Optimal Control of Autonomous Vehicles Approaching A Traffic Light

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1 1 Optimal Control of Autonomous Vehicles Approaching A Traffic Light Xiangyu Meng Christos G Cassras arxiv:1896v1 [eesssp] 6 Feb 18 Abstract This paper devotes to the development of an optimal acceleration/speed profile for autonomous vehicles approaching a traffic light The design objective is to achieve both short travel time low energy consumption as well as avoid idling at a red light This is achieved by taking full advantage of the traffic light information based on vehicle-to-infrastructure communication The problem is modeled as a mixed integer programming which is equivalently transformed into optimal control problems by relaxing the integer constraint Then the direct adjoining approach is used to solve both free fixed terminal time optimal control problems subject to state constraints By an elaborate analysis we are able to produce a real-time online analytical solution distinguishing our method from most existing approaches based on numerical calculations Extensive simulations are executed to compare the performance of autonomous vehicles under the proposed speed profile human driving vehicles The results show quantitatively the advantages of the proposed algorithm in terms of energy consumption travel time I INTRODUCTION The alarming state of existing transportation systems has been well documented For instance in 14 congestion caused vehicles in urban areas to spend 69 billion additional hours on the road at a cost of an extra 31 billion gallons of fuel resulting in a total cost estimated at $16 billion [1] From a control optimization stpoint the challenges stem from requirements for increased safety increased efficiency in energy consumption lower congestion both in highway urban traffic Connected automated vehicles CAVs commonly known as self-driving or autonomous vehicles provide an intriguing opportunity for enabling users to better monitor transportation network conditions to improve traffic flow Their proliferation has rapidly grown largely as a result of Vehicle-to-X or VX technology [] which refers to an intelligent transportation system where all vehicles infrastructure components are interconnected with each other Such connectivity provides precise knowledge of the traffic situation across the entire road network which in turn helps optimize traffic flows enhance safety reduce congestion minimize emissions Controlling a vehicle to improve energy consumption has been studied extensively eg see [3] [4] [5] [6] By utilizing road topography information an energy-optimal control algorithm for heavy diesel trucks is developed in [5] Based on Vehicle-to-Vehicle VV communication a minimum energy control strategy is investigated This work was supported in part by NSF under grants ECCS IIP CNS by AFOSR under grant FA by DOE under grant DOE-461 by Bosch the MathWorks The authors are with the Division of Systems Engineering Center for Information Systems Engineering Boston University Brookline MA 446 USA {xymengcgc}@buedu in car-following scenarios in [6] Another important line of research focuses on coordinating vehicles at intersections to increase traffic flow while also reducing energy consumption Depending on the control objectives work in this area can be classified as dynamically controlling traffic lights [7] as coordinating vehicles [8][9][1][11] More recently an optimal control framework is proposed in [1] for CAVs to cross one or two adjacent intersections in an urban area The state of art current trends in the coordination of CAVs is provided in [13] Our focus in this paper is on an optimal control approach for a single autonomous vehicle approaching an intersection in terms of energy consumption taking advantage of traffic light information The term ECO-AND short for Economical Arrival Departure is often used to refer to this problem Its solution is made possible by vehicle-toinfrastructure VI communication which enables a vehicle to automatically receive signals from upcoming traffic lights before they appear in its visual range For example such a VI communication system has been launched in Audi cars in Las Vegas by offering a traffic light timer on their dashboards: as the car approaches an intersection a red traffic light symbol a time-to-go countdown appear in the digital display reads how long it will be before the traffic light ahead turns green [14] Clearly an autonomous vehicle can take advantage of such information in order to go beyond current stop-go to achieve stop-free driving Along these lines the problem of avoiding red traffic lights is investigated in [15] [16] [17] [18] [19] The purpose in [15] is to track a target speed profile which is generated based on the feasibility of avoiding a sequence of red lights The approach uses model predictive control based on a receding horizon The authors in [16] studied an energy-efficient driving strategy on roads with varying traffic lights signals at intersections with the goal of avoiding a red light instead of following the host vehicle driven by a human Avoiding red lights with probabilistic information at multiple intersections was considered in [17] where the time horizon is discretized deterministic dynamic programming is utilized to numerically compute the optimal control input The work in [18] devises the optimal speed profile given the feasible target time which is within some green light interval A velocity pruning algorithm is proposed in [19] to identify feasible green windows a velocity profile is calculated numerically in terms of energy consumption Here we investigate the optimal control problem of autonomous vehicles approaching a traffic light where the objective function is a weighted sum of both travel time energy

