Cooperative Adaptive Cruise Control for Vehicle Following During Lane Changes*

Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress Control The of Toulouse, France,Federation July 9-14, 2017 Available online at www.sciencedirect.com The International International Federation of Automatic Automatic Control The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017

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IFAC PapersOnLine 50-1 (2017) 12582–12587

Cooperative Adaptive Cruise Control for Cooperative Adaptive Cruise Control for Cooperative Adaptive Cruise Control for Vehicle Following During Lane Changes Vehicle Following During Lane Changes Vehicle Following During Lane Changes  ∗

Klaus W. Schmidt ∗ ∗ Klaus W. Schmidt Klaus Klaus W. W. Schmidt Schmidt ∗ ∗ ¸ ankaya University, Ankara, ∗ Mechatronics Engineering Department, C ∗ Engineering Department, C ¸¸ ankaya University, Ankara, Mechatronics Engineering Department, C ∗ Mechatronics Turkey (e-mail: [email protected]). MechatronicsTurkey Engineering Department, C ¸ ankaya ankaya University, University, Ankara, Ankara, (e-mail: [email protected]). Turkey (e-mail: [email protected]). Turkey (e-mail: [email protected]). Abstract: This paper addresses the longitudinal vehicle behavior before and during lane Abstract: This paper addresses vehicle behavior before and during Abstract: This addresses the longitudinal vehicle behavior before during lane changes. Hereby, it is desired thatthe thelongitudinal lane-changing vehicle simultaneously follows its lane preAbstract: This paper paper addresses the longitudinal vehicle behavior before and and during lane changes. Hereby, it is desired that the lane-changing vehicle simultaneously follows its prechanges. Hereby, it is desired that the lane-changing vehicle simultaneously follows its predecessors Hereby, on the lanes before and after the lane change.vehicle Specifically, the lane changing vehicle changes. it is desired that the lane-changing simultaneously follows its predecessors on the before after the predecessor lane change.vehicle, Specifically, the lane changing decessors on lanes before and after lane Specifically, the vehicle should keep safelanes distance to and the rearmost while maintaining a smallvehicle interdecessors onaathe the lanes before and after the the predecessor lane change. change.vehicle, Specifically, the lane lane changing changing vehicle should keep safe distance to the rearmost while maintaining a small intershould keep a safe distance to the rearmost predecessor vehicle, while maintaining a small intervehicle spacing and supporting driving comfort. To this end, the paper develops an extension should keep a safe distance to thedriving rearmost predecessor vehicle, while maintaining aansmall intervehicle spacing and supporting comfort. To this end, the paper develops extension vehicle spacing and supporting driving comfort. To this end, the paper develops an extension of cooperative adaptive cruise control (CACC). Instead of following a single vehicle as in the vehicle spacing and supporting driving comfort. To this end, the paper develops an extension of cooperative adaptive cruise (CACC). Instead of following single as in the of cooperative adaptive cruise control (CACC). Instead of single vehicle as the classical realization of CACC, itcontrol is proposed to follow a virtual vehicleaaa that is vehicle evaluated based of cooperative adaptive cruise it control (CACC). Instead of following following singleis vehicle as in in the classical realization of CACC, is proposed to follow aa virtual vehicle that evaluated based classical realization of CACC, it is proposed to follow virtual vehicle that is evaluated based on distance measurements and communicated state information from the predecessor vehicles. classical realization of CACC, it is proposed to follow a virtual vehicle that is evaluated based on distance measurements and communicated state information from the predecessor vehicles. on distance and state information from the A simulation study demonstrates the practicability of the proposed method. on distance measurements measurements and communicated communicated state of information from the predecessor predecessor vehicles. vehicles. A simulation study demonstrates the practicability the proposed method. A simulation study demonstrates the practicability of the proposed method. A simulation study demonstrates the of the proposed method. © 2017, IFAC (International Federation of practicability Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Autonomous vehicles, cooperative adaptive cruise control, lane changes, vehicle Keywords: Autonomous Keywords:safety. Autonomous vehicles, vehicles, cooperative cooperative adaptive adaptive cruise cruise control, control, lane lane changes, changes, vehicle vehicle following, Keywords: Autonomous vehicles, cooperative adaptive cruise control, lane changes, vehicle following, safety. following, safety. following, safety. 