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Networked Vehicle Merging by Cooperative Tracking Control
7th IFAC Workshop on Distributed Estimation and 7th 7th IFAC IFAC Workshop Workshop on on Distributed Distributed Estimation Estimation and and 7th IFACin onSystems Distributed Estimation Control Networked Availableand online at www.sciencedirect.com Control inWorkshop Networked Systems Control in Networked Systems 7th IFAC Workshop on Distributed Estimation and Chicago, IL, USA, September 2019 Control Networked Systems16-17, Chicago, IL, USA, 16-17, Chicago,in IL, USA, September September 16-17, 2019 2019 Control in Networked Systems Chicago, IL, USA, September 16-17, 2019 Chicago, IL, USA, September 16-17, 2019
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Networked Vehicle Merging by Networked Vehicle Merging by Networked Vehicle Merging by Networked Vehicle Merging by Cooperative Tracking Control Cooperative Tracking Control Cooperative Tracking Control Cooperative ∗∗Tracking ∗∗Control ∗∗
Alexander Alexander Schwab Schwab ∗∗ Kai Kai Schenk Schenk ∗∗ Jan Jan Lunze Lunze ∗∗ Alexander Schwab ∗∗ Kai Schenk ∗∗ Jan Lunze ∗∗ Alexander Schwab Kai Schenk Jan Lunze ∗ ∗ a ∗ Institute of Automation and Computer Control, Ruhr-Universit¨ att ∗ Institute of Automation and Computer Control, Ruhr-Universit¨ Institute of Automation and Computer Control, Ruhr-Universit¨ at Bochum, (e-mail: {schwab, schenk, lunze}@atp.rub.de) ∗ schenk, Control, lunze}@atp.rub.de) Bochum, (e-mail: {schwab, Institute of Automation and Computer Ruhr-Universit¨ at Bochum, (e-mail: {schwab, schenk, lunze}@atp.rub.de) Bochum, (e-mail: {schwab, schenk, lunze}@atp.rub.de) Abstract: Abstract: This This paper paper solves solves the the vehicle vehicle merging merging problem problem by by trajectory trajectory tracking tracking of of networked networked Abstract: This paper solves vehicle the vehicle merging problemtracking by trajectory tracking ofdistributed networked vehicles to combine multiple streams. Trajectory is achieved with vehicles to combine multiple vehicle streams. Trajectory tracking is achieved with distributed Abstract: This paper solves the vehicle merging problem by trajectory tracking of networked vehicles to combine multiple vehicle streams. Trajectory tracking is achieved with distributed feed-forward controllers that the between the by the feed-forward controllers that consider consider the couplings couplings between the vehicles vehicles by modifying modifying the local local vehicles to combine multiple vehicle streams. Trajectory tracking is achieved with distributed feed-forward controllersFor thatthis consider the couplings between vehicles by the local reference trajectories. purpose, the controllers arethe connected viamodifying a communication communication are connected via a reference trajectories. For this purpose, the controllers feed-forward controllers that consider the couplings between the vehicles by modifying the local reference Different trajectories. For this purpose, the are controllers connected via that a communication network. communication strategies discussed and it the network. Different communication strategies discussedare it is is shown shown the proposed proposed reference trajectories. For this purpose, the are controllers areand connected via that a communication network. Different communication strategies are discussed and it is vehicle shown that the proposed tracking controllers solve the merging problem with arbitrary linear models. tracking controllers solve the merging problem with arbitrary linear vehicle models. network. Different communication strategies are discussed and it is shown that the proposed tracking controllers solve the merging problem with arbitrary linear vehicle models. tracking ©controllers solve the merging problem with arbitrary linear vehicle models. Copyright 2019. The Authors. Published by Elsevier Ltd. All rights reserved. 1. 1. INTRODUCTION INTRODUCTION Physical 1. INTRODUCTION Physical coupling coupling 1. INTRODUCTION Physical coupling Physical coupling Merging manoeuvres manoeuvres have have to to be be performed performed in in lane lane reducreducMerging Merging manoeuvres to be performed in lane reduction scenarios (Fig. 2) 2) have to combine combine multiple vehicle streams. tion scenarios (Fig. to multiple yyN yy11 vehicle streams. yy11 yyN u u Merging manoeuvres have to be performed in lane reducu111 uN N N N 1 tion scenarios (Fig. 2) to combine multiple vehicle streams. . . . N P 1 N C Some commercial driver assistance systems are able to Some commercial driver assistance systems are streams. able to yN y1 C ... C111 u1 P P111 y1 yN N u N PN N N tion scenarios (Fig. 2) to combine multiple vehicle CN N N y . . . Some commercial driver assistance systems are able to y 1 C 1 u 1 P1 y 1 perform autonomous autonomous lane lane changes, changes, but but they they have have to to wait wait N C N u N PN y N perform . . . Some commercial driver assistance systems are able to C1 P1 CN PN perform autonomous but they for sufficiently largelane gap changes, on the the target target lane.have Thisto is wait why for aa sufficiently large gap on lane. This is why perform autonomous lane changes, but they have to wait for a sufficiently large gap on the target lane. This is why the present paper proposes an alternative way that solves the paperlarge proposes anthe alternative wayThis that issolves for apresent sufficiently gap on target lane. why proposes an alternative way solves Communication the present mergingpaper problem cooperatively. A set set of of N that vehicles is the merging problem cooperatively. A N vehicles is Communication network network the present paper proposes an alternative way that solves Communication network the merging problem cooperatively. A set of N vehicles is modelled as as aa networked networked system system as as illustrated illustrated in in Fig. Fig. 1. 1. modelled Communication network the merging problem cooperatively. A set of N vehicles is modelled as a corresponds networked system as illustrated Fig. 1. Each vehicle corresponds to aa subsystem subsystem Pii ,, inwhich is Each vehicle to P which is Fig. modelled as a networked system as illustrated Fig. 1. 1. i in 1. Control Control structure structure of of aa networked networked system system Each vehicle corresponds to a subsystem P is Fig. i , which controlled by a networked feed-forward controller C ii .. 1. Control structure of a networked system controlled by a networked feed-forward controller C Each vehicle corresponds to a subsystem Pi , which is Fig. i Fig. 1. Control structure of a networked system controlled by a networked feed-forward controller Ci . controlled a networked . This paper considers vehicle merging trajectory trollers and This paperby considers vehiclefeed-forward merging as as a acontroller trajectoryCitracktracktrollers as as published published by by Semsar-Kazerooni Semsar-Kazerooni et et al. al. (2017) (2017) and This paper considers vehicle merging as a trajectory tracking problem. Trajectory tracking describes the problem of trollers asal. published by Semsar-Kazerooni et al. (2017) and Rupp et (2018), respectively. These concepts exhibit ing problem. Trajectory tracking describes the problem of Rupp et al. (2018), respectively. These concepts exhibit This paper considers vehicle merging as a trajectory track- trollers as published by Semsar-Kazerooni et al. (2017) and ing problem. Trajectory tracking describes the problem of controlling dynamical systems, so that the output follows Rupp et al. (2018), respectively. These concepts exhibit unfavourable behaviour like unbounded and binary control controlling systems, that the the output follows behaviour like unbounded binary exhibit control ing problem.dynamical Trajectory trackingso problem of unfavourable Rupp et al. (2018), respectively. Theseand concepts controlling sodescribes that the output follows a designated trajectory. Considering networked systems as behaviour like unbounded and binary signals in to analysis which is a designateddynamical trajectory.systems, Considering networked systems as unfavourable signals in addition addition to aa complicated complicated analysis which control is why why controlling dynamical systems, so that the output follows unfavourable behaviour like unbounded and binary control in addition to a complicated analysis which is why a designated trajectory. Considering networked systems as signals illustrated in Fig. 1, the control aim reads as the present paper describes a solution that manages the illustrated in Fig. 1, the control aim reads as the present paper describes a solution that manages the a designated trajectory. Considering networked systems as signals in addition to a complicated analysis which is why illustratedyiin Fig. 1, the control aim reads as. , N the present paperwith describes a solution that manages the merging problem networked linear controllers. (t) = y (t), t ≥ 0, i = 1, 2, . . (1) merging problem with networked linear controllers. illustratedyiiin the control present paper describes a solution that manages the t ≥ 0, aim i =reads 1, 2, . .as. , N (1) the (t)Fig. = yi1,(t), merging problem with networked linear controllers. (t) = yiii (t), t ≥ 0, trajectory i = 1, 2, . .of .,N (1) yidenoting with the system merging problem networked linear controllers. Tracking control interconnected systems is ii .. (t) = yi (t), t ≥ 0, trajectory i = 1, 2, . .of .,N Tracking control of ofwith interconnected systems is aa challenging challenging with yyiii (t) (t)yidenoting the planned planned system P P(1) with yi (t) denoting the planned trajectory of system Pii . Tracking control of interconnected systems is a challenging task and only a few publications deal with that topic. task and only a few publications deal with that topic. For For with y (t) denoting the planned trajectory of system P . Achieving control Tracking control of interconnected systems is a challenging i i Achieving control aim aim (1) (1) is is a a difficult difficult task task because because the the task and only a few publications deal with that topic. For the process of merging, it is particularly relevant that each the process of merging, it is particularly relevant that each Achieving control aim (1) is a difficult task because the (t) is arbitrary and the subsystems indesired trajectory y task and only a few publications deal with that topic. For ii (t) (1) is arbitrary and the subsystems in- the desired trajectory process of to merging, itgiven is particularly relevant that each Achieving control yaim is a difficult task because the vehicle is able follow a trajectory perfectly. Schenk i vehicle is able to follow a given trajectory perfectly. Schenk (t) is arbitrary and the subsystems indesired trajectory y fluence each other due to physical couplings. To solve this the process of merging, it is particularly relevant that each i fluence each other ydue to physical couplings. To solve this is arbitrary and the subsystems in- vehicle desired trajectory is able to follow a given trajectory perfectly. Schenk et and Schenk and Lunze presented i (t)to fluence each physical couplings. To solve this et al. al. (2018) (2018) and Schenk and Lunze (2018) (2018) presented problem, the local controllers are to is able to follow a given trajectory perfectly. Schenk problem, theother local due controllers are enabled enabled to communicate communicate fluence each other due to physical couplings. To solve this vehicle et al. (2018) and Schenk and Lunze (2018) presented two methods for the design of networked feed-forward two methods for the design of Lunze networked feed-forward problem, the localto controllers are influence enabled toof communicate with each other