2 consumption The problem is challenging due to the following reasons First finding a feasible green light interval leads to a Mixed Integer Programming MIP problem formulation In general solving MIP problems requires a significant amount of computation the optimality of the solution is not guaranteed due to the non-convexity of the problem involved with integer variables The second reason comes from state constraints related to speed limits The inclusion of bounds on state variables poses a significant challenge for most optimization methods To overcome the above difficulties we devise a two-step method Specifically we first address the problem without the traffic light constraint which means that the terminal time is free the mixed integer constraints are removed If the terminal time obtained from the free terminal time optimal control problem is within some green light interval then the problem is solved However if the terminal time falls within some red light interval then the optimal terminal time could be either the end of the previous green light interval or the beginning of the next green light interval by using the monotonicity property of the objective function Then we transform the original problem into a fixed terminal time optimal control problem We solve the fixed terminal time optimal control problem with two different terminal times comparing the corresponding performances leads to the optimal solution of the original problem All related optimal control problems with state constraints are solved by using the direct adjoining approach [] The main contributions of our paper are: Instead of solving this problem numerically as in [15] [17] an analytical solution is obtained For the free terminal time optimal control problem it is easy to characterize the type of the optimal acceleration profile Due to the on-line real-time nature of the algorithm the optimal control profile can be re-calculated as needed for example when the optimal trajectory is interrupted by other road users due to safety constraints The remainder of this paper is organized as follows The problem is formulated in Section II In Section III we present the methodology to solve the formulated problem where the solution to the free terminal time optimal control problem is described in Subsection III-A the solution to the fixed terminal time optimal control problem is presented in Subsection III-B Simulation results illustrating the use of the proposed algorithm are presented in Section IV Section V summarizes our findings concludes the paper provides directions for future work II PROBLEM FORMULATION The dynamics of the vehicle are modeled by a double integrator ẋ t = v t 1 v t = u t where x t v t u t are the position velocity acceleration of the vehicle respectively At time t the initial position velocity are given as x t = v t = v ft 1 O kt Fig 1 Traffic light signal kt +DT kt +T respectively Let us use l to denote the distance to the traffic light t p the intersection crossing time of the vehicle The traffic light switches between green red at an intersection are dictated by a rectangular pulse signal f t with a period T : { 1 for kt t kt + DT f t = for kt + DT < t < k + 1 T where f t = 1 indicates that the traffic light is green f t = indicates that the traffic light is red as shown in Fig 1 The parameter < D < 1 is the fraction of the time period T during which the traffic light is green k Z is a non-negative integer Our objective is to make the vehicle cross an intersection without stopping with the aid of traffic light information TLI as well as to minimize both travel time energy consumption Thus we formulate the following problem: Problem 1: ECO-AND Problem subject to min ut t p t + tp t u t dt x t p = l 5 v min v t v max 6 u min u t 7 kt t p kt + DT 8 for some k Z In 3 the term J t = t p t is the travel time while J u = t p t u t dt captures the energy consumption; see [1] In order to normalize these two terms for the purpose of a well-defined optimization problem first note that the maximum possible value of J t is l/v min Depending on the relationship between v min v max l there are two different cases for the maximum possible value of J u The first case is when the road length is long enough so that the vehicle can accelerate from v min to v max by using the maximum acceleration ie when l v max v min min v + 1 In this case J u = tp t t v max v min u maxdt = v max v min u max = v max v min The second case is when the road length is not long enough for the vehicle to accelerate to the maximum speed According to the dynamics 1 we have v min t p t + 1 t p t = l