1. INTRODUCTION The safety of lane change maneuvers is studied in different 1. INTRODUCTION INTRODUCTION The safety of lane change is studiedetin different 1. The safety of maneuvers is different contexts. worksmaneuvers such as (Naranjo 2008; 1. INTRODUCTION The safetyResearch of lane lane change change maneuvers is studied studiedetin inal., different contexts. Research works such as (Naranjo al., 2008; contexts. Research works such as (Naranjo 2008; et al., 2013; Nilsson et al., 2014; Du et al., 2015) Cooperative Adaptive Cruise Control (CACC) is a re- Zheng contexts. Research works such as 2014; (Naranjo et al., al., 2015) 2008; Zheng et al., 2013; Nilsson et al., Du et Cooperative Adaptive Cruise Control (CACC) is a reZheng et al., 2013; Nilsson et al., 2014; Du et al., 2015) consider trajectories for the combined longitudinal and Cooperative Adaptive Cruise Control (CACC) is a recent technology for safe vehicle following at small interZheng et al., 2013; Nilsson et al., 2014; Du et al., 2015) Cooperative Adaptive Cruise Control (CACC) is a reconsider trajectories for the combined longitudinal and cent technology for safedistance vehicle measurements following at at small small interconsider trajectories the combined longitudinal and motion duringfor a lane change. The human behavcent technology safe vehicle following vehicle spacings, for using and interstate lateral consider trajectories for the combined longitudinal and cent technology for safe vehicle following at small interlateral motion during a lane change. The human behavvehicle spacings, using distance measurements and state lateral motion during a lane change. The human behavior during overtaking maneuvers is mimicked by fuzzy vehicle spacings, using distance measurements and state information communicated among vehicles (Rjamani and lateral motion during a lane change. The human behavvehicle spacings, using distance measurements and state during overtaking is mimicked fuzzy information communicated among vehicles (Rjamani and ior ior during overtaking maneuvers is by fuzzy (Naranjo maneuvers et al., 2008), respecting by a safety information communicated among vehicles and Shladover, 2001; Shladover et al., 2015; Dey(Rjamani et al., 2016). ior during in overtaking maneuvers is mimicked mimicked by fuzzy information communicated among vehicles (Rjamani and controllers controllers in (Naranjo et al., 2008), respecting a safety Shladover, 2001; Shladover et al., 2015; Dey et al., 2016). controllers in (Naranjo et al., 2008), respecting a safety distance to the vehicle to be overtaken. Zheng et al. (2013) Shladover, 2001; Shladover et al., 2015; Dey et al., 2016). In the current literature, CACC is used in vehicle strings, controllers in (Naranjo et al., 2008), respecting a safety Shladover, 2001; Shladover et al., 2015; Dey et al., 2016). distance to the vehicle to be overtaken. Zheng et al. (2013) In the current current literature, CACC is used used in in vehicle vehicle strings, distance vehicle be Zheng al. a the polynomial change reference that In the literature, is strings, where each vehicle staysCACC on a fixed and follows a generate distance to to vehicle to tolane be overtaken. overtaken. Zheng et etpath al. (2013) (2013) In the current literature, is usedlane in vehicle strings, aa the polynomial lane change reference path that where each vehicle staysCACC onIn aathis fixed lane and follows a generate generate polynomial lane change reference path that is then tracked using model predictive control (MPC), where each vehicle stays on fixed lane and follows a unique predecessor vehicle. setting, various metha polynomial lane change reference path that where each vehicle vehicle. stays onIn athis fixed lane and follows a generate is then tracked using model predictive control (MPC), unique predecessor setting, various methis then tracked using model predictive control (MPC), whereby the reference path keeps a safe distance to the unique predecessor vehicle. In this setting, various methods for the CACC controller design are available such as in is then tracked using path modelkeeps predictive control (MPC), unique predecessor vehicle. In this setting, various methwhereby the reference a safe distance to the ods for the the CACC controller design are available suchetas asal., in predecessor whereby the reference path keeps aa safe distance to the vehicle. High-level trajectory planning and ods for CACC controller design are available such in (Dunbar and Caveney, 2012; Naus et al., 2010; Ploeg whereby the reference path keeps safe distance to the ods for the CACC controller design are available such as in predecessor vehicle. High-level trajectory planning and (Dunbar and Caveney, 2012; Naus et al., 2010; Ploeg et al., predecessor vehicle. High-level trajectory planning and low-level control of lane change maneuvers is realized by (Dunbar and Caveney, 2012; Naus et al., 2010; Ploeg et al., 2014; Kamal et al., 2014; Kianfar etal., al.,2010; 2014). Different predecessor vehicle. High-level