consider the the physical et al. (2018) and Schenk and (2018) presented with each other to consider the influence of the physical problem, the local controllers are enabled to communicate two methods the for design of tracking. networkedHowever, feed-forward controllers that allow perfect both controllers thatfor perfect both with eachand other to consider the influence of the physical couplings generate appropriate control u methods forallow the for design of tracking. networkedHowever, feed-forward couplings generate appropriate control signals signals uiii (t), (t), so so two controllers that allow for perfect tracking. However, both with eachand other to consider the influence of the physical publications require conditions on the coupling relation publications require conditions on the coupling relation couplings and generate appropriate control signals u (t), so controllers that allow for perfect tracking. However, both i that the control aim (1) is accomplished by all subsystems. that the control aim (1)appropriate is accomplished by signals all subsystems. couplings and generate control u (t), so publications require conditions on the coupling relation that are by As an contrii that are not not satisfied satisfied by vehicles. vehicles.on Asthe an important important contrithat the control aim (1) is accomplished by all subsystems. publications require conditions coupling relation that control aim is accomplished by all subsystems. are the not present satisfiedpaper by vehicles. important contriThis paper solves the tracking problem networked sysbution, is to relax the This the paper solves the(1) tracking problem for for networked sys- that bution, is able able As to an relax the conditions conditions that are the not present satisfiedpaper by vehicles. As an important contriThis paper solves the tracking problem for networked sysbution, the present paper is able to relax the conditions tems and uses the proposed tracking controller to solve the in order to make the method applicable for the merging tems paper and uses the the proposed tracking controller to solvesysthe bution, in orderthe to make thepaper method applicable merging This solves tracking problem for networked present is able to relaxfor thethe conditions in order to make the method applicable for the merging tems and uses the proposed tracking controller to solve the merging problem cooperatively in distributed manner. problem. Two other notable contributions are given by merging cooperatively in a acontroller distributed manner. other by tems andproblem uses the proposed tracking to solve the problem. in order toTwo make the notable method contributions applicable for are the given merging merging problem cooperatively in avehicles distributed manner. problem. Two other notable contributions areal. given by The solution is valid for arbitrary and does not Aschemann and Sun (2013) and Hamiche et (2016), The solution is valid for arbitrary vehicles and does not Aschemann and Sun (2013) and Hamiche et al. (2016), merging problem cooperatively in a distributed manner. problem. Two other notable contributions are given by The solution is valid for arbitrary vehicles and does not Aschemann and Sun (2013) and Hamiche et al. (2016), require a coordinating unit. It is discussed what informawhich use flatness-based controllers for tracking control require a coordinating unit. It is discussed informause flatness-based controllers for tracking control The solution is valid for arbitrary vehicles what and does not which Aschemann and Sun (2013) and Hamiche et al. (2016), require coordinating unit. is controlled discussed what informause flatness-based controllers for tracking control tion to be by the subsystems to of drive and chain. both tion has hasa to be exchanged exchanged by It the controlled subsystems to which of aa hydro-static hydro-static drive train train and aa supply supply chain. In In both require a coordinating unit. It is discussed what informawhich use flatness-based controllers for tracking control a hydro-static drive train controllers and a supply chain. In both tion hasthe to be exchanged by the controlled subsystems to of achieve control aim. contributions, decentralised are designed that achieve the control aim. contributions, decentralised controllers are designed that tion has to be exchanged by the controlled subsystems to of a hydro-static drive train and a supply chain. In both achieve the control aim. contributions, decentralised controllers are designed that share information with each other, which is comparable share information with each other, which is comparable achieve the control decentralised controllers are designed that Literature. There are Literature. There aim. are several several publications publications on on merging merging contributions, share information with each other, which is comparable to the present paper. However, since they exploit theinformation present paper. since they exploit the the Literature. There in arethe several on and merging share with However, each other, which is comparable control summarised survey by Macontrol summarised surveypublications by Rios-Torres Rios-Torres Ma- to to the present paper. However, since they exploit the very specific structure of their applications, their method Literature. There in arethe several publications on and merging verythe specific structure ofHowever, their applications, their method to present paper. since they exploit the control summarised in the survey by Rios-Torres and Malikopoulos (2017), but most solutions come with stringent specific structure of their applications, likopoulos (2017), but solutions come with stringent cannot be used used to the the more more general problem their here. method control summarised in most the survey by Rios-Torres and Ma- very cannot be to general problem here. very specific structure of their applications, their method likopoulos (2017), but most solutions come with stringent limitations. Schmidt and Posch (1983) proposed a central limitations. Schmidt Posch (1983)come proposed a central cannot be used to the more general problem here. likopoulos (2017), butand most solutions with stringent be usedliterature, to the more general problemapproaches here. limitations. Posch proposed a central cannot concept that requires either all vehicles to In the there are to concept thatSchmidt requiresand either all (1983) vehicles to be be connected connected the existing existing literature, there are different different approaches to limitations. Schmidt and Posch (1983) proposed a central In In the existing literature, there are different approaches to concept that requires either of alla vehicles to be connected to each other or the activity centralised coordinating model the longitudinal dynamics of vehicles. Some authors to each other or the activity of a centralised coordinating model the longitudinal dynamics of vehicles. Some authors concept that requires either all vehicles to be connected In the existing literature, there are different approaches to to each other or the activity of a centralised coordinating model the longitudinal dynamics of vehicles. Some authors unit. Some recent concepts utilise non-linear approaches, use integral n-th order lag systems, e. g. Ploeg et al. (2011) unit. Some recent concepts non-linearcoordinating approaches, use integral n-th order lag systems, g. PloegSome et al.authors (2011) to each other or the activityutilise of a centralised model the longitudinal dynamics of e. vehicles. unit. Some recent concepts utilise approaches, integral n-th integrators. order lag systems, e. g.contribution Ploeg et al. (2011) e. potential fields or conor double The of e. g. g. gradient-based gradient-based potential fieldsnon-linear or sliding-mode sliding-mode con- use or simply simply double integrators. The main main contribution of this this unit. Some recent concepts utilise non-linear approaches, use integral n-th order lag systems, e. g. Ploeg et al. of this e. g. gradient-based potential fields or sliding-mode con- or simply double integrators. The main contribution (2011) e.2405-8963 g. gradient-based potential fields or sliding-mode conor simply double integrators. The main contribution of this Copyright © 2019. The Authors. Published by Elsevier Ltd. All rights reserved.
Copyright © 2019 19 Copyright © under 2019 IFAC IFAC 19 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 19 Copyright © 2019 IFAC 19 10.1016/j.ifacol.2019.12.135 Copyright © 2019 IFAC 19
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4
2
5
3
1
Fig. 2. Five vehicles heading towards a lane reduction (t = 0). The dashed arrows indicate the communication links. x˙ i (t) Ai 0 xi (t) b = + i ui (t) ˙ T −c d 0 (t) 0 (t) d i i i 0 + vi−1 (t) 1 Pi : (5) T xi (t) (t) = v c 0 i i d (t) i xi (0) xi0 x i (t) di (t) = 0 1 , = . di0 di (t) di (0)
paper is the extension of the results of Schwab and Lunze (2019) to networked vehicles with general linear dynamics. Structure. This paper is organised as follows: Section 2 describes the control problem. In Section 3, the cooperative tracking controller is introduced and two possible communication strategies are discussed in Section 4. The proposed tracking controller will be applied to the vehicle merging problem in Section 5 and the solution of the merging problem is evaluated by a simulation study, which is given in Section 6.
The leader model P1 is equal to the velocity dynamics (3), i. e. P1 = Σ1 since it has no predecessor.
2. PROBLEM STATEMENT
2.2 Control aim
This paper is concerned with the merging process of a set of N vehicles driving on two lanes towards a lane reduction. Each vehicle is equipped with radar sensors to measure the inter-vehicle distances and a communication system to transmit and receive information wirelessly.
Merging refers to the process of combining multiple vehicle streams on different lanes to a single one to pass a lane reduction as illustrated in Fig. 2. For this purpose, each vehicle with a predecessor (i = 2, 3, . . . , N ) has to extend the inter-vehicle distance to its preceding vehicle to a prescribed spacing before it changes the lane. Thus, the merging process is structured in the following phases:
2.1 Vehicle models The vehicles are enumerated in ascending order beginning with i = 1 for the leader. The velocity of each vehicle is denoted by vi (t) and the position by si (t). The intervehicle distance of two consecutive vehicles is given by di (t) = si−1 (t) − si (t) − li (2) with li denoting the minimum distance including the individual vehicle length and a least permitted separation as shown in Fig. 3. si
vi
i
si−1
li
di
1. Planning phase: Before the controllers take action (t < 0), the trajectories are planned and communicated among each other. This step is assumed to be finished at t = 0. 2. Transition phase: In the time interval T = [0, te ], the local tracking controllers steer the vehicles into a desired formation. Since the leader i = 1 has no predecessor, it should follow a velocity trajectory (6) v1 (t) = v1 (t), t ∈ T and the subsequent vehicles adjust their individual inter-vehicle distance (7) di (t) = d i (t), t ∈ T , i = 2, 3, . . . , N with appropriately chosen trajectories di (t). 3. Merging phase: After the transition phase (t > te ), it is safe for all vehicles to change to a common lane.
vi−1
i−1
s
Fig. 3. Two consecutive vehicles
Since the distance (2) directly depends on the movement of the predecessor, control aim (7) is achieved cooperatively by networked controllers of the neighbouring vehicles. Problem 1. (Control aim) Given a set of trajectories d i (t), (i = 2, 3, . . . , N ), all vehicles should achieve their local aim (6) or (7) for any arbitrary leader trajectory v1 (t) in the transition interval T .
