A uniersal model-free and safe adaptie cruise control mechanism Thomas Berger and Anna-Lena Rauert Abstract We consider the problem of ehicle following, where a safety distance to the leader ehicle is guaranteed at all times and a faourite elocity is reached as far as possible. We introduce the funnel cruise controller as a noel uniersal adaptie cruise control mechanism which is model-free and achiees the aforementioned control objecties. The controller consists of a elocity funnel controller, which directly regulates the elocity when the leader ehicle is far away, and a distance funnel controller, which regulates the distance to the leader ehicle when it is close so that the safety distance is neer iolated. A sketch of the feasibility proof is gien. The funnel cruise controller is illustrated by a simulation of three different scenarios which may occur in daily traffic. Index Terms Autonomous ehicles, adaptie cruise control, nonlinear systems, funnel control, safety guarantees AMS subject classifications 93C, 93C4, 7Q5 I. INTRODUCTION With traffic steadily increasing, simple cruise control see e.g. [], which holds the elocity on a constant leel, is becoming less useful. A controller which additionally allows a ehicle to follow the ehicle in front of it while continually adjusting speed to maintain a safe distance is a suitable alternatie. Various methods which achiee this are aailable in the literature, see e.g. the surey [2] on adaptie cruise control systems. A common method is the use of proportional-integral-deriatie PID controllers, see e.g. [], [3], [4], [5], which howeer are not able to guarantee any safety. Another popular method is model predictie control MPC, where the control action is defined by repeated solution of a finite-horizon optimal control problem. A two-mode MPC controller is deeloped in [6], where the controller switches between elocity and distance control. The MPC method introduced in [7] incorporates the fuel consumption and drier desired response in the cost function of the optimal control problem. In [8] a method which guarantees both safety and comfort is deeloped. It consists of a nominal controller, which is based on MPC, and an emergency controller which takes oer when MPC does not proide a safe solution. Control methods based on control barrier functions which penalize the iolation of gien constraints hae been deeloped in [9], []. While safety constraints are automatically guaranteed by this approach, it may be hard to find a suitable This work was supported by the German Research Foundation Deutsche Forschungsgemeinschaft ia the grant BE 6263/-. Thomas Berger and Anna-Lena Rauert are with the Fachbereich Mathematik, Uniersität Hamburg, Bundesstraße 55, 246 Hamburg, Germany, thomas.berger@uni-hamburg.de, anna-lena.rauert@studium.uni-hamburg.de control barrier function. Another recent method is correct-byconstruction adaptie cruise control [], which is also able to guarantee safety. Howeer, the computations are based on a so called finite abstraction of the system which is already expensie and changes of the system parameters require a complete re-computation of the finite abstraction. Drawbacks of the aforementioned approaches are that either safety cannot be guaranteed as in [3], [4], [5] or the model must be known exactly as in [9], [6], [8], [], [], [7]. Howeer, the requirements on drier assistance systems are increasing steadily. It is expected that in the near future autonomous ehicles will completely take oer all driing duties. Therefore, a cruise control mechanism is desired which achiees both: under any circumstances in particular, in emergency situations the prescribed safety distance to the preceding ehicle is guaranteed and at the same time the parameters of the model, such as aerodynamic drag or rolling friction, must not be known exactly, i.e., the control mechanism is model-free. The latter property also guarantees that the controller is inherently robust, in particular with respect to uncertainties, modelling errors or external disturbances. Another requirement on the controller is that it should be simple in its design and of low complexity, and that it only requires the measurement of the elocity and the distance to the leading ehicle. We stress that in a lot of other approaches as e.g. [6], [3], [8], [7] the position, elocity and/or acceleration of the leading ehicle must be known at each time. In the present paper we propose a noel control design which satisfies the aboe requirements. Our control design is based on the funnel controller which was deeloped in [2], see also the surey [3] and the recent paper [4]. The funnel controller is an adaptie