trajectory planning and (Dunbar and Caveney, 2012; Naus et Ploeg et al., low-level control of lane change maneuvers is realized by 2014; Kamal et al., al.,setting, 2014; Kianfar Kianfar et al., al., 2014).the Different low-level control of lane change maneuvers is realized MPC by Nilsson et al. (2014). Hereby, a safe distance to 2014; Kamal et 2014; et 2014). Different from the existing this paper considers usage low-level control of lane changeHereby, maneuvers is distance realized by by 2014; Kamal et al.,setting, 2014; Kianfar et al., 2014).the Different MPC by Nilsson et al. (2014). a safe to from the existing this paper considers usage MPC by Nilsson et al. (2014). Hereby, a safe distance to the front and rear vehicles on the same lane is kept. Du from the existing setting, this paper considers the usage of CACC for controlling the longitudinal vehicle motion MPC by Nilsson et al. (2014). Hereby, a safe distance to from the existing setting,the thislongitudinal paper considers themotion usage the front and rear vehicles on the same lane is kept. Du of CACC for controlling vehicle the front and rear vehicles on the same lane is kept. Du et al. (2015) provide a mixed integer programming (MIP) of CACC for controlling the longitudinal vehicle motion before and during lane changes. In this case, a vehicle the front and rear vehicles on the same lane is kept. Du of CACC forduring controlling the longitudinal vehiclea motion al. (2015) for provide aa mixed integer (MIP) before andsimultaneously lane changes. changes. In predecessor this case, case, vehicle et et al. provide integer programming (MIP) planning a safe path programming that can be tracked before and during In this vehicle needs to follow the et al. (2015) (2015) for provide a mixed mixed integer programming (MIP) before andsimultaneously during lane lane changes. In predecessor this case, aa vehicles vehicle formulation formulation planning a safe path that can be tracked needs to follow the vehicles formulation for planning a safe path that can be tracked by a PD controller. The MIP includes safety constraints needs to simultaneously follow the predecessor vehicles on its current lane and on its target lane. In particular, formulation for planning a safe path that can be tracked needs to simultaneously follow the predecessor vehicles by PD controller. The MIP includes safety constraints on itsdesired currenttolane lane and on onfollow its target target lane. In In particular, PD The MIP includes safety constraints for aaasurrounding vehicles enforced for vehicles on on current and its lane. it isits smoothly the potentially changing by by PD controller. controller. The that MIPare includes safety constraints on itsdesired currenttolane and onfollow its target lane. In particular, particular, for surrounding vehicles that are enforced for vehicles on it is smoothly the potentially changing for surrounding vehicles that are enforced for vehicles on the same lane as the lane changing vehicle. Different from it is desired to smoothly follow the potentially changing rearmost predecessor vehicle at a safe inter-vehicle spacing. for surrounding vehicles that are enforced for vehicles on it is desired to smoothly follow the potentially changing the same lane as the lane changing vehicle. Different from rearmost predecessor predecessor vehicle vehicle at at aa safe inter-vehicle spacing. spacing. the same lane as the lane changing vehicle. Different from proposed method, these approaches consider safety rearmost safe inter-vehicle the same lane as the lane changing vehicle. Different from rearmost predecessor vehicle at a safe inter-vehicle spacing. proposed method, these approaches consider This paper addresses the stated problem by extending the the proposed method, these approaches consider safety only with respect to vehicles the same lane as thesafety lane the proposed method, these on approaches consider safety This paper addresses the stated stated problem byThe extending This paper addresses the problem by extending only with respect to vehicles on the same lane as the lane CACC by a virtual vehicle (VV) model. VV is only with respect to vehicles on the same lane as the lane This paper addresses the stated problem byThe extending changing vehicle. In addition, global position information only with respect to vehicles on the same lane as the lane CACC by a virtual vehicle (VV) model. VV is CACC aa virtual vehicle (VV) model. VV is changing vehicle. In addition, global position information suppliedby with reference trajectories that areThe computed changing vehicle. In addition, global position information CACC by virtual vehicle (VV) model. The VV is is required when performing path tracking. changing vehicle. In addition, global position information supplied with reference trajectories that are computed supplied reference trajectories are computed required when performing path tracking. based on with distance measurements andthat state information is required