The longitudinal position of a vehicle is determined by t si (t) = vi (τ ) dτ + si0 0
with the initial position si0 . The dynamical behaviour of each vehicle (i = 1, 2, . . . , N ) is described by x˙ i (t) = Ai xi (t) + bi ui (t) Σi : (3) xi (0) = xi0 vi (t) = cT i xi (t),
In order to render Problem 1 solvable, it is assumed that the velocity dynamics (3) is completely controllable. Hence, the extended model (5) is also completely controllable by the control input ui (t) since it represents a series connection of Σi with an integrator. Furthermore, it is assumed that the references v1 (t) and d i (t), (i = 2, 3, . . . , N ) are sufficiently often continuously differentiable.
and the inter-vehicle distance of subsequent vehicles (i = 2, 3, . . . , N ) is governed by the velocity difference d˙i (t) = vi−1 (t) − vi (t). (4) A combination of (3) and (4) yields the extended model for the follower vehicles (i = 2, 3, . . . , N ) 20
2019 IFAC NecSys Chicago, IL, USA, September 16-17, 2019 Alexander Schwab et al. / IFAC PapersOnLine 52-20 (2019) 19–24
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(if) Set d˜i (t) = d˜ i (t) in (12) and replace d˜ i (t) by eqn. (8) shows the identity di (t) = d i (t).
To focus the contribution on the key issues, the following simplifications are made: All vehicles execute the transition phase simultaneously, i. e. the interval T is equal for all vehicles. However, the proposed method can be extended to individual transition intervals Ti if long vehicle streams are considered (Schwab and Lunze, 2019). Furthermore, disturbances and model uncertainties are neglected. In applications in which both are present, an additional feedback part has to be added to each local controller Ci that stabilises the tracking error di (t)−d i (t).
(only if) Set di (t) = d i (t) in (12) and replace d i (t) by eqn. (8) shows the identity d˜i (t) = d˜ i (t). Theorem 2 separates the original Problem 1 into the following N − 1 local tracking problems that can be solved by each vehicle (i = 2, 3, . . . , N ) individually: Problem 3. (Local tracking) Given a reference trajectory d˜ i (t) for t ∈ T , find a control signal ui (t) that ensures perfect tracking for the isolated vehicle P˜i in eqn. (9): d˜i (t) = d˜ (t), t ∈ T .
In the next section, a solution of Problem 1 is presented, which will be used in Section 5 to solve the merging problem by choosing appropriate distance trajectories d i (t).
i
3. COOPERATIVE TRACKING CONTROL
3.2 Feed-forward control 3.1 Separation of the tracking problem Each vehicle i must be able to solve Problem 3 in order to solve the overall merging problem. This local tracking problem is solved by the flatness-based controller that is described by Schenk and Lunze (2018) and reviewed here.
From the local perspective of a vehicle, it is difficult to ensure the desired inter-vehicle distance because the predecessor affects the distance as well, as it can be seen in eqn. (4). To overcome this problem, a new local reference t ˜ di (t) = di (t) − vi−1 (τ ) dτ, t ∈ T (8)
vi−1
0
is introduced for each isolated vehicle, where it is assumed for the moment that vi−1 (t) is locally known to vehicle i. The signal d˜ i (t) describes that part of the desired distance d i (t) that must be achieved by a vehicle i individually. Here, individual means that the predecessor vehicle is virtually standing still. In that case, the vehicle i is called an isolated vehicle and the behaviour is the same as in (5) but with the velocity vi−1 (t) = 0: ˜˙ i (t) x ˜ i (t) x Ai 0 bi = ˜˙ ˜i (t) + 0 ui (t) −cT 0 d d (t) i i P˜i : (9) ˜ ˜ x x (t) (0) xi0 i i , = . d˜i (t) = (0 1) ˜ di0 di (t) d˜i (0)
d i −
Ci
xi0 d˜ i
Σi1
zi Fi
Σi2
ui
Fig. 4. Structure of the feed-forward controller Figure 4 shows the structure of the resulting feed-forward controller and, furthermore, illustrates that the control signal ui (t) is determined in two steps. In the first step, the reference d˜ i (t) is used to determine the new reference signal zi (t) that is the solution of the initial value problem (q ) d˜ i (t) = bi0 zi (t) + bi1 z˙i (t) + · · · + biq zi i (t) Σi1 : (13) (j) (j) zi0 = zi (0), j = 0, 1, . . . , qi − 1
Theorem 2. (Trajectory tracking) The control signals ui (t), i = 2, 3, . . . , N , solve Problem 1 if and only if they achieve tracking d˜i (t) = d˜ i (t) for the isolated vehicles (9).
with d˜ i (t) as the known input. In the second step, the derivatives of the reference zi (t) up to the ni -th order allow to determine the feed-forward signal (n )
Σi2 : u i (t) = ai0 zi (t) + ai1 z˙i (t) + · · · + ani zi i (t). (14) (See Schenk and Lunze (2018) for details on how to determine the parameters ai0 , . . . , ani , bi0 , . . . , biq and the (j) initial values zi0 , j = 0, 1, . . . , qi − 1.)
Proof. Integration of the relative velocity (4) yields t t T di (t) = di0 − ci xi (τ ) dτ + vi−1 (τ ) dτ (10) 0
0
for the distance between two vehicles. The output d˜i (t) of the isolated vehicle (9) can be explicitly expressed as t ˜ i (τ ) dτ. d˜i (t) = di0 − cT (11) x i
Theorem 4. (Schenk and Lunze (2018)) Let ri denote the relative degree of the isolated vehicle model (9). The feed-forward signal u i (t) given in (14) ensures perfect tracking d˜i (t) = d˜ i (t) if and only if the following conditions are satisfied: (j) (j) d˜ (0) = d˜ (0), j = 0, 1, . . . , ri − 1.
0
˜ i (t) in eqn. (9) are The states xi (t) in eqn. (3) and x influenced by the control signal ui (t) in the same way and since both have the same initial condition xi0 , they remain ˜ i (t). Next, replace d˜i (t) in (11) by d˜ i (t) and equal xi (t) = x ˜ i (t) by xi (t). Then, applying the result to (10) yields x t di (t) = d˜i (t) + vi−1 (τ ) dτ. (12)
i
i
Theorem 2 and Theorem 4 are used to design the networked feed-forward controllers Ci , (i = 1, 2, . . . , N ), shown in Fig. 1, which solve Problem 1. From the local perspective of a vehicle i, the procedure is as follows:
0
Expression (12) for the distance allows to finish the proof. 21
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com vi−2
d i−1
In general, the velocity vi (t) is not known a-priori since the local goal refers to the inter-vehicle distance except for the leader vehicle i = 1. However, the velocity trajectory can be calculated recursively with the information from the preceding vehicle according to vi (t) = vi−1 (t) − d˙ i (t), (18) which follows from the distance dynamics (4).
vi−2
Ci−1
ui−1
Pi−1
discrete vi−1
vi−1
vi−1
com vi−1
continuous
d i
−
The communication phase is executed as follows: The leader i = 1 does not need any additional information because the local trajectory v1 (t) is known. The subsequent vehicle i = 2 knows its local aim d 2 (t) but not the associated velocity v2 (t). Thus, the leader transmits v1 (t) to vehicle i = 2, which in turn uses the recursion formula (18) to calculate v2 (t). This procedure is executed step by step along the string as illustrated in Fig. 6 until every vehicle obtained all required information.
Ci Pi Fi
ui
Σi
vi −
di
Fig. 5. Overall control structure 1. Vehicle i uses the velocity vi−1 (t) of its predecessor to determine the reference d˜ i (t) as described by (8). 2. Vehicle i solves the initial value problem (13) to get zi (t) that allows to determine the control signal u i (t), described by eqn. (14), which ensures that the output of the isolated vehicle (9) steers along d˜ i (t).
v2 (t) ...
3
v1 (t)
2nd step
2
1st step
1
Fig. 6. Communication of trajectories at t = 0
The next section discusses two different communication strategies which enable each vehicle to obtain the velocity of its predecessor vehicle as this information is required for the presented approach to work.
Theorem 5. (Cooperative trajectory tracking) Assuming that the controller C1 achieves its local control aim (6), all subsequent vehicles (i = 2, 3, . . . , N ) also achieve their individual control aim (7) if the vehi cles receive their predecessors velocity trajectory vi−1 (t) according to (17) and all vehicles solve Problem 3.
4. COMMUNICATION STRATEGY 4.1 Implementation of the tracking controller
Proof by induction. The base case v1 (t) = v1 (t), t ∈ T is met by assumption. For the inductive step, it has to be shown that trajectory tracking of a subsequent vehicle is implied by trajectory tracking of its predecessor, i. e. vi (t) = vi (t) =⇒ vi+1 (t) = vi+1 (t). The tracking error (16) with the transmitted signal (17) is t vi (τ ) − vi (τ ) dτ = 0 (19) di+1 (t) − di+1 (t) =
In order to implement the proposed control approach with the modified reference (8), the velocity vi−1 (t) of the predecessor vehicle has to be known in the interval T . The communication network is used to receive the com signal vi−1 (t) from the predecessor, which describes the information of its velocity that reaches the subsequent vehicle i and, hence, the approach (8) reads as t com ˜ vi−1 (τ ) dτ. (15) di (t) = di (t) −
0
0
A combination of (15) and (12) yields the tracking error t com ˜ ˜ vi−1 (τ ) − vi−1 (τ ) dτ, di (t)−di (t) = di (t)−di (t)+
because of the induction hypothesis and the fact that the vehicles solve their isolated tracking problem and, hence, equal to zero. Rearranging the distance dynamics (4) yields vi+1 (t) = vi (t) − d˙i+1 (t)
4.2 Discrete communication
Note that communication is only necessary with the individual predecessor (cf. Fig. 6) and due to the recursive construction of the trajectories with (18), no additional communication links are required.
0
(16) which will be used to evaluate the two communication strategies shown in Fig. 5: Either the trajectory vi−1 (t), t ∈ T is communicated discretely at t = 0 or there is a communication channel that transmits the measured velocity vi−1 (t) continuously during the transition interval T .
