Distributed Cooperative MPC For Autonomous Driving In .
Distributed Cooperative MPC for AutonomousDriving in Different Traffic ScenariosFatemeh Mohseni, Erik Frisk and Lars NielsenThe self-archived postprint version of this journal article is available at LinköpingUniversity Institutional Repository (DiVA):http://urn.kb.se/resolve?urn urn:nbn:se:liu:diva-172226N.B.: When citing this work, cite the original publication.Mohseni, F., Frisk, E., Nielsen, L., (2020), Distributed Cooperative MPC for Autonomous Driving inDifferent Traffic Scenarios, IEEE Transactions on Intelligent Vehicles, , al publication available ht: Institute of Electrical and Electronics Engineershttp://www.ieee.org/index.html
1Distributed Cooperative MPC for AutonomousDriving in Different Traffic ScenariosFatemeh Mohseni, Erik Frisk, and Lars NielsenAbstract—A cooperative control approach for autonomousvehicles is developed in order to perform different complextraffic maneuvers, e.g., double lane-switching or intersectionsituations. The problem is formulated as a distributed optimalcontrol problem for a system of multiple autonomous vehicles andthen solved using a nonlinear Model Predictive Control (MPC)technique, where the distributed approach is used to make theproblem computationally feasible in real-time. To provide safety, acollision avoidance constraint is introduced, also in a distributedway. In the proposed method, each vehicle computes its owncontrol inputs using estimated states of neighboring vehicles. Inaddition, a compatibility constraint is defined that takes collisionavoidance into account but also ensures that each vehicle doesnot deviate significantly from what is expected by neighboringvehicles. The method allows us to construct a cost function forseveral different traffic scenarios. The asymptotic convergence ofthe system to the desired destination is proven, in the absence ofuncertainty and disturbances, for a sufficiently small MPC controlhorizon. Simulation results show that the distributed algorithmscales well with increasing number of vehicles.Index Terms—Autonomous Vehicles, Collision Avoidance, Cooperative Vehicle Systems, Model Predictive Control.I. I NTRODUCTIONUTONOMOUS vehicle technologies have the potentialto fundamentally change the transportation system with apossibility to reduce energy consumption, pollution, accidentsand congestion costs, see e.g., [1]. In addition, groupingvehicles leads to a reduction of space used by vehicles onthe road, and can significantly address the above traffic issues.Consequently, the concept of cooperation and control algorithms that apply to multiple autonomous vehicle systems havebecome an important topic for researchers in the automationand control field.Among the existing motion planning methods, see [2], pathplanning approaches that are based on optimization, more explicitly Model Predictive Control (MPC), have been successfulin autonomous driving. Handling different limitations of boththe state and input control signals while planning maneuversbased on the trade-off between different control objectives isthe main advantages of MPC [3]. Therefore, it is an appropriatescheme for complex and constrained environments, and is wellsuited to the complex traffic situations considered in this paper.Recently, formulating a cooperative control design amongmultiple agents assigned to a specific task that can navigate autonomously without collision has received significant attention,see e.g., [4]. Further, distributed control among multi-vehiclesystems can improve tractability and provide better scalabilitythan centralized approaches, see [5] and the recent survey [6].These methods have less communication and computation costcompared to centralized methods, see e.g., [7], and therefore,Athey are appealing to be used for control of a system thatcontains multiple autonomous vehicles.When more than one autonomous vehicle work in the samearea, the problem of vehicle collision has to be consideredin control design in order to provide safety of passengers.Recently, many researchers have worked on improved safety intransportation systems, see e.g., [8], [9] and our conclusion isthat it is important to consider the non-holonomic property ofvehicles in the control design. In addition, it is also importantto predict future behavior of other vehicles to be able toguarantee safety for the entire maneuver among autonomousvehicles. In a previous paper [10], a first attempt was madeto handle this situation by a collision avoidance cost in anMPC criterion. In comparison, in our now proposed method, acollision avoidance constraint is introduced in a predictive waythat takes the uncertain deviation between the actual predictiveand estimated trajectories of vehicles into consideration. Thisconstraint together with a compatibility constraint and thetuning parameters for the terminal region guarantee convergenceand collision avoidance properties. The compatibility constraintmotivated by [11]–[14] has an important role in guaranteeingboth collision avoidance and the convergence of the system. Thepredictive collision avoidance feature of our proposed methodis one of the contributions, and makes our method differ fromsome of the existing works that use MPC for multi-vehicleautonomous driving, e.g., [10], [13].There exist different traffic scenarios that if they can beperformed properly by autonomous vehicles, then differenttraffic issues, e.g., accidents and congestion costs can bereduced significantly. One of the challenging scenarios inautonomous vehicle technology that is also considered in thiswork is intersection maneuvers, see, e.g., [15]. In [16], by usingLagrangian formulation, an intersection maneuver for cooperative vehicles was defined in the form of a convex optimizationproblem. Their proposed method has been demonstrated towork well with minimizing the trip time. In [17], by using a newreservation-based approach, a mechanism has been proposedfor autonomous multi-agent system coordination in intersectionsituations. The authors showed that their proposed method canperform well and improve the intersection control.The main problem studied in this paper is how to controlmultiple vehicles in a safe and distributed computationallyefficient way in complex traffic situations, accounting forcapabilities of maneuvering and in the presence of limitations,e.g., road boundaries, speed limitations or other traffic rules andmoving non-autonomous vehicles. Existing methods have beendefined for one specific scenario, e.g., [13], [16], [17]; however,a key contribution of this work is defining an approach that isgeneral enough to be applied for different traffic maneuvers.
