 28 29 30 31 − p o w e r [ k W ] P P P d i s t a n ce [ m ] d d d s,2 d s,3 s p ee d [ m / s ] v v v Figure 16. Simulation results obtained using the proposed controller for athree-vehicle platoon while driving along the Sector A highlighted in Figure 1.The three vehicles are identical with parameters shown in Table I. The firstplot shows the road topography. For the explanation of the other plots referto the caption of Figure 14. possible. Second, since the speed limit is reached despitethe decrease of speed at the beginning of the downhill, theplatoon coordinator requires the first vehicle to coast andthe follower vehicles to brake slightly to maximize theefficiency. In fact, in this case, to require the first vehicleto fuel slightly and brake at the end of the downhill wouldbe contradictory.In conclusion the platoon, under the control of the integratedcontrol architecture, shows a fuel-efficient and smooth behav-ior. X. C
ONCLUSIONS AND FUTURE WORKS
A. Conclusions
In this paper we have presented a novel control architecturebased on look-ahead control for fuel-efficient and safe HDVplatooning.The use of a look-ahead control framework for HDVplatooning has been first motivated by the analysis of realexperiments. In particular in this analysis we concluded thatthe use topography information in order to predict the behaviorof the vehicles and coordination between the vehicles can bebeneficial for both fuel-efficiency and safety reasons.This led to the design of a novel control architecture forplatooning. Such architecture is divided into two layers. Acentralized higher layer, denoted as platoon coordinator, isresponsible for the coordination of the platoon by defininga speed profile that is feasible and fuel-efficient for the entireplatoon by exploiting preview topography information. Suchspeed profile is communicated to each block of the decentral-ized lower layer, denoted as vehicle controller layer. Within each vehicle controller a model predictive control routinetracks the reference speed profile and generates the real-timedesired acceleration for the low-level vehicle controller.The performance of such control architecture has beenevaluated through the analysis of numerical experiments. Indetails, the performance of the two layers has been studiedboth separately and in conjunction. B. Future works
In the modeling of the vehicle powertrain we have assumedthat the gear ratio can be chosen on a continuous interval ona unlimited span. However, this is not typically the case incommercial HDVs, where usually the transmission is handledby a gearbox that introduces fixed gear ratios and power lossesduring the gear shifts. Therefore in some future works wewant to investigate how the presence of a gearbox shouldbe managed in a optimal way. The optimal engine speedas function of the generated power shown in Figure 6 andthe knowledge of the current speed can be used to computethe instantaneous optimal gear ratio. However the power lossand the delay during the gear shift make the problem ofwhen the HDVs should change gear (e.g., independently orsimultaneously) and which gear they should engage not trivial.Secondly, we would like to investigate how external distur-bances, as traffic ahead or a vehicle cutting in the platoon,can be handled in an autonomous way within the platooncontroller framework. So far, in fact, such disturbances havebeen assumed to be handled manually by the drivers. Howeverthe prediction of local traffic would allow the platoon to movefuel-efficiently and safely in it.A
CKNOWLEDGMENT
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