The local controllers Ci are designed to achieve the local goals (6) and (7). Consequently, the behaviour of the vehicles is predetermined at time t = 0 and, hence, it is sufficient to transmit the local trajectory once to the subsequent vehicle: com vi−1 (t) = vi−1 (t), t ∈ T . (17)
(t) = vi (t) − d˙ i+1 (t) = vi+1 using (19) and the trajectory recursion (18), which completes the inductive step. The theorem is proved by (19), which shows that control aim (7) is achieved for all subsequent vehicles.
A discrete communication strategy benefits from the low data traffic because a trajectory in, e. g. polynomial form Ki di (t) = p k tk (20) k=0
22
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is represented by Ki + 1 coefficients, which have to be transmitted solely. Furthermore, once the trajectories are exchanged at t = 0, all vehicles are able to achieve their individual control aims (6) or (7) locally without any further communication. However, if one of the vehicles is not able to track its trajectory, because of a disturbance or similar, the subsequent vehicle will also not be able to track its trajectory since it is using wrong information.
distance cannot jump and that all vehicles move with the same velocity (21) initially (cf. Theorem 4). Requirement (24) states the final distance δei , which is required to exceed the vehicle length δei > li and (25) makes sure that all velocities are equal again after the transition phase at t = te . These requirements can be satisfied with a polynomial (20) of order Ki = 2ri + 1.
4.3 Continuous communication
The trajectory v1 (t) of the leader is designed analogously to (22)–(25). The initial velocity is chosen in consistency with (21), i. e. v1 (0) = v¯. The derivatives at t = 0 and t = te are set to zero to ensure stationarity and the final value v1 (te ) can be chosen arbitrarily. Remark 6. To guarantee collision avoidance, the distance of two consecutive vehicles on the same lane has to exceed a lower bound sj (t) − si (t) ≥ li , t ∈ T , which has to be ensured with an appropriate trajectory planning algorithm. This problem is discussed by Schwab and Lunze (2019) and, thus, will not be addressed here.
The second possibility is to send the measured velocity of the predecessor continuously in the transition interval T : com vi−1 (t) = vi−1 (t), t ∈ T . In this case, vehicle i can achieve its local aim (7) for any arbitrary predecessor velocity vi−1 (t), even if the predecessor i − 1 does not track its own trajectory. However, this approach suffers from high requirements on the communication link because the predecessor velocity vi−1 (t) has to be known throughout the whole transition interval T . Delays and packet loss in the communication link may cause deviation from the trajectory.
6. SIMULATION STUDY
Furthermore, the local controllers Ci have to compute the first ni derivatives (cf. (14)) of the trajectories in order to generate appropriate control signals. With discrete communication, the derivatives can be calculated accurately given the trajectory and coefficients. In the continuous com case, vi−1 (t) has to be differentiated numerically, which is inaccurate with noisy communication channels or sensors.
To evaluate the effectiveness of the proposed method, the set of N = 5 vehicles heading towards a lane reduction depicted in Fig. 2 is considered in this simulation study. The model parameters of the vehicles (3) are given by 0 1 0 A1 = , b1 = −1 −2 1
cT 1 = (1 0) , 0 1 0 0 A2 = A5 = 0 0 1 , b2 = b5 = 0 −1 −2 −3 1 T T c2 = c5 = (1 2 0) , 0 1 0 0 0 0 0 1 0 0 A3 = A4 = , b3 = b4 = 0 0 0 1 0 −1 −2 −3 −4 1 T T 2 0 0) c3 = c4 = (1 . Furthermore, the initial travel velocity and the minimum distance are chosen to be v¯ = 20 m/s and li = 20 m, (i = 2, ..., 5), respectively.
5. MERGING OF NETWORKED VEHICLES
5.1 The merging problem The proposed cooperative control scheme is applied to the vehicle merging problem (Fig. 2). The vehicles are heading towards an upcoming lane reduction and, hence, have to rearrange themselves so that they can change to a common lane safely. For this purpose, the proposed feed-forward controllers presented in Section 3 are used to adjust the inter-vehicle distances before the actual merging takes place. The vehicles should use the zipper method, i. e. they pass the lane reduction alternately from both lanes. 5.2 Trajectory planning phase
In the following, the proposed networked controller is applied to the merging problem. In addition to the nominal case (Fig. 7, left column), a braking manoeuvre of the leading vehicle is investigated in the second scenario (Fig. 7, right column). In order to compare the discrete and the continuous communication strategies described in Section 4, the vehicles communicate in different ways. Vehicle i = 1 sends its locally measured velocity continuously to vehicle i = 2, while the subsequent vehicles i = 3, 4, 5 receive the trajectory vi−1 (t), t ∈ T of the individual predecessors velocity completely at t = 0.
The trajectories d i (t) are planned locally and have to be known before the transition phase begins. Consequently, it is assumed that the trajectory planning phase is finished at t = 0 and to simplify the trajectory planning, it is assumed that all vehicles move initially with same constant velocity vi (t) = v¯, t ≤ 0, i = 1, 2, . . . , N. (21) The following requirements are imposed on the trajectory of the vehicles i = 2, 3, . . . , N : d i (0) = si−1 (0) − si (0) − li (22) (j)
di (0) = 0, j = 1, 2, . . . , ri d i (te ) = δei
Figure 7 shows the velocities vi (t), inter-vehicle distances di (t) with minimum distance li and the positions si (t) of the individual vehicles during the transition phase T = [0, 40 s]. The coloured solid lines and the dotted ones represent the vehicle behaviour and the planned trajectories, respectively.
(23) (24)
(j)
di (te ) = 0, j = 1, 2, . . . , ri . (25) The first two requirements (22) and (23) ensure state consistency and follow from the fact that the inter-vehicle 23
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vi (t) in m/s
20 15 10 5 0
consequently, the subsequent vehicles i = 4 and i = 5 are again able to achieve their local control aims.
P1 P2 P3 P4 P5 di (t) + li in m
80 60 40 20 0 -20 -40 800 600 400 200 0
vi (t) in m/s
10
20 30 t in s
This paper has presented a solution to the vehicle merging problem with cooperative tracking controllers, which are able to communicate via a given communication network to compensate for the physical interaction between the vehicles. It is shown that the proposed control method achieves trajectory tracking for different communication strategies.
di (t) + li in m
si (t) in m
0
7. CONCLUSION
In a simulation study, the performance of the two communication strategies is evaluated in nominal and hazardous situations caused by emergency braking, for example. It is shown that the continuous communication of the velocity enables the subsequent vehicles to track their individual trajectory even in the hazardous case. However, this strategy suffers from high requirements on the communication link. Future work may consider a combination of both strategies, where the continuous communication is triggered by a deviation from the planned trajectory in a selforganised manner combining all advantages.
si (t) in m
40
0
10
20 30 t in s
40
Fig. 7. Transition in nominal and hazardous scenarios
REFERENCES
The initial condition coincides qualitatively with the depiction in Fig. 2. During the transition interval T , the leading vehicle i = 1 reduces its velocity by half to enter the lane reduction. At the same time, the following vehicles i = 2, . . . , 5 adapt their individual inter-vehicle distance to pass the lane reduction with the zipper method. For that purpose, the local tracking controllers increase or reduce the inter-vehicle distance to a common value δei +li = 60 m at te = 40 s for all vehicles. Even though the vehicles influence each other by physical couplings, the cooperative networked controllers clearly achieve trajectory tracking in the specified time interval (cf. Fig. 7, left column). Afterwards, it is safe for all vehicles to change to a common lane in order to pass the lane reduction. Since the velocities are equal to the trajectories vi (t) = vi (t), i = 1, 2, . . . , N, all vehicles achieve the local control aim (6) or (7) independently of the applied communication strategy.
Aschemann, H. and Sun, H. (2013). Decentralised flatnessbased control of a hydrostatic drive train subject to actuator uncertainty and disturbances. IEEE Conf. on Methods and Models in Autom. and Robotics, 759–764. Hamiche, K., Abouaissa, H., Goncalves, G., and Hsu, T. (2016). Real-time decentralized flatness-based control of dynamic supply chain systems. IEEE Conf. on Contr., Decision and Information Technologies, 607–612. Ploeg, J., Scheepers, B.T., Van Nunen, E., Van de Wouw, N., and Nijmeijer, H. (2011). Design and experimental evaluation of cooperative adaptive cruise control. IEEE Conf. on Int. Transp. Systems, 260–265. Rios-Torres, J. and Malikopoulos, A.A. (2017). A survey on the coordination of connected and automated vehicles at intersections and merging at highway on-ramps. IEEE Trans. on Int. Transp. Systems, 18(5), 1066–1077. Rupp, A., Stolz, M., and Horn, M. (2018). Decentralized cooperative merging using sliding mode control. IFAC Symp. on Contr. in Transp. Systems, 51(9), 349–354. Schenk, K., G¨ ulbitti, B., and Lunze, J. (2018). Cooperative fault-tolerant control of networked control systems. IFAC Symp. on Fault Detection, Supervision and Safety for Technical Processes, 570–577. Schenk, K. and Lunze, J. (2018). Tracking control of networked and interconnected systems. IFAC Workshop on Distributed Estimation and Contr. in Networked Systems, 51(23), 40–45. Schmidt, G.K. and Posch, B. (1983). A two-layer control scheme for merging of automated vehicles. In IEEE Conf. on Decision and Contr., 495–500. Schwab, A. and Lunze, J. (2019). Cooperative vehicle merging with guaranteed collision avoidance. IFAC Workshop on Contr. of Transp. Systems. Semsar-Kazerooni, E., Elferink, K., Ploeg, J., and Nijmeijer, H. (2017). Multi-objective platoon maneuvering using artificial potential fields. IFAC World Congress, 50(1), 15006–15011.