3 3 By solving the above quadratic equation we are able to get v t p t = min + l v min Therefore in this case: J u = tp t u maxdt = v min + l v min We can now specify the two weighting parameters as follows: = ρ vmin l v if l v max v min 1 ρ min v max v min = + 1 v max v min 1 ρ otherwise v min +umaxl vmin capturing the normalized trade-off between the travel time energy consumption by setting ρ 1 When ρ = the problem reduces to minimizing the energy consumption only; when ρ = 1 we seek to minimize the travel time only In 6-7 the parameters v min v max > are the minimum maximum allowable speeds for road vehicles respectively while the parameters u min are the maximum allowable deceleration acceleration respectively Note that hen u < the vehicle decelerates due to braking when u > the vehicle accelerates Finally the integer constraint 8 reflects the requirement that t p belongs to an interval when the light is green see Fig 1 III MAIN RESULTS Problem 1 is a Mixed Integer Programming MIP problem Existing approaches to such problems turn out to be computationally very deming for off-line computation not to mention obtaining analytical solutions in a real-time on-line context We propose a two-step approach which allows us to efficiently obtain an analytical solution on-line The first step is to solve Problem 1 without the integer constraint 8 If the optimal arrival time t p is within some green light interval then the problem is solved However if kt + DT < t p < kt + T for some k then we solve Problem 1 twice with the constraint 8 replaced by t p = kd +DT t p = kt +T respectively We compare the performance obtained with different terminal times the solution produced by the one with better performance naturally yields the optimal solution Let us first introduce a lemma which will be used frequently throughout the following analysis Lemma 1: Consider the vehicle s dynamics 1 with the initial conditions x v If the control input u t = u is constant during the time interval [t t 1 ] then v t 1 = v + u t 1 t x t 1 = x + v t 1 t + 1 u t 1 t J u = u t 1 t ; If the control input u t = u t 1 t with a constant u then v t 1 = v + 1 u t 1 t x t 1 = x + v t 1 t u t 1 t 3 J u = 1 3 u t 1 t 3 The proof is given in Appendix A In the following we first seek the optimal solution to Problem 1 without the constraint 8 which is termed free terminal time optimal control problem A Free Terminal Time Optimal Control Problem The free terminal time optimal control problem is given below Problem : Free Terminal Time Optimal Control Problem subject to min ut t p t + tp t u t dt x t p = l 11 v min v t v max 1 u min u t 13 where are given in Section II From the objective function 9 it can be seen that a minimum energy consumption solution should avoid braking that is u t for t [t t p ] We will show this fact in the following lemma Lemma : The optimal solution u t to Problem satisfies u t for all t [ t tp] The proof is given in Appendix B In addition it follows from this lemma that whenever v τ = v max which may not be possible in some cases we must have u t = for all t [τ t p ] Based on these observations we can derive necessary conditions for the solution to Problem which are summarized in the following theorem Theorem 1: Let x t v t u t t p be an optimal solution to Problem assume that Then the optimal control u t satisfies u t = arg min u + ut v t τ u 14 t p where τ is the first time on the optimal path when v τ = v max if τ < t p ; τ = t p otherwise Proof: Here we use the direct adjoining approach in [] to obtain necessary conditions for the optimal solution u t t p The Hamiltonian H v u λ Lagrangian L v u λ µ η are defined as H v u λ = u + + λ 1 v + λ u 15 L v u λ µ η =H v u λ + µ u + η 1 v min v + η v v max 16

4 4 respectively where λt = [λ 1 t λ t] T ηt = [η 1 t η t] T µ t µ t [u t ] = 17 η 1 t η t η 1 t [v min v t] + η t [v t v max ] = 18 Note that we did not include the constraint u t u min since we have already established that the optimal control u t in the free terminal time optimal control problem in Lemma Let us temporarily assume that both According to Pontryagin s minimum principle the optimal control u t must satisfy u t = arg min H v t ut λ t 19 ut which allows us to express u t in terms of the costate λ t resulting in { u t = min λ } t with λ t due to Lemma The Lagrange multiplier µ t is such that L u = u t + λ t + µ t = 1 u=u t Since we can always find µ t to make 17 1 hold under the minimum principle 17 1 can be considered as redundant conditions For the costate λ 1 t we have λ 1 t = L t = x which means λ 1 t = λ 1 is a constant The costate λ t satisfies λ t = L t = λ 1 + η 1 t η t v First let us use a proof by contradiction to show that if v t = v min then t = t Assume that v t = v min for t t Then we must have v t = v min for all t [ t tp] This is because acceleration always precedes cruising at constant speed in the optimal control profile If not the vehicle would travel a longer time for the same trip using the same amount of energy According to the system dynamics in u t = for all t [ t tp] Based on the minimum principle λ t = for all t [ t tp] From 18 we know that η t = for all t [ t tp] Since the terminal time tp is unspecified there is a necessary transversality condition for t p to be optimal namely H v t p u t p λt p = that is u t p + ρt + λ 1 v t p + λ t p u t p = 3 Since u t p = we must have λ1 < according to 3 Then we obtain λ t > from which contradicts λ t = for t [ t t p] We have thus established that if v t = v min then t = t Next we will show that λ t has no discontinuities Since it is impossible that v t = v min for t t the costate trajectory λ t may jump only at some time τ when v τ = v max The condition H τ = H τ + can be written as u τ + λ τ u τ = u τ + + λ τ + u τ + 4 where τ + τ denote the left-h side the righth side limits respectively We know from Lemma that u t = for t [ τ t p] Therefore from 4 we obtain u τ + λ τ u τ = 5 According to we either have u τ = λτ or u τ = When u τ = λτ 5 becomes u τ = which implies u τ = λ τ = When u