controller of high-gain type and has been successfully applied e.g. in temperature control of chemical reactor models [5], control of industrial serosystems [6], [7], [8], DC-link power flow control [9], oltage and current control of electrical circuits [2], and control of peak inspiratory pressure [2]. In our design we will distinguish two different cases. If the preceding ehicle is far away, i.e., the distance to it is larger than the safety distance plus some constant, then a elocity funnel controller will be actie which simply regulates the elocity of the ehicle to the desired pre-defined elocity. If the preceding ehicle is close, then a distance funnel controller will be actie which regulates the distance to the preceding ehicle to stay within a predefined performance funnel in front of the safety distance. The combination of these two controllers results in a funnel cruise controller which guarantees safety at all times. We like to stress that
the distance funnel controller does not directly regulate the position of the ehicle, but a certain weighting between position and elocity; hence, a relatie degree one controller suffices. A. Nomenclature R = [, Lloc I Rn the set of locally essentially bounded functions f : I R n, I R an interal L I R n the set of essentially bounded functions f : I R n f = ess sup t I f t W k, I R n the set of k-times weakly differentiable functions f : I R n such that f,..., f k L I R n C V R n the set of continuous functions f : V R n, V R m f W restriction of the function f : V R n to W V disty,x = inf x X y x, the distance of y R n to a set X R n B. Framework and system class In the present paper we consider the framework of one ehicle following another, see Fig.. Follower xt t x l t x safe t Fig. : Vehicle following framework. Leader By x l we denote the position of the leader ehicle, while x and denote the position and elocity of the follower ehicle. The change in momentum of the latter is gien by the difference of the force F generated by the contact of the wheels with the road and the forces due to graity F g including the changing slope of the road, the aerodynamic drag F a and the rolling friction F r. Detailed modelling of these forces can be ery complicated since all the indiidual components of the ehicle hae to be taken into account. Therefore, we use the following simple models which are taken from [, Sec. 3.]: F g : R R, t mgsinθt, F a : R R R, t, 2 ρtc da 2, F r : R R, mgc r sgn, where m in kg denotes the mass of the following ehicle, g = 9.8m/s 2 is the acceleration of graity, θt [ π 2 rad, π 2 rad] and ρt in kg/m3 denote the slope of the road and the bounded density of air at time t, resp., C d denotes the dimensionless shape-dependent aerodynamic drag coefficient and C r the dimensionless coefficient of rolling friction, and A in m 2 is the frontal area of the ehicle. Since the discontinuous nature of the rolling friction causes some problems in the theoretical treatment we approximate the sgn function by the smooth error function using the property that erfz = 2 z e t2 dt, z R, π z R : lim erfαz = sgnz. α Therefore, we will use the following model for the rolling friction: F r : R R, mgc r erfα. for sufficiently large parameter α >. The force F which is generated by the engine of the ehicle is usually gien as torque cure depending on the engine speed times a signal which controls the throttle position, see [, Sec. 3.]. Since the latter can be calculated from any gien force F and elocity taking the current gear into account, here we assume that we can directly control the force F, i.e., the control signal is ut = Ft. The equations of motion for the ehicle are then gien by ẋt = t, m t = ut F g t F a t,t Fr t,.2 with the initial conditions C. Control objectie x = x R, = R..3 Roughly speaking, the control objectie is to design a control input ut such that t is as close to a gien faourite speed ref t as possible, while at the same time a safety distance to the leading ehicle is guaranteed, i.e., x l t xt x safe t. The safety distance x safe t should preent collision with the leading ehicle and is typically a function of the ehicle elocity, but could also be a constant or a function of other ariables. In the literature different concepts are used, see e.g. [22], [23] and the references therein. A common model for the safety distance that we use in the present paper is x safe t = λ t + λ 2.4 with positie constants λ in s and λ 2 in m. The parameter λ models the time gap between the leader and follower ehicle and λ 2 is the minimal distance when the elocity is zero. If for instance λ =.5s, then it would take the following ehicle.5 s to arrie at the leading ehicle s present position. We assume that the distance x l t xt to the leader ehicle as well as the elocity t can be measured, i.e., they are aailable for the controller design. Apart from that, the controller design should be model-free, i.e., knowledge of the parameters m,θt,ρt,c d,c r and A as well as of the initial alues x, is not required. This makes the controller robust