The second scenario (Fig. 7, right column) considers the case that vehicle i = 1 has to perform an unplanned braking manoeuvre, because of an emerging obstacle, in the time interval [10 s, 15 s]. As a consequence, the velocity of the leader i = 1 (blue curve in Fig. 7, top right) is lower than the planned trajectory in some time interval. Since vehicle i = 2 receives the velocity v1 (t) continuously in the transition interval, it also reduces its velocity as a response to its braking predecessor, so that the local control aim d2 (t) = d 2 (t) is achieved nevertheless (red curve in Fig. 7, middle right). Thus, the continuous communication introduces a robustness towards disturbances of the predecessor vehicle. The next vehicle i = 3 is not able to follow the planned trajectory (yellow curve in Fig. 7, middle right), because it received the whole trajectory v2 (t) at t = 0. Thus, the local controller C3 is working with wrong information and behaves as if nothing happened downstream and, 24
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Networked Vehicle Merging by Networked Vehicle Merging by Networked Vehicle Merging by Networked Vehicle Merging by Cooperative Tracking Control Cooperative Tracking Control Cooperative Tracking Control Cooperative ∗∗Tracking ∗∗Control ∗∗
Alexander Alexander Schwab Schwab ∗∗ Kai Kai Schenk Schenk ∗∗ Jan Jan Lunze Lunze ∗∗ Alexander Schwab ∗∗ Kai Schenk ∗∗ Jan Lunze ∗∗ Alexander Schwab Kai Schenk Jan Lunze ∗ ∗ a ∗ Institute of Automation and Computer Control, Ruhr-Universit¨ att ∗ Institute of Automation and Computer Control, Ruhr-Universit¨ Institute of Automation and Computer Control, Ruhr-Universit¨ at Bochum, (e-mail: {schwab, schenk, lunze}@atp.rub.de) ∗ schenk, Control, lunze}@atp.rub.de) Bochum, (e-mail: {schwab, Institute of Automation and Computer Ruhr-Universit¨ at Bochum, (e-mail: {schwab, schenk, lunze}@atp.rub.de) Bochum, (e-mail: {schwab, schenk, lunze}@atp.rub.de) Abstract: Abstract: This This paper paper solves solves the the vehicle vehicle merging merging problem problem by by trajectory trajectory tracking tracking of of networked networked Abstract: This paper solves vehicle the vehicle merging problemtracking by trajectory tracking ofdistributed networked vehicles to combine multiple streams. Trajectory is achieved with vehicles to combine multiple vehicle streams. Trajectory tracking is achieved with distributed Abstract: This paper solves the vehicle merging problem by trajectory tracking of networked vehicles to combine multiple vehicle streams. Trajectory tracking is achieved with distributed feed-forward controllers that the between the by the feed-forward controllers that consider consider the couplings couplings between the vehicles vehicles by modifying modifying the local local vehicles to combine multiple vehicle streams. Trajectory tracking is achieved with distributed feed-forward controllersFor thatthis consider the couplings between vehicles by the local reference trajectories. purpose, the controllers arethe connected viamodifying a communication communication are connected via a reference trajectories. For this purpose, the controllers feed-forward controllers that consider the couplings between the vehicles by modifying the local reference Different trajectories. For this purpose, the are controllers connected via that a communication network. communication strategies discussed and it the network. Different communication strategies discussedare it is is shown shown the proposed proposed reference trajectories. For this purpose, the are controllers areand connected via that a communication network. Different communication strategies are discussed and it is vehicle shown that the proposed tracking controllers solve the merging problem with arbitrary linear models. tracking controllers solve the merging problem with arbitrary linear vehicle models. network. Different communication strategies are discussed and it is shown that the proposed tracking controllers solve the merging problem with arbitrary linear vehicle models. tracking ©controllers solve the merging problem with arbitrary linear vehicle models. Copyright 2019. The Authors. Published by Elsevier Ltd. All rights reserved. 1. 1. INTRODUCTION INTRODUCTION Physical 1. INTRODUCTION Physical coupling coupling 1. INTRODUCTION Physical coupling Physical coupling Merging manoeuvres manoeuvres have have to to be be performed performed in in lane lane reducreducMerging Merging manoeuvres to be performed in lane reduction scenarios (Fig. 2) 2) have to combine combine multiple vehicle streams. tion scenarios (Fig. to multiple yyN yy11 vehicle streams. yy11 yyN u u Merging manoeuvres have to be performed in lane reducu111 uN N N N 1 tion scenarios (Fig. 2) to combine multiple vehicle streams. . . . N P 1 N C Some commercial driver assistance systems are able to Some commercial driver assistance systems are streams. able to yN y1 C ... C111 u1 P P111 y1 yN N u N PN N N tion scenarios (Fig. 2) to combine multiple vehicle CN N N y . . . Some commercial driver assistance systems are able to y 1 C 1 u 1 P1 y 1 perform autonomous autonomous lane lane changes, changes, but but they they have have to to wait wait N C N u N PN y N perform . . . Some commercial driver assistance systems are able to C1 P1 CN PN perform autonomous but they for sufficiently largelane gap changes, on the the target target lane.have Thisto is wait why for aa sufficiently large gap on lane. This is why perform autonomous lane changes, but they have to wait for a sufficiently large gap on the target lane. This is why the present paper proposes an alternative way that solves the paperlarge proposes anthe alternative wayThis that issolves for apresent sufficiently gap on target lane. why proposes an alternative way solves Communication the present mergingpaper problem cooperatively. A set set of of N that vehicles is the merging problem cooperatively. A N vehicles is Communication network network the present paper proposes an alternative way that solves Communication network the merging problem cooperatively. A set of N vehicles is modelled as as aa networked networked system system as as illustrated illustrated in in Fig. Fig. 1. 1. modelled Communication network the merging problem cooperatively. A set of N vehicles is modelled as a corresponds networked system as illustrated Fig. 1. Each vehicle corresponds to aa subsystem subsystem Pii ,, inwhich is Each vehicle to P which is Fig. modelled as a networked system as illustrated Fig. 1. 1. i in 1. Control Control structure structure of of aa networked networked system system Each vehicle corresponds to a subsystem P is Fig. i , which controlled by a networked feed-forward controller C ii .. 1. Control structure of a networked system controlled by a networked feed-forward controller C Each vehicle corresponds to a subsystem Pi , which is Fig. i Fig. 1. Control structure of a networked system controlled by a networked feed-forward controller Ci . controlled a networked . This paper considers vehicle merging trajectory trollers and This paperby considers vehiclefeed-forward merging as as a acontroller trajectoryCitracktracktrollers as as published published by by Semsar-Kazerooni Semsar-Kazerooni et et al. al. (2017) (2017) and This paper considers vehicle merging as a trajectory tracking problem. Trajectory tracking describes the problem of trollers asal. published by Semsar-Kazerooni et al. (2017) and Rupp et (2018), respectively. These concepts exhibit ing problem. Trajectory tracking describes the problem of Rupp et al. (2018), respectively. These concepts exhibit This paper considers vehicle merging as a trajectory track- trollers as published by Semsar-Kazerooni et al. (2017) and ing problem. Trajectory tracking describes the problem of controlling dynamical systems, so that the output follows Rupp et al. (2018), respectively. These concepts exhibit unfavourable behaviour like unbounded and binary control controlling systems, that the the output follows behaviour like unbounded binary exhibit control ing problem.dynamical Trajectory trackingso problem of unfavourable Rupp et al. (2018), respectively. Theseand concepts controlling sodescribes that the output follows a designated trajectory. Considering networked systems as behaviour like unbounded and binary signals in to analysis which is a designateddynamical trajectory.systems, Considering networked systems as unfavourable signals in addition addition to aa complicated complicated analysis which control is why why controlling dynamical systems, so that the output follows unfavourable behaviour like unbounded and binary control in addition to a complicated analysis which is why a designated trajectory. Considering networked systems as signals illustrated in Fig. 1, the control aim reads as the present paper describes a solution that manages the illustrated in Fig. 1, the control aim reads as the present paper describes a solution that manages the a designated trajectory. Considering networked systems as signals in addition to a complicated analysis which is why illustratedyiin Fig. 1, the control aim reads as. , N the present paperwith describes a solution that manages the merging problem networked linear controllers. (t) = y (t), t ≥ 0, i = 1, 2, . . (1) merging problem with networked linear controllers. illustratedyiiin the control present paper describes a solution that manages the t ≥ 0, aim i =reads 1, 2, . .as. , N (1) the (t)Fig. = yi1,(t), merging problem with networked linear controllers. (t) = yiii (t), t ≥ 0, trajectory i = 1, 2, . .of .,N (1) yidenoting with the system merging problem networked linear controllers. Tracking control interconnected systems is ii .. (t) = yi (t), t ≥ 0, trajectory i = 1, 2, . .of .,N Tracking control of ofwith interconnected systems is aa challenging challenging with yyiii (t) (t)yidenoting the planned planned system P P(1) with yi (t) denoting the planned trajectory of system Pii . Tracking control of interconnected systems is a challenging task and only a few publications deal with that topic. task and only a few publications deal with that topic. For For with y (t) denoting the planned trajectory of system P . Achieving control Tracking control of interconnected systems is a challenging i i Achieving control aim aim (1) (1) is is a a difficult difficult task task because because the the task and only a few publications deal with that topic. For the process of merging, it is particularly relevant that each the process of merging, it is particularly relevant that each Achieving control aim (1) is a difficult task because the (t) is arbitrary and the subsystems indesired trajectory y task and only a few publications deal with that topic. For ii (t) (1) is arbitrary and the subsystems in- the desired trajectory process of to merging, itgiven is particularly relevant that each Achieving control yaim is a difficult task because the vehicle is able follow a trajectory perfectly. Schenk i vehicle is able to follow a given trajectory perfectly. Schenk (t) is arbitrary and the subsystems indesired trajectory y fluence each other due to physical couplings. To solve this the process of merging, it is particularly relevant that each i fluence each other ydue to physical couplings. To solve this is arbitrary and the subsystems in- vehicle desired trajectory is able to follow a given trajectory perfectly. Schenk et and Schenk and Lunze presented i (t)to fluence each physical couplings. To solve this et al. al. (2018) (2018) and Schenk and Lunze (2018) (2018) presented problem, the local controllers are to is able to follow a given trajectory perfectly. Schenk problem, theother local due controllers are enabled enabled to communicate communicate fluence each other due to physical couplings. To solve this vehicle et al. (2018) and Schenk and Lunze (2018) presented two methods for the design of networked feed-forward two methods for