Towards Cooperative Motion Planning for Automated Vehicles in Mixed Traffic
TTowards Cooperative Motion Planningfor Automated Vehicles in Mixed Traffic
Maximilian Naumann and Christoph Stiller Abstract — While motion planning techniques for au-tomated vehicles in a reactive and anticipatory mannerare already widely presented, approaches to cooperativemotion planning are still remaining. In this paper,we present an approach to enhance common motionplanning algorithms, that allows for cooperation withhuman-driven vehicles. Unlike previous approaches, weintegrate the prediction of other traffic participants intothe motion planning, such that the influence of theego vehicle’s behavior on the other traffic participantscan be taken into account. For this purpose, a newcost functional is presented, containing the cost forall relevant traffic participants in the scene. Finally,we propose a path-velocity-decomposing sampling-basedimplementation of our approach for selected scenarios,which is evaluated in a simulation.
I. I
NTRODUCTION
In the field of intelligent vehicles, tremendous progresshas been achieved in the last decades [1]. With the firstsuccessful experiments of close-to-production cars in realtraffic [2], automated driving has gained more and moreattention in public.In order to improve the reliability and thus the safetyof automated vehicles, but also to increase their effi-ciency, cooperation is focused on in recent research. Here,cooperation through explicit communication of (fused)sensor information and desired driving behaviour [3] aswell as negotiation of possible solutions [4], [5], [6] orcentralized approaches [7] are frequently addressed.However, as reported in [8], cooperative behavior doesnot require V2X-communication. Furthermore, as auto-mated vehicles will share the road with human-drivencars at least at the beginning, cooperation with humandrivers in non-V2X-equipped cars is essential. Also, anatural, cooperative, human-like behavior of automatedvehicles potentially increases their social acceptance.To the best of the authors’ knowledge, previous motionplanning approaches treated other traffic participantsas obstacles which are to be avoided, similar to staticobstacles like parked cars [2], [9]. While such approachescan deal with many everyday situations, such as driv-ing autonomously or following other vehicles, some ma-neuvers, such as overtaking with oncoming traffic or *We gratefully acknowledge support of this work by the Tech Centera-drive The authors are with FZI Research Center for InformationTechnology, Mobile Perception Systems, 76131 Karlsruhe, Germany [email protected] The author is also with Karlsruhe Institute of Technology (KIT),Institute of Measurement and Control, 76131 Karlsruhe, Germany [email protected] (a) without signposted right of way(b) with signposted right of wayFig. 1: Narrowing, with and without signposted right of way. passing a narrowing (cf. Figure 1), require combinato-rial approaches, as already reported by [2]. Still, evenwith combinatorial considerations as proposed by [10],[11], cooperative behavior cannot be implemented: If themotion prediction of other traffic participants is doneisolated from the motion planning for the ego-vehicle,the behavior can be foresighted, but not cooperative ina bidirectional manner [8]. According to a study aboutGerman road traffic, cooperative behavior on averageonly occurs in the scale of one cooperative action perhour per traffic participant [12]. Thus, their treatmentby a separate method, besides the conventional motionplanning, is reasonable.This paper addresses the problem of cooperative mo-tion planning without V2X-communication. We propose acost functional for trajectory ensembles, consisting of onetrajectory per vehicle. Thereby, we acknowledge the factthat not only the behavior of other traffic participantsaffects us, but also our behavior affects the others in aclosed loop. We consider the motion planning problem asthe problem to find a globally optimal solution for a spe-cific situation, knowing that every traffic participant hasa different viewpoint considering optimality. The costsdepend on vehicle dynamics, passenger comfort, drivingintention and trajectory clearance, as well as the trafficregulations, as further outlined in Section II. In thisapproach, the prediction of other traffic participants isintegrated into the motion planning. As the assumptionof cooperative behavior might be violated by some trafficparticipants, this risk is assessed and the trajectory isonly driven if a safe "plan B" [13] trajectory is still pos-sible in case of unexpected behavior. An implementationof this approach is presented in Section III. The proposedalgorithm is finally evaluated in Section IV. a r X i v : . [ c s . R O ] A ug I. G
LOBAL O PTIMUM A PPROACH
This section introduces the main building blocks ofour approach to cooperative motion planning. Centralto this approach is the assumption that all traffic par-ticipants are aware of each other and therefore reacton each other’s behavior in a closed loop. Subsequently,the trajectories for all relevant traffic participants areconsidered as one trajectory ensemble, and the qualityof the solution depends on the trajectory of every partic-ipant separately as well as on the pairwise relation ofthe trajectories among each other.This section is structured as follows: First, the repre-sentation of one trajectory in the ensemble is introduced.Subsequently, the cost functional is introduced. Next,before a solution is selected, the limitations to thisapproach are treated by a "plan B".