when path supplied with reference trajectories areinformation computed is is required whenisperforming performing path tracking. tracking. based onfrom distance measurements andthat state A different view taken by Hatipoglu et al. (2003); Mukai based on distance measurements and state information received the predecessor vehicles. It is guaranteed based onfrom distance measurements and state information A different view is taken by Hatipoglu et al.Here, (2003); received the predecessor vehicles. It is guaranteed A different view is taken by Hatipoglu et (2003); Mukai and Kawabe (2006); Dang et al. (2015). it Mukai is asreceived from the predecessor vehicles. It is guaranteed by designfrom that the the trajectory of vehicles. the VV It is guaranteed A different view is taken by Hatipoglu et al. al.Here, (2003); Mukai received predecessor and Kawabe (2006); Dang et al. (2015). it is asby design that the trajectory of the VV and Kawabe (2006); Dang et al. (2015). Here, it is sumed that the longitudinal and lateral motion during by design that the trajectory of the VV Kawabe (2006); Dang et and al. (2015). Here, it during is asasby •design that to thethe trajectory of the VVrearmost predeces- and sumed that the longitudinal lateral motion converges trajectory of the that the and lateral motion during asumed lane change canlongitudinal be considered separately. Accordingly, sumed that the longitudinal and lateral motion during converges to the trajectory of the rearmost predeces•• converges to the trajectory of the rearmost predeceslane change can be considered separately. Accordingly, sor vehicletoifthe the distance of between the predecessor lane can be separately. Accordingly, • converges the rearmost predeces- aaaHatipoglu et al. realize the longitudinal motion lane change change can(2003) be considered considered separately. Accordingly, sor vehicle if the thetrajectory distance between between the predecessor predecessor sor vehicle if distance the Hatipoglu et al. (2003) realize the longitudinal motion vehicles is large Hatipoglu et al. (2003) realize the longitudinal motion sor vehicle if the distance between the predecessor by intelligent cruise control and focus on the generation Hatipoglu et al. (2003) realize the longitudinal motion vehicles is large is large by intelligent cruise control and focus on the generation • vehicles smoothly changes between the predecessor vehicles if by intelligent cruise control and focus on the generation vehicles is large of steering commands for the lateral motion. Mukai and by intelligent cruise control and focus on the generation • smoothly changes between the predecessor vehicles if •• smoothly changes between the predecessor vehicles if of of steering commands for the lateral motion. motion. Mukai and the predecessor vehicles overtake each other. steering commands for the lateral Mukai and smoothly changes between the predecessor vehicles if Kawabe (2006) address the longitudinal motion and proof steering commands for the lateral motion. Mukai and the predecessor predecessor vehicles vehicles overtake overtake each each other. other. the Kawabe (2006) address the longitudinal motion and proKawabe (2006) address the longitudinal motion and prothe predecessor vehicles overtake each other. pose a method for the high-level trajectory planning based A simulation study shows that the lane-changing vehicle Kawabe (2006)for address the longitudinal motion andbased propose a method the high-level trajectory planning A simulation study shows shows the pose aa method for high-level planning based system modeling andtrajectory model-predictive A simulation study that the lane-changing lane-changing vehicle on safely and smoothly followsthat the VV using CACC. vehicle posehybrid method for the the high-level trajectory planningcontrol based A simulation study shows the lane-changing hybrid system modeling and model-predictive control safely and smoothly smoothly followsthat the VV VV using CACC. CACC. vehicle on on hybrid system modeling and model-predictive control (MPC) without a low-level realization. Hereby, the safety safely and follows the using on hybrid system modeling and model-predictive control safely and smoothly follows the VV using CACC. (MPC) without low-level realization. realization. Hereby, the safety  This work was supported by the Scientific and Technological (MPC) without aaa low-level Hereby, safety distance to two surrounding vehicles is taken intothe account. (MPC) without low-level realization. Hereby, the safety  distance to two surrounding vehicles is taken into account. This work was supported by the Scientific and Technological  Research Council of Turkey (TUBITAK) [Award 115E372]. distance to two surrounding vehicles is taken into account. This work was supported by the Scientific and Technological  This work was supported by the Scientific and Technological distance to two surrounding vehicles is taken into account. Research Council of Turkey (TUBITAK) [Award 115E372].