the design of Lunze networked feed-forward problem, the localto controllers are influence enabled toof communicate with each other consider the the physical et al. (2018) and Schenk and (2018) presented with each other to consider the influence of the physical problem, the local controllers are enabled to communicate two methods the for design of tracking. networkedHowever, feed-forward controllers that allow perfect both controllers thatfor perfect both with eachand other to consider the influence of the physical couplings generate appropriate control u methods forallow the for design of tracking. networkedHowever, feed-forward couplings generate appropriate control signals signals uiii (t), (t), so so two controllers that allow for perfect tracking. However, both with eachand other to consider the influence of the physical publications require conditions on the coupling relation publications require conditions on the coupling relation couplings and generate appropriate control signals u (t), so controllers that allow for perfect tracking. However, both i that the control aim (1) is accomplished by all subsystems. that the control aim (1)appropriate is accomplished by signals all subsystems. couplings and generate control u (t), so publications require conditions on the coupling relation that are by As an contrii that are not not satisfied satisfied by vehicles. vehicles.on Asthe an important important contrithat the control aim (1) is accomplished by all subsystems. publications require conditions coupling relation that control aim is accomplished by all subsystems. are the not present satisfiedpaper by vehicles. important contriThis paper solves the tracking problem networked sysbution, is to relax the This the paper solves the(1) tracking problem for for networked sys- that bution, is able able As to an relax the conditions conditions that are the not present satisfiedpaper by vehicles. As an important contriThis paper solves the tracking problem for networked sysbution, the present paper is able to relax the conditions tems and uses the proposed tracking controller to solve the in order to make the method applicable for the merging tems paper and uses the the proposed tracking controller to solvesysthe bution, in orderthe to make thepaper method applicable merging This solves tracking problem for networked present is able to relaxfor thethe conditions in order to make the method applicable for the merging tems and uses the proposed tracking controller to solve the merging problem cooperatively in distributed manner. problem. Two other notable contributions are given by merging cooperatively in a acontroller distributed manner. other by tems andproblem uses the proposed tracking to solve the problem. in order toTwo make the notable method contributions applicable for are the given merging merging problem cooperatively in avehicles distributed manner. problem. Two other notable contributions areal. given by The solution is valid for arbitrary and does not Aschemann and Sun (2013) and Hamiche et (2016), The solution is valid for arbitrary vehicles and does not Aschemann and Sun (2013) and Hamiche et al. (2016), merging problem cooperatively in a distributed manner. problem. Two other notable contributions are given by The solution is valid for arbitrary vehicles and does not Aschemann and Sun (2013) and Hamiche et al. (2016), require a coordinating unit. It is discussed what informawhich use flatness-based controllers for tracking control require a coordinating unit. It is discussed informause flatness-based controllers for tracking control The solution is valid for arbitrary vehicles what and does not which Aschemann and Sun (2013) and Hamiche et al. (2016), require coordinating unit. is controlled discussed what informause flatness-based controllers for tracking control tion to be by the subsystems to of drive and chain. both tion has hasa to be exchanged exchanged by It the controlled subsystems to which of aa hydro-static hydro-static drive train train and aa supply supply chain. In In both require a coordinating unit. It is discussed what informawhich use flatness-based controllers for tracking control a hydro-static drive train controllers and a supply chain. In both tion hasthe to be exchanged by the controlled subsystems to of achieve control aim. contributions, decentralised are designed that achieve the control aim. contributions, decentralised controllers are designed that tion has to be exchanged by the controlled subsystems to of a hydro-static drive train and a supply chain. In both achieve the control aim. contributions, decentralised controllers are designed that share information with each other, which is comparable share information with each other, which is comparable achieve the control decentralised controllers are designed that Literature. There are Literature. There aim. are several several publications publications on on merging merging contributions, share information with each other, which is comparable to the present paper. However, since they exploit theinformation present paper. since they exploit the the Literature. There in arethe several on and merging share with However, each other, which is comparable control summarised survey by Macontrol summarised surveypublications by Rios-Torres Rios-Torres Ma- to to the present paper. However, since they exploit the very specific structure of their applications, their method Literature. There in arethe several publications on and merging verythe specific structure ofHowever, their applications, their method to present paper. since they exploit the control summarised in the survey by Rios-Torres and Malikopoulos (2017), but most solutions come with stringent specific structure of their applications, likopoulos (2017), but solutions come with stringent cannot be used used to the the more more general problem their here. method control summarised in most the survey by Rios-Torres and Ma- very cannot be to general problem here. very specific structure of their applications, their method likopoulos (2017), but most solutions come with stringent limitations. Schmidt and Posch (1983) proposed a central limitations. Schmidt Posch (1983)come proposed a central cannot be used to the more general problem here. likopoulos (2017), butand most solutions with stringent be usedliterature, to the more general problemapproaches here. limitations. Posch proposed a central cannot concept that requires either all vehicles to In the there are to concept thatSchmidt requiresand either all (1983) vehicles to be be connected connected the existing existing literature, there are different different approaches to limitations. Schmidt and Posch (1983) proposed a central In In the existing literature, there are different approaches to concept that requires either of alla vehicles to be connected to each other or the activity centralised coordinating model the longitudinal dynamics of vehicles. Some authors to each other or the activity of a centralised coordinating model the longitudinal dynamics of vehicles. Some authors concept that requires either all vehicles to be connected In the existing literature, there are different approaches to to each other or the activity of a centralised coordinating model the longitudinal dynamics of vehicles. Some authors unit. Some recent concepts utilise non-linear approaches, use integral n-th order lag systems, e. g. Ploeg et al. (2011) unit. Some recent concepts non-linearcoordinating approaches, use integral n-th order lag systems, g. PloegSome et al.authors (2011) to each other or the activityutilise of a centralised model the longitudinal dynamics of e. vehicles. unit. Some recent concepts utilise approaches, integral n-th integrators. order lag systems, e. g.contribution Ploeg et al. (2011) e. potential fields or conor double The of e. g. g. gradient-based gradient-based potential fieldsnon-linear or sliding-mode sliding-mode con- use or simply simply double integrators. The main main contribution of this this unit. Some recent concepts utilise non-linear approaches, use integral n-th order lag systems, e. g. Ploeg et al. of this e. g. gradient-based potential fields or sliding-mode con- or simply double integrators. The main contribution (2011) e.2405-8963 g. gradient-based potential fields or sliding-mode conor simply double integrators. The main contribution of this Copyright © 2019. The Authors. Published by Elsevier Ltd. All rights reserved.
Copyright © 2019 19 Copyright © under 2019 IFAC IFAC 19 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 19 Copyright © 2019 IFAC 19 10.1016/j.ifacol.2019.12.135 Copyright © 2019 IFAC 19
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4
2
5
3
1
Fig. 2. Five vehicles heading towards a lane reduction (t = 0). The dashed arrows indicate the communication links. x˙ i (t) Ai 0 xi (t) b = + i ui (t) ˙ T −c d 0 (t) 0 (t) d i i i 0 + vi−1 (t) 1 Pi : (5) T xi (t) (t) = v c 0 i i d (t) i xi (0) xi0 x i (t) di (t) = 0 1 , = . di0 di (t) di (0)
paper is the extension of the results of Schwab and Lunze (2019) to networked vehicles with general linear dynamics. Structure. This paper is organised as follows: Section 2 describes the control problem. In Section 3, the cooperative tracking controller is introduced and two possible communication strategies are discussed in Section 4. The proposed tracking controller will be applied to the vehicle merging problem in Section 5 and the solution of the merging problem is evaluated by a simulation study, which is given in Section 6.
The leader model P1 is equal to the velocity dynamics (3), i. e. P1 = Σ1 since it has no predecessor.
2. PROBLEM STATEMENT
2.2 Control aim
This paper is concerned with the merging process of a set of N vehicles driving on two lanes towards a lane reduction. Each vehicle is equipped with radar sensors to measure the inter-vehicle distances and a communication system to transmit and receive information wirelessly.
Merging refers to the process of combining multiple vehicle streams on different lanes to a single one to pass a lane reduction as illustrated in Fig. 2. For this purpose, each vehicle with a predecessor (i = 2, 3, . . . , N ) has to extend the inter-vehicle distance to its preceding vehicle to a prescribed spacing before it changes the lane. Thus, the merging process is structured in the following phases:
2.1 Vehicle models The vehicles are enumerated in ascending order beginning with i = 1 for the leader. The velocity of each vehicle is denoted by vi (t) and the position by si (t). The intervehicle distance of two consecutive vehicles is given by di (t) = si−1 (t) − si (t) − li (2) with li denoting the minimum distance including the individual vehicle length and a least permitted separation as shown in Fig. 3. si
vi
i
si−1
li
di
1. Planning phase: Before the controllers take action (t < 0), the trajectories are planned and communicated among each other. This step is assumed to be finished at t = 0. 2. Transition phase: In the time interval T = [0, te ], the local tracking controllers steer the vehicles into a desired formation. Since the leader i = 1 has no predecessor, it should follow a velocity trajectory (6) v1 (t) = v1 (t), t ∈ T and the subsequent vehicles adjust their individual inter-vehicle distance (7) di (t) = d i (t), t ∈ T , i = 2, 3, . . . , N with appropriately chosen trajectories di (t). 3. Merging phase: After the transition phase (t > te ), it is safe for all vehicles to change to a common lane.
vi−1
i−1
s
Fig. 3. Two consecutive vehicles
Since the distance (2) directly depends on the movement of the predecessor, control aim (7) is achieved cooperatively by networked controllers of the neighbouring vehicles. Problem 1. (Control aim) Given a set of trajectories d i (t), (i = 2, 3, . . . , N ), all vehicles should achieve their local aim (6) or (7) for any arbitrary leader trajectory v1 (t) in the transition interval T .