A. Behavior Policy
Cooperative motion planning is aware of the inter-action of traffic participants. Therefore, wrong assump-tions concerning the behavior of other traffic participantsmight cause undesired behavior. Even though, theoret-ically, any feasible behavior is possible, the authorsmake the following assumption:
Every traffic participantfollows the traffic regulations, as long as this compliantbehavior is physically feasible.
Consequently, assuming perfect perception, a collisioninvolving our vehicle can only be caused by violating thetraffic regulation without foreseeable reason while ourreaction at the time of violation is insufficient to avoidthe collision.Arising from this assumption, we pursue the followingpolicies: • If we have to give way, we can exclude a collisionindependent of others’ behavior. • If we have the right of way or the situation is notclearly regulated, we can exclude a collision if othersbehave rule compliant.
B. Trajectory Representation
For the representation of a single, deterministic tra-jectory, the established method of [14] is chosen: Thetrajectory x ( t ) = ( x ( t ), y ( t )) T is a mapping (cid:82) → (cid:82) , withtangent angle ψ and curvature κ . A trajectory ensembleconsists of one trajectory per traffic participant: X = ( x , x ,...), where the superscript describes the partici-pants identifier. C. Cost Functional
As proposed in [8], the quality of a solution, given by atrajectory ensemble, is determined by a cost functional.The lowest costs denote the best solution. Costs exceed-ing a certain value represent an infeasible solution. Thecost functional is the sum of the costs of every trafficparticipant i G total = (cid:88) i G i . The costs G i pursue two main goals: They ensure thefeasibility of the trajectory but also rate its comfort andeffectiveness for a single car. For this reason, the prop-erties of the trajectory, such as velocity and acceleration,are rated with multiple evaluation functionals:The feasibility costs exceed a certain bound if a tra-jectory is physically not feasible. The pleasantness costsreflect the wish of the passenger to travel steady andcomfortable, including the perceived safety of the jour-ney. Furthermore, the costs should motivate compliancewith the traffic regulations. Not yielding is avoided byupscaling the costs of the vehicle that has the right ofway in the pairwise trajectory costs.In this approach, the ability to cooperate is associatedwith the ability to estimate the cost or quality of asolution for other traffic participants.With the above information, the costs G i per partic-ipant can be split into costs G i ,0 that only concern theown trajectory and costs G i , j that consider the relationto other trajectories: G i = G i ,0 + (cid:88) j G i , j
1) Formulation of the trajectory properties:
Analog to[14] the properties of the trajectory that are examinedby the evaluation functionals are • the velocity v ( t ) = ˙ x ( t ) • the acceleration a ( t ) = ¨ x ( t ) • the jerk j ( t ) = ... x ( t ) • the distance to the left and right drivingcorridor bound d left ( x ( t )) and d right ( x ( t )) • the yaw rate ω ( t ) = ˙ ψ ( t ) and • the curvature κ ( t ) .Additionally, properties of trajectory pairs describetheir distance to each other. The shortest spatial distanceis described by d min ( x ( t ), x ( t )) = min t (cid:161) d ( x ( t ), x ( t ), t ) (cid:162) ,where d denotes a distance measure between states ofdifferent vehicles.To account for the perceived safety, but also to obey thetraffic regulations, another property is introduced. Here,we can make use of time-referenced measures, as theyequal a velocity-referenced spatial distance measure. Ingeneral, a collision is only possible if paths overlap. Whendetermining the criticality, respectively the collision risk,of two trajectories, their closest point in time and spaceis crucial. Regarding a violation of the right of way,Cooper investigated the post encroachment time (PET) forspecific scenarios [15]. Based on the latter, also regardingthe potential collision zone, we propose the time of zoneclearance (TZC) as a measure for the criticality of twotrajectories with overlapping paths: The TZC is the timethat elapses between the first vehicle leaving potentialcollision zone and the second vehicle entering this area,independent of the right of way (cf. Figure 2). a) blue vehicle drove first (b) black vehicle drove firstFig. 2: The TZC is the time that the second vehicle takesto enter the red potential collision zone, assuming constantvelocity. Given the paths are overlapping and given the trajec-tories are not colliding, the TZC is calculated as follows:TZC = TZC( x first ( t ), x second ( t )) = gap along pathvelocity of the second vehicle = s second ( t second , in ) − s second ( t first , out ) v second ( t first,out )with s second being the path of the vehicle that passesthe collision zone second, v second being the scalar velocityalong this path, t first , out being the time at which the firstvehicle clears the collision zone and t second , in being thetime at which the second vehicle enters the collision zone.Constant velocity is chosen as passengers cannot foreseethe planned trajectory and as it reflects possible actions(maximum deceleration or acceleration) best.If the paths do not overlap, the TZC is defined to beinfinite, if the trajectories collide, it is less or equal zero.