2Compared to [10], a new traffic scenario is included whichincludes an analysis of robustness towards non-autonomousvehicles in the traffic scenario, and a new formulation ofthe control objective is introduced. The control problem isreformulated with new and extended compatibility constraintsthat allows the vehicles to deviate from intended trajectoriesand still guarantee collision avoidance. The reformulationallows a treatment as trajectory stabilization instead of as apoint stabilization problem, which is essential to completethe collision free convergence result. With this method itis possible to quantify the computational gains with thedistributed approach in common traffic situations like doublelane-switching and intersections maneuvers.i.e., collaborating to ensure safe driving for all included vehicles.To make the computational time scale well with increasingscenario complexity, and also to make each vehicle moreautonomous and not dependent on any centralized infrastructure,the solution must be distributed.When controlling an autonomous vehicle in general traffic,with significant situation uncertainties, it is essential thatthe proposed approach is robust against disturbances forexample, changes in vehicles behavior and vehicles that do notcollaborate and communicate their driving intent. In addition,the control design must deal with actuator limitations, vehiclemotion constraints, and the other constraints described above.Based on this problem context, a model predictive control(MPC) approach is explored where a finite-horizon optimalcontrol problem is solved repeatedly. A centralized approach isII. V EHICLE M ODEL AND CONSTRAINTSThe system dynamics and kinematic constraints that are used straight forward to formulate and provides good performancein our proposed method are defined in this section. It is assumed w.r.t the global cost function and constraints, but at the costthat the model of each vehicle is described by an ODE and for of computation effort. Therefore, first the problem is definedthe ith vehicle, i 1, . . . , n, the state and control vectors are in a centralized way and then, since the aim is to solve thezi (t) (xri (t), yri (t), θi (t), ψi (t)) and ui (t) (vi (t), ωi (t)) problem in a computationally more efficient way, the centralizedrespectively, where pri (t) (xri (t), yri (t)) and θi (t) are the problem is decomposed into a set of distributed optimal controlmid-point of the rear wheel axle in the Cartesian coordinate problems each associated with one vehicle.To limit the scope of the work and focus on the controls; thisframe and the orientation angle respectively; vi (t) is the speed;ψi (t) is the steering angle of the front wheels with respect to the work does not consider the uncertainty in sensing and assumesvehicle’s body and ωi (t) denotes the steering rate. In addition, that reliable positional information from cameras, lidars orpi (t) (xi (t), yi (t)) refers to the position of vehicle’s center radars are available. In addition, communication delays areand the parameter L denotes the wheelbase length, see Fig. 1. not taken into account and only forward paths are of interest.Further, the mission is defined by a higher-level task planner.Each vehicle is modeled by the kinematic equations vi (t) cos(θi (t))IV. D ISTRIBUTED C OOPERATIVE MPC (DCMPC) FOR vi (t) sin(θi (t)) (1)żi (t) f (zi (t), ui (t)) M ULTIPLE V EHICLES 1 v (t) tan(ψ (t)) .L iiωi (t)A. Optimization formulationThroughout the rest of this paper, the nonholonomic vehiclemodel (1) is used. A more detailed model for each vehiclemay include also load transfer, the rotation angles of eachwheel as generalized coordinates and the body slip, as wellas typical non-idealities such as tire deformation. However,in this paper regular traffic situations are considered with noaggressive maneuvers, and so, model (1) captures the essenceof the vehicle motion and is well suited.The autonomous vehicle motion planning and stabilizationproblem contains limitations resulting from obstacles and inputrestrictions dealing with mechanical limits. These constraintsmust be satisfied in control design; therefore, the input limitations have to be considered simultaneously. So, in addition to(1), the constraints on the control variables are also consideredto reflect physical and mechanical limitationsvmini vi (t) vmaxi , ωi (t) ωmaxi .