The longitudinal position of a vehicle is determined by t si (t) = vi (τ ) dτ + si0 0
with the initial position si0 . The dynamical behaviour of each vehicle (i = 1, 2, . . . , N ) is described by x˙ i (t) = Ai xi (t) + bi ui (t) Σi : (3) xi (0) = xi0 vi (t) = cT i xi (t),
In order to render Problem 1 solvable, it is assumed that the velocity dynamics (3) is completely controllable. Hence, the extended model (5) is also completely controllable by the control input ui (t) since it represents a series connection of Σi with an integrator. Furthermore, it is assumed that the references v1 (t) and d i (t), (i = 2, 3, . . . , N ) are sufficiently often continuously differentiable.
and the inter-vehicle distance of subsequent vehicles (i = 2, 3, . . . , N ) is governed by the velocity difference d˙i (t) = vi−1 (t) − vi (t). (4) A combination of (3) and (4) yields the extended model for the follower vehicles (i = 2, 3, . . . , N ) 20
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(if) Set d˜i (t) = d˜ i (t) in (12) and replace d˜ i (t) by eqn. (8) shows the identity di (t) = d i (t).
To focus the contribution on the key issues, the following simplifications are made: All vehicles execute the transition phase simultaneously, i. e. the interval T is equal for all vehicles. However, the proposed method can be extended to individual transition intervals Ti if long vehicle streams are considered (Schwab and Lunze, 2019). Furthermore, disturbances and model uncertainties are neglected. In applications in which both are present, an additional feedback part has to be added to each local controller Ci that stabilises the tracking error di (t)−d i (t).
(only if) Set di (t) = d i (t) in (12) and replace d i (t) by eqn. (8) shows the identity d˜i (t) = d˜ i (t). Theorem 2 separates the original Problem 1 into the following N − 1 local tracking problems that can be solved by each vehicle (i = 2, 3, . . . , N ) individually: Problem 3. (Local tracking) Given a reference trajectory d˜ i (t) for t ∈ T , find a control signal ui (t) that ensures perfect tracking for the isolated vehicle P˜i in eqn. (9): d˜i (t) = d˜ (t), t ∈ T .
In the next section, a solution of Problem 1 is presented, which will be used in Section 5 to solve the merging problem by choosing appropriate distance trajectories d i (t).
i
3. COOPERATIVE TRACKING CONTROL
3.2 Feed-forward control 3.1 Separation of the tracking problem Each vehicle i must be able to solve Problem 3 in order to solve the overall merging problem. This local tracking problem is solved by the flatness-based controller that is described by Schenk and Lunze (2018) and reviewed here.
From the local perspective of a vehicle, it is difficult to ensure the desired inter-vehicle distance because the predecessor affects the distance as well, as it can be seen in eqn. (4). To overcome this problem, a new local reference t ˜ di (t) = di (t) − vi−1 (τ ) dτ, t ∈ T (8)
vi−1
0
is introduced for each isolated vehicle, where it is assumed for the moment that vi−1 (t) is locally known to vehicle i. The signal d˜ i (t) describes that part of the desired distance d i (t) that must be achieved by a vehicle i individually. Here, individual means that the predecessor vehicle is virtually standing still. In that case, the vehicle i is called an isolated vehicle and the behaviour is the same as in (5) but with the velocity vi−1 (t) = 0: ˜˙ i (t) x ˜ i (t) x Ai 0 bi = ˜˙ ˜i (t) + 0 ui (t) −cT 0 d d (t) i i P˜i : (9) ˜ ˜ x x (t) (0) xi0 i i , = . d˜i (t) = (0 1) ˜ di0 di (t) d˜i (0)
d i −
Ci
xi0 d˜ i
Σi1
zi Fi
Σi2
ui
Fig. 4. Structure of the feed-forward controller Figure 4 shows the structure of the resulting feed-forward controller and, furthermore, illustrates that the control signal ui (t) is determined in two steps. In the first step, the reference d˜ i (t) is used to determine the new reference signal zi (t) that is the solution of the initial value problem (q ) d˜ i (t) = bi0 zi (t) + bi1 z˙i (t) + · · · + biq zi i (t) Σi1 : (13) (j) (j) zi0 = zi (0), j = 0, 1, . . . , qi − 1
Theorem 2. (Trajectory tracking) The control signals ui (t), i = 2, 3, . . . , N , solve Problem 1 if and only if they achieve tracking d˜i (t) = d˜ i (t) for the isolated vehicles (9).
with d˜ i (t) as the known input. In the second step, the derivatives of the reference zi (t) up to the ni -th order allow to determine the feed-forward signal (n )
Σi2 : u i (t) = ai0 zi (t) + ai1 z˙i (t) + · · · + ani zi i (t). (14) (See Schenk and Lunze (2018) for details on how to determine the parameters ai0 , . . . , ani , bi0 , . . . , biq and the (j) initial values zi0 , j = 0, 1, . . . , qi − 1.)
Proof. Integration of the relative velocity (4) yields t t T di (t) = di0 − ci xi (τ ) dτ + vi−1 (τ ) dτ (10) 0
0
for the distance between two vehicles. The output d˜i (t) of the isolated vehicle (9) can be explicitly expressed as t ˜ i (τ ) dτ. d˜i (t) = di0 − cT (11) x i
Theorem 4. (Schenk and Lunze (2018)) Let ri denote the relative degree of the isolated vehicle model (9). The feed-forward signal u i (t) given in (14) ensures perfect tracking d˜i (t) = d˜ i (t) if and only if the following conditions are satisfied: (j) (j) d˜ (0) = d˜ (0), j = 0, 1, . . . , ri − 1.
0
˜ i (t) in eqn. (9) are The states xi (t) in eqn. (3) and x influenced by the control signal ui (t) in the same way and since both have the same initial condition xi0 , they remain ˜ i (t). Next, replace d˜i (t) in (11) by d˜ i (t) and equal xi (t) = x ˜ i (t) by xi (t). Then, applying the result to (10) yields x t di (t) = d˜i (t) + vi−1 (τ ) dτ. (12)
i
i
Theorem 2 and Theorem 4 are used to design the networked feed-forward controllers Ci , (i = 1, 2, . . . , N ), shown in Fig. 1, which solve Problem 1. From the local perspective of a vehicle i, the procedure is as follows:
0
Expression (12) for the distance allows to finish the proof. 21
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com vi−2
d i−1
In general, the velocity vi (t) is not known a-priori since the local goal refers to the inter-vehicle distance except for the leader vehicle i = 1. However, the velocity trajectory can be calculated recursively with the information from the preceding vehicle according to vi (t) = vi−1 (t) − d˙ i (t), (18) which follows from the distance dynamics (4).
vi−2
Ci−1
ui−1
Pi−1
discrete vi−1
vi−1
vi−1
com vi−1
continuous
d i
−
The communication phase is executed as follows: The leader i = 1 does not need any additional information because the local trajectory v1 (t) is known. The subsequent vehicle i = 2 knows its local aim d 2 (t) but not the associated velocity v2 (t). Thus, the leader transmits v1 (t) to vehicle i = 2, which in turn uses the recursion formula (18) to calculate v2 (t). This procedure is executed step by step along the string as illustrated in Fig. 6 until every vehicle obtained all required information.
Ci Pi Fi
ui
Σi
vi −
di
Fig. 5. Overall control structure 1. Vehicle i uses the velocity vi−1 (t) of its predecessor to determine the reference d˜ i (t) as described by (8). 2. Vehicle i solves the initial value problem (13) to get zi (t) that allows to determine the control signal u i (t), described by eqn. (14), which ensures that the output of the isolated vehicle (9) steers along d˜ i (t).
v2 (t) ...
3
v1 (t)
2nd step
2
1st step
1
Fig. 6. Communication of trajectories at t = 0
The next section discusses two different communication strategies which enable each vehicle to obtain the velocity of its predecessor vehicle as this information is required for the presented approach to work.
Theorem 5. (Cooperative trajectory tracking) Assuming that the controller C1 achieves its local control aim (6), all subsequent vehicles (i = 2, 3, . . . , N ) also achieve their individual control aim (7) if the vehi cles receive their predecessors velocity trajectory vi−1 (t) according to (17) and all vehicles solve Problem 3.
4. COMMUNICATION STRATEGY 4.1 Implementation of the tracking controller
Proof by induction. The base case v1 (t) = v1 (t), t ∈ T is met by assumption. For the inductive step, it has to be shown that trajectory tracking of a subsequent vehicle is implied by trajectory tracking of its predecessor, i. e. vi (t) = vi (t) =⇒ vi+1 (t) = vi+1 (t). The tracking error (16) with the transmitted signal (17) is t vi (τ ) − vi (τ ) dτ = 0 (19) di+1 (t) − di+1 (t) =
In order to implement the proposed control approach with the modified reference (8), the velocity vi−1 (t) of the predecessor vehicle has to be known in the interval T . The communication network is used to receive the com signal vi−1 (t) from the predecessor, which describes the information of its velocity that reaches the subsequent vehicle i and, hence, the approach (8) reads as t com ˜ vi−1 (τ ) dτ. (15) di (t) = di (t) −
0
0
A combination of (15) and (12) yields the tracking error t com ˜ ˜ vi−1 (τ ) − vi−1 (τ ) dτ, di (t)−di (t) = di (t)−di (t)+
because of the induction hypothesis and the fact that the vehicles solve their isolated tracking problem and, hence, equal to zero. Rearranging the distance dynamics (4) yields vi+1 (t) = vi (t) − d˙i+1 (t)
4.2 Discrete communication
Note that communication is only necessary with the individual predecessor (cf. Fig. 6) and due to the recursive construction of the trajectories with (18), no additional communication links are required.
0
(16) which will be used to evaluate the two communication strategies shown in Fig. 5: Either the trajectory vi−1 (t), t ∈ T is communicated discretely at t = 0 or there is a communication channel that transmits the measured velocity vi−1 (t) continuously during the transition interval T .
The local controllers Ci are designed to achieve the local goals (6) and (7). Consequently, the behaviour of the vehicles is predetermined at time t = 0 and, hence, it is sufficient to transmit the local trajectory once to the subsequent vehicle: com vi−1 (t) = vi−1 (t), t ∈ T . (17)
(t) = vi (t) − d˙ i+1 (t) = vi+1 using (19) and the trajectory recursion (18), which completes the inductive step. The theorem is proved by (19), which shows that control aim (7) is achieved for all subsequent vehicles.