2) Formulation of the evaluation functionals:
As inthis work the costs are also calculated for human-drivencars in order to predict their behavioral decisions, theyshould reflect humans’ understanding of the quality ofa trajectory. Therefore, the previously introduced scalartrajectory properties f ( X ) are investigated. Vectorialproperties, such as the acceleration, are therefore splitinto their longitudinal and lateral part, using a motionmodel.The costs of a trajectory are subdivided into threezones: • comfort zone Z comf • discomfort zone Z disc • infeasibility zone Z inf each for positive ( + ) and negative ( − ) deviation from theoptimum f opt . The functionals G ( f ) expressing the costsinduced by a trajectory property f are called evaluationfunctionals .For the sake of steadiness and piecewise differentia-bility, all costs are starting from zero at their lowerbound but do not vanish at the start of the next zone. Accordingly, the total costs G are defined as G ( f ) = G comf , f ∈ Z comf G comf + G disc , f ∈ Z disc G comf + G disc + G inf , f ∈ Z inf .The comfort component induces only little costs G + comf ( f ) = a + · (cid:161) ∆ f + comfort (cid:162) depending on the distance of f to the optimal value ∆ f + comf = ∆ f + comf ( X ) = f ( X ) − f opt .Consequently, given a comfort threshold T comf and as-suming a comfortable deviation ∆ f + cmargin , the parameter a + is to be set to a + = T comf (cid:179) ∆ f + cmargin (cid:180) .The costs G − comf for comfortable negative deviation arecalculated correspondingly with the parameter a − .The discomfort costs rise quadratic, but direction-dependent: G + disc ( f ) = b + · (cid:161) ∆ f + disc (cid:162) depending on the distance of f to the upper start of thediscomfort zone f + disc , start ∆ f + disc = ∆ f + disc ( X ) = f ( X ) − f + disc .For logical reasons, the parameter b + should be notablyhigher than a + . Negative deviations are treated corre-spondingly with the parameter b − .Before the property represents the infeasibility of atrajectory, the infeasibility costs rise exponentially G + inf ( f ) = c + · (cid:161) ∆ f + inf (cid:162) · e | ∆ f + inf | depending on the distance of f to the upper infeasiblevalue f + inf minus a margin f + margin from which the costsstart rising ∆ f + inf = ∆ f + inf ( X ) = f ( X ) − (cid:179) f + inf − ∆ f + margin (cid:180) .Consequently, given an infeasibility threshold T inf andassuming a margin ∆ f + imargin , the parameter c + is to beset to c + = T inf (cid:179) ∆ f + imargin (cid:180) · e | ∆ f + imargin | .Further, the infeasibility zone Z inf includes the marginin this notation. Again, negative deviations are treatedcorrespondingly with the parameter c − . comfort G discomfort G infeasible G f opt f +disc , start f +inf , start Fig. 3: Composition of the cost function G for a single trajectoryproperty f : Very low costs around the optimum value f opt ,increasing rapidly in close vicinity of f inf .