to modelling errors, uncertainties, noise and disturbances. Summarizing, the objectie is to design a nonlinear and time-arying control law of the form ut = F t,t,x l t xt.5 such that, when applied to a system.2, in the closed-loop system we hae that for all t O x l t xt x safe t, O2 t ref t is as small as possible such that O is not iolated. D. Funnel control for relatie degree one systems The final control design will consist of two different funnel controllers for appropriate relatie degree one systems. The first ersion of the funnel controller was deeloped in [2] and this ersion will be sufficient for our purposes. We consider nonlinear relatie degree one systems goerned by functional differential equations of the form ẏt = f dt,tyt + γ ut, y [ h,] = y C [ h,] R,.6 where h is the memory of the system, γ > is the high-frequency gain and d L R R p, p N, is a disturbance; f C R p R q R, q N; T : C [ h, R L loc R R q is an operator with the following properties: a T maps bounded trajectories to bounded trajectories, i.e, for all c >, there exists c 2 > such that for all ζ C [ h, R, sup ζ t c sup T ζ t c 2. t [ h, t [, b T is causal, i.e., for all t and all ζ, ξ C [ h, R: ζ [ h,t] = ξ [ h,t] = T ζ [,t] a.e. = T ξ [,t], where a.e. stands for almost eerywhere. c T is locally Lipschitz continuous in the following sense: for all t there exist τ,δ,c > such that for all ζ, ζ C [ h, R with ζ [ h,t] = and ζ [t,t+τ] < δ we hae T ζ + ζ T ζ [t,t+τ] c ζ [t,t+τ]. The funnel controller for systems.6 is of the form ut = ktet, et = yt y ref t, kt = φt 2 et 2,.7 where y ref W, R R is the reference signal, and guarantees that the tracking error et eoles within a prescribed performance funnel F φ := { t,e R R φt e < },.8 which is determined by a function φ belonging to Φ:= φ W, φs > for all s > and R R for all ε > : /φ [ε, W, [ε, R. The funnel boundary is gien by the reciprocal of φ, see Fig. 2. The case φ = is explicitly allowed, meaning that no restriction is put on the initial alue since φ e < ; the funnel boundary /φ has a pole at t = in this case.,e λ φt Fig. 2: Error eolution in a funnel F φ with boundary φt. An important property is that each performance funnel F φ is bounded away from zero, i.e., because of boundedness of φ there exists λ > such that /φt λ for all t >. We stress that the funnel boundary is not necessarily monotonically decreasing. Widening the funnel oer some later time interal might be beneficial, e.g., when periodic disturbances are present or the reference signal changes strongly. For typical choices of funnel boundaries see e.g. [24, Sec. 3.2]. In [2], the existence of global solutions of the closedloop system.6,.7 is inestigated. To this end, y : [ h,ω R is called a solution of.6,.7 on [ h,ω, ω, ], if y [ h,] = y and y [,ω is weakly differentiable and satisfies.6,.7 for almost all t [,ω; y is called maximal, if it has no right extension that is also a solution. Note that uniqueness of solutions of.6,.7 is not guaranteed in general. The following result is proed in [2]. Theorem.: Consider a system.6 with initial trajectory y C [ h,] R, a reference signal y ref W, R R and a funnel function φ Φ such that φ y y ref <. Then the controller.7 applied to.6 yields a closed-loop system which has a solution, and eery maximal solution y : [,ω R has the properties: i ω = ; ii all inoled signals y, k and u are bounded; iii the tracking error eoles uniformly within the performance funnel in the sense ε > t > : et φt ε. E. Organization of the present paper In Section II we present a noel funnel cruise controller which satisfies the control objecties as stated in Section I- C. The controller is basically the conjunction of a elocity t