A discrete communication strategy benefits from the low data traffic because a trajectory in, e. g. polynomial form Ki di (t) = p k tk (20) k=0
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is represented by Ki + 1 coefficients, which have to be transmitted solely. Furthermore, once the trajectories are exchanged at t = 0, all vehicles are able to achieve their individual control aims (6) or (7) locally without any further communication. However, if one of the vehicles is not able to track its trajectory, because of a disturbance or similar, the subsequent vehicle will also not be able to track its trajectory since it is using wrong information.
distance cannot jump and that all vehicles move with the same velocity (21) initially (cf. Theorem 4). Requirement (24) states the final distance δei , which is required to exceed the vehicle length δei > li and (25) makes sure that all velocities are equal again after the transition phase at t = te . These requirements can be satisfied with a polynomial (20) of order Ki = 2ri + 1.
4.3 Continuous communication
The trajectory v1 (t) of the leader is designed analogously to (22)–(25). The initial velocity is chosen in consistency with (21), i. e. v1 (0) = v¯. The derivatives at t = 0 and t = te are set to zero to ensure stationarity and the final value v1 (te ) can be chosen arbitrarily. Remark 6. To guarantee collision avoidance, the distance of two consecutive vehicles on the same lane has to exceed a lower bound sj (t) − si (t) ≥ li , t ∈ T , which has to be ensured with an appropriate trajectory planning algorithm. This problem is discussed by Schwab and Lunze (2019) and, thus, will not be addressed here.
The second possibility is to send the measured velocity of the predecessor continuously in the transition interval T : com vi−1 (t) = vi−1 (t), t ∈ T . In this case, vehicle i can achieve its local aim (7) for any arbitrary predecessor velocity vi−1 (t), even if the predecessor i − 1 does not track its own trajectory. However, this approach suffers from high requirements on the communication link because the predecessor velocity vi−1 (t) has to be known throughout the whole transition interval T . Delays and packet loss in the communication link may cause deviation from the trajectory.
6. SIMULATION STUDY
Furthermore, the local controllers Ci have to compute the first ni derivatives (cf. (14)) of the trajectories in order to generate appropriate control signals. With discrete communication, the derivatives can be calculated accurately given the trajectory and coefficients. In the continuous com case, vi−1 (t) has to be differentiated numerically, which is inaccurate with noisy communication channels or sensors.
To evaluate the effectiveness of the proposed method, the set of N = 5 vehicles heading towards a lane reduction depicted in Fig. 2 is considered in this simulation study. The model parameters of the vehicles (3) are given by 0 1 0 A1 = , b1 = −1 −2 1
cT 1 = (1 0) , 0 1 0 0 A2 = A5 = 0 0 1 , b2 = b5 = 0 −1 −2 −3 1 T T c2 = c5 = (1 2 0) , 0 1 0 0 0 0 0 1 0 0 A3 = A4 = , b3 = b4 = 0 0 0 1 0 −1 −2 −3 −4 1 T T 2 0 0) c3 = c4 = (1 . Furthermore, the initial travel velocity and the minimum distance are chosen to be v¯ = 20 m/s and li = 20 m, (i = 2, ..., 5), respectively.
5. MERGING OF NETWORKED VEHICLES
5.1 The merging problem The proposed cooperative control scheme is applied to the vehicle merging problem (Fig. 2). The vehicles are heading towards an upcoming lane reduction and, hence, have to rearrange themselves so that they can change to a common lane safely. For this purpose, the proposed feed-forward controllers presented in Section 3 are used to adjust the inter-vehicle distances before the actual merging takes place. The vehicles should use the zipper method, i. e. they pass the lane reduction alternately from both lanes. 5.2 Trajectory planning phase
In the following, the proposed networked controller is applied to the merging problem. In addition to the nominal case (Fig. 7, left column), a braking manoeuvre of the leading vehicle is investigated in the second scenario (Fig. 7, right column). In order to compare the discrete and the continuous communication strategies described in Section 4, the vehicles communicate in different ways. Vehicle i = 1 sends its locally measured velocity continuously to vehicle i = 2, while the subsequent vehicles i = 3, 4, 5 receive the trajectory vi−1 (t), t ∈ T of the individual predecessors velocity completely at t = 0.
The trajectories d i (t) are planned locally and have to be known before the transition phase begins. Consequently, it is assumed that the trajectory planning phase is finished at t = 0 and to simplify the trajectory planning, it is assumed that all vehicles move initially with same constant velocity vi (t) = v¯, t ≤ 0, i = 1, 2, . . . , N. (21) The following requirements are imposed on the trajectory of the vehicles i = 2, 3, . . . , N : d i (0) = si−1 (0) − si (0) − li (22) (j)
di (0) = 0, j = 1, 2, . . . , ri d i (te ) = δei
Figure 7 shows the velocities vi (t), inter-vehicle distances di (t) with minimum distance li and the positions si (t) of the individual vehicles during the transition phase T = [0, 40 s]. The coloured solid lines and the dotted ones represent the vehicle behaviour and the planned trajectories, respectively.
(23) (24)
(j)
di (te ) = 0, j = 1, 2, . . . , ri . (25) The first two requirements (22) and (23) ensure state consistency and follow from the fact that the inter-vehicle 23
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vi (t) in m/s
20 15 10 5 0
consequently, the subsequent vehicles i = 4 and i = 5 are again able to achieve their local control aims.
P1 P2 P3 P4 P5 di (t) + li in m
80 60 40 20 0 -20 -40 800 600 400 200 0
vi (t) in m/s
10
20 30 t in s
This paper has presented a solution to the vehicle merging problem with cooperative tracking controllers, which are able to communicate via a given communication network to compensate for the physical interaction between the vehicles. It is shown that the proposed control method achieves trajectory tracking for different communication strategies.
di (t) + li in m
si (t) in m
0
7. CONCLUSION
In a simulation study, the performance of the two communication strategies is evaluated in nominal and hazardous situations caused by emergency braking, for example. It is shown that the continuous communication of the velocity enables the subsequent vehicles to track their individual trajectory even in the hazardous case. However, this strategy suffers from high requirements on the communication link. Future work may consider a combination of both strategies, where the continuous communication is triggered by a deviation from the planned trajectory in a selforganised manner combining all advantages.
si (t) in m
40
0
10
20 30 t in s
40
Fig. 7. Transition in nominal and hazardous scenarios
REFERENCES
The initial condition coincides qualitatively with the depiction in Fig. 2. During the transition interval T , the leading vehicle i = 1 reduces its velocity by half to enter the lane reduction. At the same time, the following vehicles i = 2, . . . , 5 adapt their individual inter-vehicle distance to pass the lane reduction with the zipper method. For that purpose, the local tracking controllers increase or reduce the inter-vehicle distance to a common value δei +li = 60 m at te = 40 s for all vehicles. Even though the vehicles influence each other by physical couplings, the cooperative networked controllers clearly achieve trajectory tracking in the specified time interval (cf. Fig. 7, left column). Afterwards, it is safe for all vehicles to change to a common lane in order to pass the lane reduction. Since the velocities are equal to the trajectories vi (t) = vi (t), i = 1, 2, . . . , N, all vehicles achieve the local control aim (6) or (7) independently of the applied communication strategy.
Aschemann, H. and Sun, H. (2013). Decentralised flatnessbased control of a hydrostatic drive train subject to actuator uncertainty and disturbances. IEEE Conf. on Methods and Models in Autom. and Robotics, 759–764. Hamiche, K., Abouaissa, H., Goncalves, G., and Hsu, T. (2016). Real-time decentralized flatness-based control of dynamic supply chain systems. IEEE Conf. on Contr., Decision and Information Technologies, 607–612. Ploeg, J., Scheepers, B.T., Van Nunen, E., Van de Wouw, N., and Nijmeijer, H. (2011). Design and experimental evaluation of cooperative adaptive cruise control. IEEE Conf. on Int. Transp. Systems, 260–265. Rios-Torres, J. and Malikopoulos, A.A. (2017). A survey on the coordination of connected and automated vehicles at intersections and merging at highway on-ramps. IEEE Trans. on Int. Transp. Systems, 18(5), 1066–1077. Rupp, A., Stolz, M., and Horn, M. (2018). Decentralized cooperative merging using sliding mode control. IFAC Symp. on Contr. in Transp. Systems, 51(9), 349–354. Schenk, K., G¨ ulbitti, B., and Lunze, J. (2018). Cooperative fault-tolerant control of networked control systems. IFAC Symp. on Fault Detection, Supervision and Safety for Technical Processes, 570–577. Schenk, K. and Lunze, J. (2018). Tracking control of networked and interconnected systems. IFAC Workshop on Distributed Estimation and Contr. in Networked Systems, 51(23), 40–45. Schmidt, G.K. and Posch, B. (1983). A two-layer control scheme for merging of automated vehicles. In IEEE Conf. on Decision and Contr., 495–500. Schwab, A. and Lunze, J. (2019). Cooperative vehicle merging with guaranteed collision avoidance. IFAC Workshop on Contr. of Transp. Systems. Semsar-Kazerooni, E., Elferink, K., Ploeg, J., and Nijmeijer, H. (2017). Multi-objective platoon maneuvering using artificial potential fields. IFAC World Congress, 50(1), 15006–15011.
The second scenario (Fig. 7, right column) considers the case that vehicle i = 1 has to perform an unplanned braking manoeuvre, because of an emerging obstacle, in the time interval [10 s, 15 s]. As a consequence, the velocity of the leader i = 1 (blue curve in Fig. 7, top right) is lower than the planned trajectory in some time interval. Since vehicle i = 2 receives the velocity v1 (t) continuously in the transition interval, it also reduces its velocity as a response to its braking predecessor, so that the local control aim d2 (t) = d 2 (t) is achieved nevertheless (red curve in Fig. 7, middle right). Thus, the continuous communication introduces a robustness towards disturbances of the predecessor vehicle. The next vehicle i = 3 is not able to follow the planned trajectory (yellow curve in Fig. 7, middle right), because it received the whole trajectory v2 (t) at t = 0. Thus, the local controller C3 is working with wrong information and behaves as if nothing happened downstream and, 24