3) Formulation of the cost functional:
With the evalua-tion functionals, the cost functional for a single property f is composed as follows (cf. Figure 3): G ( f ) = G + comf · σ ( ∆ f + comf ) + G − comf · σ ( − ∆ f − comf ) + G + disc · σ ( ∆ f + disc ) + G − disc · σ ( − ∆ f − disc ) + G + inf · σ ( ∆ f + inf ) + G − inf · σ ( − ∆ f − inf ),where σ denotes the step function. The right of way of i over j is acknowledged by adding the comfort-relatedcosts of vehicle i , upscaled with factor u , to the pairwisetrajectory costs, if i has the right of way: G i , j , row = u (cid:161) G i ,0,comf + G i ,0,disc (cid:162) .A suitable choice of u ensures that the right of way isheeded, but its violation is still feasible, as stated inSection II-A.The full cost functional is composed as follows: G total ( X ) = (cid:88) i (cid:195) G i ,0 ( x i ) + (cid:88) j G i , j ( x i , x j ) (cid:33) with singleton trajectory costs for vehicle iG i ,0 ( x i ) = G v + G a + G j + G ω + G κ + G offset and pairwise trajectory costs for vehicle i due to vehicle jG i , j ( x i , x j ) = G TZC + G d min + G row . D. Plan B
In order to obey our previously introduced policy,plan B trajectories are to be checked, as proposed in[13]. By doing so, we avoid maneuvering into situationsthat lead to collisions, if we made wrong assumptionsconcerning the behavior of other traffic participants. Astheir execution is unlikely, we accept discomfortable butfeasible trajectories. This corresponds to a neglection ofthe comfort terms in the upper cost functional. As with the previous trajectories, plan B trajectories can be cal-culated via a local continuous method [14], a sampling-based method such as RRT ∗ [16] or other approaches. E. Selection of Solution
As for passenger comfort, the evaluation of the TZCshould already cause high discomfort costs at around2 s , a security margin is induced intrinsically by thisapproach. Thus, even a very small optimum, representedby a small range of minimal costs, does not equal aphysically optimal trajectory, that would pass objectsas close as possible in space-time. Rather, it alreadycontains those security margins that are consideredcomfortable by humans and that consequently should befeasible with measurement uncertainties in the range ofhuman perception errors. Hence, the optimum point canbe chosen independent of its wideness, as long as a validplan B protects the approach against consequences ofwrong assumptions.III. I MPLEMENTATION
In the following, a first approach for cooperative mo-tion planning in specific situations, based on the previ-ously introduced cost functional, is presented.
A. Path-Velocity Decomposition
Several potentially cooperative situations have highlyconstraint driving corridors for the traffic participants,independent of the order and number of traffic partici-pants. Consequently, we make use of the path-velocitydecomposition (PVD) , as introduced by [17]. The calcu-lation of paths in static environments has already beenwidely investigated. Hence, valid paths are consideredpredefined (cf. Figure 4) and the implementation focuseson the velocity profiles along the paths.
B. Sampling
As the optimization problem is non-convex, but thecontrol variable for the velocity of each vehicle is onlyone-dimensional, a classical sampling approach is cho-sen. Therefore, the trajectories x ( t ) are approximated bydiscretization in equidistant time steps: x i = x ( t i ), t i = t + i ∆ t .For each car, multiple trajectories are sampled: Start-ing with an initial position and velocity, a random jerksequence determines the velocity profiles and thus thetrajectory. Next, the overall costs of each trajectory en-semble are calculated. For the solutions with the lowestcosts, the plan B trajectory is checked until a valid plan Bis found. C. Plan B
Instead of using a different planning method with theassumption or classical prediction of disadvantageousbehavior of others, we again make use of the PVD: Giventhe paths, a collision is only possible in particular areasthat can be determined a priori. Thus, unlike in [13],o trajectory has to be planned. Rather, the plan B-consideration can be seen as a “what could I do if”-consideration. The key questions are: In every time step,what could the other vehicle do that leads to a collisionwith us? And what could we do to avoid this? Thisconsideration can be split into the following cases:
1) Other vehicle drives first:
If the other vehicle drivesfirst, it can only cause a collision by deceleration. Inreaction, we can decelerate as well. If we can manageto stop before the collision zone, we have a valid plan B.
2) Ego vehicle drives first:
If the ego vehicle drivesfirst, the other vehicle can only cause a collision byacceleration. In reaction, we can also accelerate, to stilldrive first, or decelerate to stop before the other vehiclecollides with us. As the path is regarded as predefined,changing the path is not considered.