Research Council of Turkey (TUBITAK) [Award 115E372]. Research Council of Turkey (TUBITAK) [Award 115E372]. Copyright © 2017, 2017 IFAC 13093 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 13093 Copyright ©under 2017 responsibility IFAC 13093 Peer review of International Federation of Automatic Control. Copyright © 2017 IFAC 13093 10.1016/j.ifacol.2017.08.2199

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i+1 i Proceedings of the 20th IFAC World Congress Klaus W. Schmidt et al. / IFAC PapersOnLine 50-1 (2017) 12582–12587 Toulouse, France, July 9-14, 2017

An extension of adaptive cruise control (ACC) to the case of lane changes is proposed by Dang et al. (2015). In this work, the scenario where the lane-changing vehicle has to simultaneously follow two vehicles on different lanes is addressed. Safe following is achieved by a combination of MPC and fuzzy logic based on distance measurements. This is different from our paper, where vehicle following is realized by CACC, which requires the usage of a virtual predecessor vehicle that provides consistent virtual distance measurements and additional state information. The remainder of the paper is organized as follows. Section 2 provides the necessary background on CACC and Section 3 motivates the problem of simultaneously following two predecessor vehicles. In Section 4, the virtual vehicle is defined and its properties are demonstrated. Conclusions are given in Section 5. 2. PRELIMINARIES

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by Radar or di is measured Gi-1 Kfb,iLIDAR in analogy to adaptive ucruise control (ACC). In addition, information from other i-1 vehicles can be received via vehicle-to-vehicle (V2V) comHi munication. In vehicle this paper, we employ the frequently used i predecessor following (PF) strategy, where vehicle i − 1 sends state information to vehicle i (Ploeg et al., 2014; Shladover et al., 2015; Dey et al., 2016). There are different policies for the desired vehicle spacing. We focus on the constant-time-gap policy with (2) dr,i = ri + hi vi as the desired gap with the headway time hi and the desired spacing at standstill ri (for vi = 0). 2.2 Control Architecture 1

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Fig. 3. Feedback loop for CACC on vehicle i. The constant-time-gap policy in (4) is realized by the transfer function Hi (s) = 1 + hi s and the spacing error is ei = di − dr,i = qi−1 − qi − ri − hi vi . (4) Kfb,i is a feedback controller that controls the spacing error and Kff,i is a feedforward filter that receives the signal ui−1 from vehicle i − 1 via V2V communication as in (Ploeg et al., 2014, 2015). Here, a communication delay represented by Di (s) = e−s θi is encountered. N

This paper employs vehicle following as depicted in Fig. 1 based on the concept of cooperative adaptive cruise control (CACC) (Shladover et al., 2015; Liu and Xu, 2015; Ploeg et al., 2015; Dey et al., 2016).

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3.1 Motivation The usage of CACC in the existing literature is limited to the case where vehicles follow each other on a fixed lane, where the predecessor of each vehicle is uniquely known. However, this setting does not capture the desired vehicle behavior before and during lane changes, where the lane changing vehicle, denoted as ego vehicle (EV), is faced with predecessor vehicles on the lanes before and after the lane change. The EV needs to simultaneously follow both predecessor vehicles before and during the lane change. In order to illustrate the stated scenario, we consider Fig. 2. Here, the EV intends to perform a lane change. Its two predecessor vehicles are L1 and L2. It is further assumed that EV can measure the distance to both L1 and L2 and receives uL1 , uL2 from L1, L2 via V2V communication. Part (a) 9 shows the 8 most basic7 configuration, where L1 10 6 5 and L2 assume the same position and travel at the same velocity. In that case, EV can alternatively perform CACC with L1 or L2. The situation is different in part (b) and part (c). In part (b), EV should follow the closer vehicle L1, whereas EV should follow L2 in part (c). 3.2 Safety and Comfort Requirements We next point out the main issues when simultaneously following two vehicles. To this end, we perform a simple simulation experiment. L1 and L2 start at the same initial position with the same initial velocity 20 m/sec. L1 performs a speed-up/slow-down maneuver between the times 1 sec and 5 sec, whereas L2 performs a similar maneuver between the times 3.2 sec and 9 sec such that L2 overtakes L1 at time 6.2 sec. For illustration, we pursue a 1 3 2 naive approach, where EV always performs CACC using the signals of the rearmost vehicle (RV). The simulation results are shown in Fig. 4. We plot the position, velocity (c) and acceleration of L1, L2, RV and EV. Since velocities around 20 m/sec are chosen in the experiments, the figure shows ”position−20 m/sec·t” instead of the actual position for better visibility.

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4. SIMULTANEOUS VEHICLE FOLLOWING USING A VIRTUAL VEHICLE 4.1 Virtual Vehicle Definition We define a virtual vehicle (VV) that generates suitable signals for applying CACC on the EV. VV is based on the plant model in (3) as is shown in Fig. 6.