funnel controller and a distance funnel controller, both formulated for appropriate relatie degree one systems. Those controllers are presented separately before the final controller design is stated and feasibility is proed. In Section III the performance of the controller is illustrated for some typical model parameters and scenarios from daily traffic. Some conclusions are gien in Section IV. II. FUNNEL CRUISE CONTROL In this section we present our noel funnel cruise control design to achiee O and O2, which consists of a elocity funnel controller and a distance funnel controller.. We first present those controllers separately before we state the unified controller design. Velocity Control Follower Follower φ d t Distance Control. x safe t Leader Leader Fig. 3: Illustration of the distance funnel controller. A. Velocity funnel controller When the leader ehicle is far away we do not need to care about the control objectie O and simply need to regulate the elocity to the faourite elocity ref W, R R. For this purpose we can treat the elocity as the output of system.2 and hence the elocity tracking error is gien by e t = t ref t. Since the first equation in.2 can be ignored in this case it does not play a role for the inputoutput behaior, we may define dt := F g t,ρt for t and f : R 3 R, d,d 2, m d + 2 d 2C d A 2 + F r. Since ρ is assumed to be bounded we obtain that d is bounded and the second equation in.2 can be written as t = m ut f dt,t, 2. and hence belongs to the class of systems.6 with the identity operator T =. Then Theorem. yields feasibility of the elocity funnel controller u t = k te t, e t = t ref t, k t = φ t 2 e t 2, 2.2 where φ Φ, when applied to system.2 with initial conditions.3 such that φ ref <. We stress that since x has been ignored for the controller design aboe, the first equation in.2 may cause it to grow unboundedly. Howeer, since x l is assumed to be bounded, x l x will eentually get small enough so that the distance funnel controller discussed in the following section becomes actie. In the end, this will guarantee boundedness of x. B. Distance funnel controller If the leader ehicle is close, then the main objectie of the controller is to ensure that O is guaranteed, so that t may be much smaller than ref t if necessary. To this end, we introduce a performance funnel, defined by φ d Φ with φ d, which lies directly in front of the safety distance to the leader ehicle, see Fig. 3. The aim is then to regulate the position xt to the middle of this performance funnel gien by x l t x safe t φ d t, where x l W, R R. The corresponding distance tracking error is hence gien by e d t = xt x l t + x safe t + φ d t. In order to reformulate system.2 in the form.6 with appropriate output y and reference signal y ref we recall that x safe t = λ t + λ 2 and define yt := λ t + xt, y ref t := x l t λ 2 φ d t. Then e d t = yt y ref t and we further find that, inoking the first equation in.2, hence xt = e λ t x + Now define ẋt = λ xt + λ yt, x = x, t λ e λ t s ysds =: T yt, t. Ty := T y,y for all y C R R. It is straightforward to check that the operator T : C [, R Lloc R R 2, which is parameterized by x R, has the properties a c as stated in Section I-D. Using equation 2. as well as d and f defined in Section II-A we obtain where ẏt = λ t + ẋt = λ m ut λ f dt,t λ xt + λ yt, = λ m ut λ f dt, λ T yt + λ yt λ T yt + λ yt, = λ m ut f d dt,tyt, 2.3 f d : R 4 R, d,d 2,ζ,ζ 2 λ f d,d 2, λ ζ + λ ζ 2 + λ ζ λ ζ 2. Clearly, 2.3 belongs to the class of systems.6. Then Theorem. yields feasibility of the distance funnel con-
troller u d t = k d te d t, e d t = xt x l t k d t = φ d t 2 e d t 2, + x safet + φ d t, 2.4 when applied to system.2 with initial conditions.3 such that φ d λ + x x l + λ 2 + φ d <. C. Final control design and its feasibility In Sections II-A and II-B we hae seen that the separate elocity and distance funnel controllers are feasible when the initial conditions lie within the funnel boundaries at t = ; the control objectie O is ignored in the case of elocity control and the control objectie O2 is ignored in the case of distance control. Howeer, it is our aim to simultaneously satisfy O and O2, i.e., always guarantee the safety distance and regulate the elocity to the faourite elocity as far as possible. This means that two additional scenarios must be possible: if the follower ehicle, while using the elocity funnel controller 2.2, enters the performance funnel in front of the safety distance, i.e., xt = x l t x safe t φ d t, then the controller should switch to the distance funnel controller 2.4; when the distance funnel controller is actie it should still be guaranteed that t < ref t + φ t, but it is possible that t ref t φ t when the leader decelerates; in the latter case it is not possible to switch back to 2.2. A controller which combines 2.2 and 2.4 and takes the aboe conditions into account faces an immediate problem: The controller 2.4 has a singularity when xt = x l t x safe t φ d t since k d t at such points. Likewise, k t for points where t = ref t φ t, i.e., when a strong deceleration is necessary. To resole these problems, the minimum of the control signals u t and u d t is chosen in the region where the elocity performance funnel and the distance performance funnel