(2)III. P ROBLEM M OTIVATIONThe main objective of this paper is to control multiplevehicles in an optimal and computationally efficient cooperativeway in order to resolve different traffic situations, for example,double lane-switching and intersections maneuvers. Here,controlling multiple vehicles requires that vehicles in the trafficscenario communicate and send information about their intent,Assume n vehicles, and let zi and ui represent the stateand control vectors from (1) and let pfi (t) (xfi (t), yif (t)) bethe desired final destination of the ith vehicle. With a positionnPerror kP (.; tk ) P f k kpi (.; tk ) pfi k included in thei 1objective function, the optimal control problem is formulated asa point stabilization problem. However, as shown in [18], for anonholonomic dynamic model, the point stabilization problemis more difficult than path following and trajectory tracking.Therefore, the point stabilizing problem is transformed into apath tracking problem by defining a reference nominal path foreach vehicle denoted by pdi (t). Note that pdi (t) (xdi (t), yid (t))is a position trajectory but as this position trajectory also definesa path we will refer to the position trajectory as a path.A first step is to find an admissible desired nominal path foreach vehicle. Here, the method defined in [19] that considersthe continuous curvature reference path generation method isused in order to find an admissible nominal path and notethat the collision is not considered at this stage. Collisionavoidance will be added in the MPC problem that will result inthe actual optimal path. After determining the desired nominalpath pdi (t) (xdi (t), yid (t)) that satisfies the dynamical vehiclemodel, we can derive the remaining state and input trajectories.The deviation from the desired nominal path for each vehicleis called adaption and is denoted by pai (t) pi (t) pdi (t), and
3the desired nominal state and control input vectors are denotedby zid and udi respectively. Furthermore, the correspondingadaption vectors are redefined as zia (t) zi (t) zid (t),and uai (t) ui (t) udi (t). By using the above adaptionvectors, the Centralized Cooperative MPC (CCMPC) problemcan be defined as follows. First, the path, state, control andcorresponding adaption vectors of n vehicles are concatenatedinto vectors as P (t), Z(t), U (t), P a (t), Z a (t) and U a (t).Then, for the weighting scalars γz , γu and γt , the centralizedformulation of problem is formulated as follows.Problem 1. With given prediction horizon hp for a systemof n vehicles and at any update time tk , k 0, 1, . . . , findJ (tk , P a (t), U a (t)) min J(tk , P a (t), U a (t)), forYriΦiDiLf priψiψjvipiθiLrLDjpjLLrLfθjOXFig. 1. Variables, and notations related to vehicles and zones.U (.)Ztk hpJ (γz kP a (τ ; tk )k2 γu kU a (τ ; tk )k2 )dτ tkγt g(P a (tk hp ; tk )),(3)subject to the constraintsŻ(t; tk ) f (Z(t; tk ), U (t; tk )), Z(t; tk ) Z, U (t; tk ) Ugcolij (pi (t; tk ), pj (t; tk )) 0, i, j n, j 6 i(4)where t [tk , tk hp ]. The first two terms in (3) are definedfor performing the maneuver, and the last term is the terminalcost which is needed to guarantee the convergence of the system.This terminal cost is a continuous differentiable function,satisfying g(0) 0 and g(P a (t)) 0 for any P a (t) 6 0, andnnXXis defined as g gi kpai (tk hp ; tk )k2 . Furthermore,i 1i 1(4) represents the collision avoidance constraint and for alli, j n where j 6 i is defined asgcolij (pi (t; tk ), pj (t; tk )) : α kpi (t; tk ) pj (t; tk )k (5)where α is a safe distance between every two vehicles.B. Decoupling of the systemA main issue in control of multiple vehicles is collisionavoidance. To formulate a distributed solution, first note that ina system of multiple vehicles, it is not required to immediatelyact in response to the information from vehicles that are faraway. In Problem 1, a collision constraint (5) is enforcedbetween any two vehicles. Therefore, the communication andcomputational cost of implementing a centralized MPC growswith the number of vehicles. Thus, it is attractive to producea distributed scheme of MPC that enables autonomy of theindividual vehicles and improves the tractability of the MPC.Let i, j be two arbitrary identical vehicles. Considera circular dominance zone with radius Di p(Lf /2)2 (L/2 Lr )2 , where Lf and Lr denote thevehicle frontal width and overhang length respectively, aprotection zone with radius Di Φi , where Φi is a safetythreshold, and a neighborhood zone with radius ri , see Fig. 1.Definition 1 (Neighbors). The set of neighbors of vehicle i isdenoted by Ni and is defined asNi : {j : kpi pj k ri , i 6 j}(6)where ri 2(vmax hp Di ) Φi , and vmax , is the maximumspeed limit. Note that the size of the neighborhood zone isdefined as a function of maximum speed limit and predictionhorizon in order to take into account the worst-case head-oncollision over the MPC prediction horizon.To avoid collision between vehicle i and neighbor j, thefollowing inequality should hold:kpi pj k 2Di Φi , j Ni .