D. Implications on the cost functional
Since in this implementation, trajectories are dis-cretized in time, derivatives are approximated by finitedifferences. Thus, the functionals of section II-C turninto functions. Consequently, trajectory properties thatdepend on a single minimum, such as TZC, can be largelyaffected if this minimum is not sampled. In order toavoid this, either the sampling rate must be sufficientlyhigh, or the point of the exact minimum has to beinterpolated. As this implementation is not based onlinear optimization but on sampling, we interpolate thecrucial points.The jerk is not considered to avoid high order deriva-tives. Also, the curvature itself is not considered asthe predefined path guarantees the compliance with thesteering geometry. However, it is used to calculate thelateral acceleration values. Furthermore, the shortestspatial distance d min either lies in the collision zone andis considered by the TZC, or it is not relevant. Hence, itis neglected as well. E. Selection of Solutions
As explained in section II-E, criticality protection isensured via the costs of the TZC and the check for aplan B. Consequently, the solution with the lowest costsand a valid plan B is selected and its ego trajectory isexecuted, as long as its costs do not exceed the feasibility-threshold. In case no solution has a valid plan B, anemergency braking maneuver is triggered. Note: In in-car applications, the parallel running classical, reactivemotion-planner would have to take over control in thiscase. IV. E
VALUATION
In this section, the method outlined in section III isevaluated for two scenarios, a left turn at a T-intersectionand passing through a narrowing of the road (cf. Figures1 and 4). (a) (b)Fig. 4: Left turn at T-junction, with and without signpostedright of way and predefined paths.
A. Simulation
For both scenarios, each with and without signpostedright of way, but sharing the same paths, velocity profileswere sampled. From the resulting trajectories, ensembleswith one trajectory per vehicle were generated. In orderto reduce computational cost, trajectories that did notreach the end of the collision zone were excluded fromthe cost calculation. Furthermore, colliding trajectoryensembles were excluded. The remaining ensembles wereanalyzed with respect to • comfort costs • discomfort costs • infeasibility costs • traffic regulation costs. B. Analysis
As depicted in Figure 5 and 6, the initial states werechosen in a way that the optima of both vehicles overlapin the collision zone. In the T-junction scenario, the rightof way is regulated with and without traffic signs. Aviolation of the right of way causes high costs so theoptimal solution is following the rules. The trajectory ofthe vehicle that has right of way is not interfered (cf.Figure 5 (2) and (3)).In the narrowing scenario, the right of way is notregulated without traffic signs. Here, due to equal costparameters, the vehicle that is closer to the narrowingpasses first. Still, traffic signs can overrule this globallymost comfortable solution and shift the optimum (cf.Figure 1 and 6 (3)).If a collision can only be avoided by one of the vehicles,as the other is too close to the collision zone, the optimalsolution is the collision avoidance. Even though thisviolates the traffic regulations, the infeasibility costsoverrule discomfort costs and traffic regulation costs.Further, if we do not interfere a vehicle that has theright of way, its costs G row are constantly high, but notraised by our behavior. In this case, the optimal solutionis that we pass first, without violation of traffic rules.V. C ONCLUSIONS AND F UTURE W ORK
In this paper, we presented a new approach to coop-erative motion planning, able to cooperate with human ig. 5: Minimum cost trajectories in the T-junction scenariowith the collision zone marked in grey: (1) for each vehiclesolely on the road, (2) when upper vehicle has right of way(Fig. 4a), (3) when lower vehicle has right of way (Fig. 4b).Fig. 6: Minimum cost trajectories in the narrowing scenariowith the collision zone marked in grey: (1) for each vehiclesolely on the road, (2) when no right of way predefined (Fig. 1a),(3) when left vehicle has right of way (Fig. 1b). drivers and automated vehicles without requiring V2X-communication. While the approach is valid for two-dimensional motion planning, our first implementationcovers several scenarios deploying PVD.The preliminary results for the simulated scenariosdemonstrate that the method produces safe and com-fortable cooperative trajectories in a narrowing and atypical intersection scenario. Individual trajectory costshave been extended by costs accounting for mutualcomfort and safety of any pair of trajectories. Othertraffic participants have been taken into account byincorporating their individual costs. The total trajectorycosts for each participant have been segmented into threeareas representing comfortable driving, uncomfortable driving and collision/infeasibility.Future work includes real time implementation andon-road experiments with our vehicle "BerthaOne". Sev-eral parametrizations will be used for the cost functional,considering different vehicle types and driver behaviors.Furthermore, probabilistic trajectories will be accommo-dated to account for inherent uncertainties in perceptionand behavior. R
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