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It has to be noted that the undesired behavior is observed due to the naive approach of directly following the RV. Nevertheless, the identified problem is general and needs to be addressed if the EV has to simultaneously follow two predecessor vehicles. In particular, a smooth transition should be achieved in case the RV changes to support driving safety and comfort. As opposed to the naive approach, this requires a fusion of the distance measurements with L1 and L2 as well as the communicated state information.

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N to Fig. 4 indicates that the naive approach can lead N undesired behavior. For example, a large spacing error of EV is observed after (a) L2 overtakes L1, resulting in reduced (b) driving safety (see Fig. 5). The reason for this effect is the velocity difference between L1 and L2 at the time of overtaking. EV first follows L2 with a larger7 velocity 8 11 10 9 and then switches to L1 with a smaller velocity. In order to reduce the resulting spacing error, EV has to break sharply, which also reduces the driving comfort.

Here, q and u denote the position and input signal of VV, respectively. In addition, VV employs reference signals qr 1 1 2 (position), vr (velocity), ar (acceleration)N and ur (input N signal). These reference signals are computed based on the distance(a)measurements qi−1 − qi and the (b) communicated signals q˙i , q¨i and ui , i ∈ {L1, L2}. The definition of suitable reference signals according to the observations in Section 7 10 6 3.2 is the main9 subject of8 the remainder of the paper.5 In order to track these reference signals, proportional feedback of the deviations qr −q, vr − q˙ and ar − q¨ is applied. From the implementation perspective, VV is computed locally by the EV and the signals u and q are used for performing CACC according to the architecture in Fig. 3. 4.2 Reference Position for the Virtual Vehicle Considering the reference position qr of VV, it is desired that q(t) ≤ min{qL1 (t), qL2 (t)} in order to ensure driving safety. Let ∆q = qL1 − qL2 and q(t) = 0.5 · (qL1 (t) + qL2 (t)) be the position difference and average position of L1 and L2, respectively. Then, we define the reference position (5) qr (t) = q(t) + gq (∆q) ∆q with a symmetric function gq (∆q) = −qq (−∆q) such that   gq (∆q) = 0.5 if ∆q < −q g (∆q) = −0.5 if ∆q > q (6)  |g q (∆q)| ≤ 0.5 otherwise. q

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Here, q is a safety parameter that indicates if L1 and L2 are close to each other (|∆q| ≤ q ). Hence, qr (t) = qL1 (t) if L1 is sufficiently behind L2 (∆q < −q ) and qr (t) = qL2 (t) in the reverse case (∆q > q ). If both vehicles are close to each other (−q ≤ ∆q ≤ q ), a reference position close to the rearmost vehicle is assumed. Consider for example that L1 is the rearmost vehicle with −q ≤ ∆q ≤ 0. Then, the distance between qr and qL1 is evaluated as qL1 (t) + qL2 (t) + gq (∆q(t))∆q(t) − qL1 (t) qr (t) − qL1 (t) = 2 = (gq (∆q(t)) − 0.5) ∆q(t). Hence, the maximum deviation from the desired minimum position qL1 (t) is given by ∆qr,max = max (gq (∆q) − 0.5) ∆q. (7) −q ≤∆q≤0

Due to symmetry, the same result is obtained when considering L2 as the rearmost vehicle with 0 ≤ ∆q ≤ q .

In order to ensure a smooth transition for changes of the rearmost vehicle, we further require that gq (∆q) has three continuous derivatives w.r.t. ∆q, that is, gq (∆q) ∈ C 3 , respecting that the vehicle model has degree three.

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|∆q| ) ˙ (1 − gq (∆q) ˙ = gv (∆q, ∆q) αv (∆q) ˙     +gq (∆q) |∆q| αv (∆q) ˙

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otherwise (9) (10)

That is, if the distance |∆q| between both predecessor vehicles is large compared to their velocity difference ˙ vr is equal to the velocity of the rearmost (|∆q| ≥ αv (∆q)), vehicle. Otherwise, vr approaches the velocity of the slower vehicle especially for small ratios |∆q|/αv (∆q). ˙ As an effect of the latter measure, VV is able to slow down if a faster rearmost vehicle is going to overtake a slower vehicle. In particular, a faster slow-down is achieved for larger velocity differences ∆q˙ since av (∆q) ˙ is assumed to be monotonically increasing. For illustration, Fig. 8 shows gv (∆q, ∆q) ˙ for q = 1 m and av (∆q) ˙ = 1 + 0.3 · ∆q˙2 .