(7)Note that if two vehicles i and j are already apart by at leastri at time tk , it is not necessary to consider collision betweenthem over the prediction horizon [tk , tk hp ].C. Estimation of the neighbors behaviorNow suppose n DCMPC optimal control problems, one foreach vehicle, are all solved at common time instants tk . Foreach optimal control problem, the collision constraint containsthe connection between the vehicle and its neighbors at time tk .Therefore, when the local optimal control problems are solved,each vehicle must know the state trajectories of all its neighborsover the time interval [tk , tk hp ]. However, this informationis not available at instant tk for the neighboring vehicles andeach vehicle must estimate the state trajectories of its neighborsover [tk , tk hp ]. With these estimates, the vehicle has allthe information needed for solving its local optimal controlproblem. The trajectories that each vehicle predicts for itsneighbors are called the estimated trajectories. Further, beforetime t tk each vehicle has transmitted its control signalcomputed at tk 1 for the time interval [tk , tk 1 hp ]. Thiscontrol signal will be used as the first part of the estimatedcontroller when the local DCMPC problems are solved attime t tk . In addition, since the control signal for the timeinterval [tk 1 hp , tk hp ] is not known, a stabilizing feedbackcontroller is used for the second part of the estimated controller.Definition 2 (Estimated control). At every time instant τ [tk , tk hp ], the estimated control for each vehicle is definedby the optimal predictive input and a terminal controller, i.e.,(u i (τ ; tk 1 ), τ [tk , tk 1 hp ]ûi (τ ; tk ) (8)uKτ [tk 1 hp , tk hp ]i (τ ; tk ),In Fig. 2, the red curve is the optimal control signalcomputed at time t tk 1 . Then, when computing the optimal
4uiborders of the region. However, following any of the existingmethods does not result in a unique terminal state region for agiven system. The size of the region depends on the selectionof the terminal controller parameters and the non-linearity ofthe system to be controlled.Optimal controller computed at tk 1Optimal controller computed at tkEstimated controller used at tkTerminal controller used at tkhctk 1hctktk 1tk 1 hptk hp tD. Compatibility and collision avoidance constraintsFig. 2. An illustration of the optimal and estimated control signals.control input at time t tk , each vehicle needs to know thetrajectory of its neighboring vehicles also for the time interval[tk 1 hp , tk hp ] which is unknown. Therefore, the terminalcontroller uKi (τ ; tk ), a dynamic feedback controller here foundvia standard full-state linearization, see for example [20], [21],is used to compute an estimate of the optimal trajectories, thegreen curve in Fig 2. The dashed line indicates the estimatedcontrol signal and contains a part of the red curve and theterminal controller. Finally, the resulting optimal controller forthe whole interval [tk , tk hp ] can be then computed whichis the blue curve. According to the estimated control, theestimated predictive trajectory is given by(zi (τ ; tk 1 ),τ [tk , tk 1 hp ]ẑi (τ ; tk ) (9)Kzi (τ ; tk ), τ [tk 1 hp , tk hp ]where ziK (τ ; tk ) is the estimated terminal trajectory.Definition 3 (Terminal set). For every vehicle i, and forξi (tk ) (0, ), and τ tk hp , the terminal set isΩi (tk ) : {zi : kzia (τ ; tk )k ξi (tk )}.(10)Assumption 1. For every vehicle, the largest ξi (tk ) 0 andthe terminal feedback controller gains are chosen such thatkpi (τ ; zi (tk )) pj (τ ; zi (tk ))k 2Di Φi , j Ni , (11)dγti gi (pai (τ ; zi (tk ))) dt
Distributed Cooperative MPC for Autonomous Driving in Different Traffic Scenarios Fatemeh Mohseni, Erik Frisk, and Lars Nielsen Abstract—A cooperative control approach for autonomous vehicles is developed in order to perform different complex traffic maneuvers
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