intersect, i.e., when t,e t F φ and t,e d t F φd ; see also Fig. 4 for an illustration. φ e = u = u u = min{u d,u } The oerall funnel cruise controller is of the form.5 and gien in 2.5. For later purposes we also define the relatiely open set D gien in 2.6. In the remainder of this section we gie the feasibility result for the controller 2.5 and sketch its proof. Theorem 2.: Consider a system.2 with initial conditions.3, a faourite elocity ref W, R R, position of the leader ehicle x l W, R R, safety distance x safe as in.4 with λ,λ 2 > and funnel functions φ,φ d Φ such that φ d and,x, D. Then the funnel cruise controller 2.5 applied to.2 yields a closed-loop system which has a solution, and eery maximal solution x, : [,ω R 2 has the properties: i ω = ; ii all inoled signals x, and u are bounded; iii the solution eoles uniformly within the set D, i.e., ε > t : dist t,xt,t, D > ε, where D denotes the boundary of D in R R 2. Sketch of the Proof: Since the input u as in 2.5 is continuous in x, on the set D from 2.6, the existence of a maximal solution of the closed-loop system is a consequence of Carathéodory s existence theorem, see e.g. [25,, Thm. XX]. Let x, : [,ω R 2, ω, ], be a maximal solution, then we may also infer that the closure of the graph of x, is not a compact subset of D. Clearly, x and are bounded since φd, x l and ref are bounded and t, xt, t D for all t [, ω. Therefore, if iii holds on [,ω, then it implies i and ii. In order to show iii we consider three different cases according to the three parts in the definition of D in 2.6, see also Fig. 4. By standard arguments as used in the proof of Theorem. see [2] we may exclude that the graph of x, can reach the boundary of D in the first two cases; in other words, k is bounded on the first part where u = u and k d is bounded on the second part where u = u d. At the boundary between the first part and the third part u d has a pole, and at the boundary between the second part and the third part u has a pole; but u is continuous since u = min{u,u d }. To show that in this last part the graph of x, cannot reach the boundary of D we diide it again into four distinct parts and consider the minimum separately on each of them. Using similar arguments as before, the result can then be obtained. u = u d φ d e d = Fig. 4: Illustration of the final control design. x III. SIMULATIONS We illustrate the funnel cruise controller 2.5 for three different scenarios which may occur in daily traffic. The first standard scenario is that the follower ehicle, with a constant faourite elocity ref, is far away from the leader, catches up and follows it for some time until the leader accelerates past ref. The second scenario illustrates that safety is guaranteed een in the case of a full brake of the leader ehicle. In the last scenario the leader ehicle has a strongly arying acceleration..
k te t, e d t φ d t t,e t F φ, ut = k d te d t, e t φ t t,e d t F φd, min{ k te t, k d te d t}, t,e t F φ t,e d t F φd, k t = φ t 2 e t 2, e t = t ref t, k d t = φ d t 2 e d t 2, e dt = xt x l t + x safe t + φ d t. λ + x x l t λ 2 2φ d t t, ref t F φ D := t,x, R R 2 ref t φ t t,λ + x x l t + λ 2 + φ d t F φd t, ref t F φ t,λ + x x l t + λ 2 + φ d t F φd 2.5. 2.6 m θt ρt C d C r A 3kg rad.3kg/m 3.32. 2.4m 2 TABLE I: Parameter alues for the ehicle model. For the simulations we will use some typical parameter alues for the model.2 which are taken from [] and summarized in Table I. For the approximated friction model. we choose the parameter α =. The initial conditions.3 are chosen as x = m and = 5ms and the constants in.4 as λ =.5s and λ 2 = 2m. For all three scenarios we choose the faourite elocity ref t = 36ms and the funnel functions φ t = 22.5e.2t +.2, φd t =.25. Scenario : We hae chosen x l and l = ẋ l so that initially the leader ehicle has a larger elocity than the follower, which is hence free to accelerate and catch up using the elocity funnel controller. When the distance is between x safe t+2φ d t and x safe t, the distance funnel controller will ensure that the safety distance is not iolated. After a period of safe following, where t < ref t φ t, the leader accelerates to a elocity larger than ref t and the elocity funnel controller will again take oer. The simulation of the controller 2.5 for the system.2 with parameters as in Table I and the aboe described scenario oer the time interal 5s has been performed in MATLAB soler: ode5s, rel. tol.:, abs. tol.: and is depicted in Fig. 5. Fig. 5a shows the distance x l x to the leader and the distance funnel in front of the safety distance. The elocities and l are depicted in Fig. 5b together with the elocity funnel. Fig. 5c shows the input