An example function for the stated requirements is gq (∆q) =   −1.25 ∆q + 2.5 ( ∆q )3 − 2.5 ( ∆q )4 + 0.75 ( ∆q )5 if ∆q ≥ 0 q

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(8) as depicted in Fig. 7 (a). It can be further seen in Fig. 7 (b), which shows (qr (t) − min{qL1 (t), qL2 (t)})/q , that ∆qr,max in (7) depends on the chosen safety parameter q . For example, ∆qr,max < 6 cm if q = 1 m and ∆qr,max < 30 cm if q = 5 m.

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Fig. 7. (a) Example choice of gq (∆q); (b) corresponding deviation from minimum position normalized by q . 4.3 Reference Velocity for the Virtual Vehicle Considering the reference velocity vr , the velocity difference between L1 and L2 when performing an overtaking maneuver has to be taken into account as discussed in Section 3.2. For example, if L1 is the rearmost vehicle (see Fig. 2 (b)) and approaches L2 at a velocity q˙L1 > q˙L2 , it is desired that VV gradually adjusts its velocity from q˙L1 to q˙L2 . Hereby, it is intuitive that more adjustment is needed if |∆q | is small or the velocity difference |∆q| ˙ is large.

In order to realize the specified requirements, we use a monotonically increasing function αv (∆q) ˙ with αv (0) = q that indicates the requirement of a velocity adjustment if |∆q| < αv (∆q). ˙ Using αv , we define the function

Considering the reference acceleration/input, we require that ar and ur follow the respective signal of the rearmost vehicle with a smooth transition in case of overtaking similar to the reference position in Section 4.2. That is, q (11) ar (t) = ¨q(t) + gq (∆q) ∆¨ and (12) ur (t) = u(t) + gq (∆q) ∆u uL1 (t) + uL2 (t) and ∆u = uL1 (t) − uL2 (t). using u(t) = 2 4.5 Virtual Vehicle Properties It is now possible to state two main properties of the VV defined in the previous sections. Theorem 1. Assume that the VV is defined as in Section 4.1 with the reference signals in (5), (9), (11) and (12). Further let K1 , K2 , K3 be chosen such that τ s3 + (K3 + 1) s2 + K2 s + K1 is Hurwitz. Then, it holds that ˙ 1. limt→∞ q(t) = min{qL1 (t), qL2 (t)} if ∆q(t) ≥ av (∆q(t)) 2. limt→∞ ... q(t) − min{qL1 (t), qL2 (t)} ≤ ∆r,max if ∆q˙ = ∆¨ q = ∆ q = 0.

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The proof of Theorem 1 is published in the form of a technical report. Theorem 1 characterizes two important scenarios for the trajectory q of VV. According to 1., q equals the trajectory of the rearmost vehicle if the distance between both predecessor vehicles is sufficiently large. Furthermore, 2. suggests that q is close to the trajectory of the rearmost vehicle if both leader vehicles travel at the same constant velocity. We further note that the computation of the VV trajectory does not require any optimization and is hence suitable for a real-time implementation. In order to evaluate the dynamic behavior of the VV in scenarios different from the ones in Theorem 1, we next perform various simulation experiments.

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Fig. 11. Spacing error signals of EV for three experiments. 4.6 Simulation Examples We consider the setting in Fig. 2 with two predecessor vehicles L1 and L2 and the EV that follows VV according to Section 4.1. We perform three different experiments. In each experiment, we plot the position, velocity and acceleration of L1, L2, VV and EV as well as the spacing error of EV according to (4). Since velocities around 20 m/sec are chosen in the experiments, our plots show ”position − 20 m/sec · t” instead of the actual position. We use a plant model with τ = 0.1 and the headway time constant is h = 0.5 as in (Ploeg et al., 2014). For the realization of the reference trajectories, we use gq (∆q) in (8), q = 1 m and av (∆q) ˙ = 1 + 0.3 · ∆q˙2 . The feedback constants in Fig. 6 are chosen as K1 = 10, K2 = 30 and K3 = 25 such that τ s3 +(K3 +1) s2 +K2 s+K1 is Hurwitz. The first experiment is identical to the experiment in Section 3.2, using VV instead of RV for CACC. The simulation results are shown in Fig. 9. It is readily observed that VV moves close to the position of the rearmost vehicle. In particular, q ≈ qL2 until L2 overtakes L1 at time 6.2 sec and q ≈ qL1 afterwards. This behavior is enabled by the realization of vr in (10). As can be seen from the figure, the velocity of VV decreases before the actual overtaking maneuver, achieving a smooth transition when the rearmost vehicle changes. In addition, it can be seen that the EV is able to follow VV (and hence the rearmost vehicle) at a safe distance using CACC with the virtual input signals u and q from VV. This observation is confirmed when looking at the spacing error according to (4) in Fig. 11, which stays below 1 m for this experiment. We recall that a spacing error with a magnitude of up to 5 m is observed in Fig. 5 when using the RV for CACC. The second experiment is analogous to the first experiment with the difference that L2 overtakes L1 at time 8.6 sec with a lower speed difference. The results are shown in Fig. 10. Again, the VV slows down before overtaking, whereby smaller accelerations are required compared to experiment 1 and smaller spacing errors are encountered (see Fig. 11). The third experiment in Fig. 12 realizes multiple overtake maneuvers with non-zero accelerations, where the overtaking vehicle accelerates and the vehicle being overtaken decelerates. Similar to the previous experiments, a smooth transition between changing RV trajectories with small spacing errors is achieved. We note that the VV is defined for the case of zero communication delay θi = 0 according to Fig. 3. In order

to provide a preliminary assessment of the case with nonzero communication delay, we consider the same setting as in experiment 1 to 3, whereby the communicated signals q˙i , q¨i , ui , i ∈ {L1, L2} are received with a delay of θ = 0.06 sec as in (Naus et al., 2010). The resulting spacing errors are also shown in Fig. 11. It can be seen that the suggested VV shows robustness to communication delays. A further investigation of this topic is the subject of future work. 5. CONCLUSION Cooperative adaptive cruise control (CACC) is a recent technology for safe vehicle following at small inter-vehicle spacings based on distance measurements and communicated state information. This paper extends CACC to the case where a vehicle follows two predecessor vehicles on different lanes simultaneously, for example when preparing or carrying out a lane change. To this end, the paper defines a virtual vehicle (VV) that can be used for safe and smooth vehicle following and that can be computed in real-time. On the one hand, VV tracks the trajectory of the rearmost predecessor vehicle if it is safe to follow this vehicle. On the other hand, a smooth transition between the predecessors is achieved in case they overtake each other. Representative simulation experiments demonstrate the applicability of the proposed method. REFERENCES Dang, R., Wang, J., Li, S.E., and Li, K. (2015). Coordinated adaptive cruise control system with lane-change assistance. IEEE Transactions on Intelligent Transportation Systems, 16(5), 2373–2383. Dey, K.C., Yan, L., Wang, X., Wang, Y., Shen, H., Chowdhury, M., Yu, L., Qiu, C., and Soundararaj, V. (2016). A review of communication, driver characteristics, and controls aspects of cooperative adaptive cruise control (CACC). IEEE Transactions on Intelligent Transportation Systems, 17(2), 491–509. Du, Y., Wang, Y., and Chan, C.Y. (2015). Autonomous lane-change controller. In IEEE Intelligent Vehicles Symposium, 386–393. Dunbar, W. and Caveney, D. (2012). Distributed receding horizon control of vehicle platoons: Stability and string stability. Automatic Control, IEEE Transactions on, 57(3), 620–633. Hatipoglu, C., Ozguner, U., and Redmill, K.A. (2003). Automated lane change controller design. IEEE Transactions on Intelligent Transportation Systems, 4(1), 13– 22.

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approach. Vehicular Technology, IEEE Transactions on, 59(9), 4268–4279. Nilsson, J., Gao, Y., Carvalho, A., and Borrelli, F. (2014). Manoeuvre generation and control for automated highway driving. In IFAC World Congress, 6301–6306. Ploeg, J., Semsar-Kazerooni, E., Lijster, G., van de Wouw, N., and Nijmeijer, H. (2015). Graceful degradation of cooperative adaptive cruise control. IEEE Transactions on Intelligent Transportation Systems, 16(1), 488–497. Ploeg, J., Shukla, D., Van De Wouw, N., and Nijmeijer, H. (2014). Controller synthesis for string stability of vehicle platoons. Intelligent Transportation Systems, IEEE Transactions on, 15(2), 854–865. Rjamani, R. and Shladover, S.E. (2001). An experimental comparative study of autonomous and cooperative control systems for automated vehicles. Journal of Transportation Research, Part C: Emerging Technologies, 9(1), 15–31. Shladover, S.E., Nowakowski, C., Lu, X.Y., and Ferlis, R. (2015). Cooperative adaptive cruise control: Definitions and operating concepts. Transportation Research Record, 2489, 145–152. Zheng, H., Huang, Z., Wu, C., and Negenborn, R.R. (2013). Model Predictive Control for Intelligent Vehicle Lane Change, 265–276.

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