signal u generated by the controller and the engine force u l of the leader ehicle. We stress that, due to the mass of the ehicles of 3kg, the forces u and u l which are between ± 4 N correspond to an acceleration between ±8m/s 2. It can be seen that the controller achiees the faourite elocity as far as possible while guaranteeing safety at all times, thus the control objecties O and O2 are satisfied. Scenario 2: We hae chosen x l and l so that after a period of safe following the leader ehicle suddenly fully brakes. This illustrates that een in such extreme cases the funnel cruise controller is able to guarantee that the safety distance is not iolated. The simulation of the controller 2.5 for the system.2 with parameters as in Table I and the aboe described scenario oer the time interal 5s has been performed in MATLAB soler: ode5s, rel. tol.:, abs. tol.: and is depicted in Fig. 6. It can be seen in Fig. 6a that the safety distance is always guaranteed. From Fig. 6b we can obsere that the elocity of the follower does not drop as sharp as the elocity l of the leader. Fig. 6c shows that the engine forces u and u l are quite comparable. Scenario 3: We hae chosen x l and l so that the leader ehicle has a strongly arying acceleration. After a period of elocity control where the follower ehicle is free to accelerate close to the faourite elocity ref, it will catch up with the leader ehicle and a period of safe following using distance funnel control follows. During this period seeral sharp acceleration and deceleration maneuers are necessary to guarantee safety in the face of the mercurial behaior of the leader ehicle. The simulation of the controller 2.5 for the system.2 with parameters as in Table I and the aboe described scenario oer the time interal 5s has been performed in MATLAB soler: ode5s, rel. tol.:, abs. tol.: and is depicted in Fig. 7. It can be seen in Fig. 7a that safety is guaranteed een in the case of the strongly arying behaior of the leader. The elocity of the follower, as shown in Fig. 7b, does not ary as strongly as l, but shows a much smoother behaior. The engine force u depicted in Fig. 7c shows sharp drops and rises, but this cannot be aoided since the funnel cruise controller 2.5 is causal and hence cannot look into the future. This is different from other approaches such as MPC see e.g. [6], [7], [8] which is able to incorporate future information since an optimal control problem is soled oer some future time interal. Howeer, the drawback of this is that the model.2 must be known
8 x l! x x safe + 2ϕ! d x safe 25 2 x / m 6 4 2 2 3 4 5 Fig. 5a: Distance to leader and distance funnel x / m 5 5 x l! x x safe + 2ϕ! d x safe 2 3 4 5 Fig. 6a: Distance to leader and distance funnel / m/s u / N 6 5 4 3 2 l ref ' ϕ! 2 3 4 5.5 -.5 x 4 Fig. 5b: Velocities and elocity funnel u l - 2 3 4 5 Fig. 5c: Input and leader engine force Fig. 5: Simulation of the funnel cruise controller 2.5 for the system.2 with parameters as in Table I in Scenario. / m/s u / N 6 4 2 l ref ' ϕ! -2 2 3 4 5 - -2-3 Fig. 6b: Velocities and elocity funnel x 4-4 u u l -5 2 3 4 5 Fig. 6c: Input and leader engine force Fig. 6: Simulation of the funnel cruise controller 2.5 for the system.2 with parameters as in Table I in Scenario 2. as good as possible. IV. CONCLUSIONS In the present paper we proposed a noel and uniersal adaptie cruise control mechanism which is model-free and guarantees safety at all times. The funnel cruise controller consists of a elocity funnel controller, which is actie when the leader ehicle is far away, and a distance funnel controller, which ensures that the safety distance is not iolated when the leader ehicle is close. We hae sketched the proof of feasibility of this controller; a comprehensie proof will be gien in future works. Three simulation scenarios illustrate the application of the funnel cruise controller. The simulations show that, although the funnel cruise controller satisfies the control objecties, the generated control input engine force of the follower ehicle usually contains sharp peaks which are not desired in terms of drier comfort. This issue should be resoled by smoothing the peaks e.g. by combining the funnel cruise controller with the PI-funnel controller with anti-windup as discussed in [26]. Furthermore, constraints on the acceleration should be incorporated in future research to aoid unrealistic high peaks in the control input generated by the controller. Another topic of future research is the inestigation of platoons of seeral ehicles, each equipped with a funnel cruise controller. An important question is under which
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