τ = 5 becomes = λ τ which contradicts the condition where = λτ Therefore only the case u τ = λ τ = is possible In other words the costate trajectory λ t has no discontinuities the following jump conditions: λ τ = λ τ + ζ 1 τ + ζ τ 6 ζ 1 τ ζ τ ζ 1 τ [v min v τ] + ζ τ [v τ v max ] = 7 are always satisfied with ζ 1 τ = ζ τ = Next we will show that λ t p = At the terminal time t p the following transversality conditions hold: that is where λ t p γ 1 γ = γ 1 v [v min v] v=v t p +γ v [v v max] v=v t p λ t p = γ1 + γ 8 γ 1 [ vmin v t p] + γ [ v t p vmax ] = 9 If v min < v tp < vmax then γ 1 = γ = which leads to λ t p = by the continuity of λ t When v tp = vmax then u tp = which results in λ t p = according to Last we will show that η 1 t = { for t [t τ η t = λ 1 for t [ ] τ t p Since H v u λ is not an explicit function of time t it follows that dh t = dt

5 5 that is [ u t + λ t] u t+[η 1 t η t] u t = 3 The first term [ u t + λ t] u t is always zero since when u t u t+λ t = according to when u t = u t = The condition 3 can thus be reduced to [η 1 t η t] u t = 31 When v = v min we have η t = from the fact that if v t = v min then t = t shown earlier from 18 Condition 31 then implies η 1 t u t = Since u t > we can get η 1 t = For t t η 1 t = since vt > v min for t t Therefore for any v we have η 1 t = It is easy to get from 18 that η t = for t [t τ For t [ τ t p] η t = λ 1 satisfies the condition 18 λ t = in Based on the above observations the differential equation becomes λ t = λ 1 3 for t [t τ From 3 we have λ 1 = ρt v t p since u t p = Solving the differential equation 3 we have λ t = v t τ 33 t p for t [t τ] In the case that v tp < vmax we simply let τ = t p in 33 The proof is completed by substituting 33 for λ t in 19 Recall that the theorem was proved under the assumption that The special cases when either = or = are considered in the following two corollaries Corollary : Let x t v t u t t p be an optimal solution to Problem when = Then the optimal control u t satisfies u t = 34 for all t [t t p] Corollary 3: Let x t v t u t t p be an optimal solution to Problem when = Then the optimal control u t satisfies { u umax for t [t t = τ for t [ ] τ t 35 p where τ is the first time on the optimal path when v τ = v max The proofs of the above two corollaries are straightforward by setting = = respectively in 14 in Theorem 1 Based on the vehicle dynamics 1 the initial conditions x t = v t = v the terminal condition x t p = l the optimal control law 14 the optimal time t p can be uniquely determined In the following we will classify the results into different cases dependent on the values of the model parameters In order to do so we define two functions: f v = l v max v vmax g v = l v v max v v max 4 3 v max v u3 maxv max v max v v max ρ u ρ t Depending on the signs of these two functions the optimal solution consisting of u t t p can be classified as shown in Table I with all detailed calculations provided in Appendix C Referring to this table the optimal control is parameterized by the following function when t a Φ t a b c = ct b when a < t < b when t b The parameters shown in Table I are defined as follows: 1 u max ρu v max v t 1 = t + t 3 = t + v 1 v u max t = t 1 + v max t 4 = t + v max v v max where l + v v 1 = ρu t max ρt u 4 ρ 1 ρu t max u ρ t v is the solution of the following equation: l = 3 v + v v v v The parameters δ 1 δ δ 3 δ 4 specifying in Table I the optimal time t p when the vehicle arrives at the traffic light in each of the four possible cases are given below: δ 1 = t + f v v max v 1 δ = t ρu u max δ 3 = t 4 + gv v max δ 4 = t + v v v Remark 1: This remark pertains to the underlying criteria for the optimal solution classification in Table I The first row determines whether or not the maximum acceleration will be used for a given initial speed v The optimality conditions tell us that the vehicle starts with the maximum acceleration when the initial speed is relatively slow The second row determines if the road length l is large enough for a vehicle to reach its maximum speed for a given initial speed v In general the optimal control contains three phases: full acceleration linearly decreasing acceleration no acceleration The first column specifies the case where all three phases are included with switches defined by t 1 t The second

6 6 TABLE I OPTIMAL SOLUTION CLASSIFICATION FOR PROBLEM u Φ t t 1 t v < 1 u v max max ρu v max 1 u max ρu v v max f v f v < g v g v < Φ t t 3 t t 4 Φ t v 1 1 Φ v max v t p δ 1 δ δ 3 δ 4 column corresponds to the case of low initial speeds shortlength roads Under optimal control in this case the vehicle starts with full acceleration but the road length is so short that the maximum speed cannot be reached Therefore the optimal control contains only the first two phases The third column corresponds to the case of large initial speeds long-length roads The vehicle starts with linearly decreasing acceleration then proceeds with no acceleration when the speed reaches the limit v max Here the optimal control contains only the last two phases The last column corresponds to the case of large initial speeds short-length roads Therefore the vehicle uses only linearly decreasing acceleration B Fixed Terminal Time Optimal Control Problem In this section we consider the case where the optimal time t p obtained in the free terminal time optimal control problem is within some red light interval that is kt + DT < t p < kt + T In this case the cidate optimal arrival time t p in Problem 1 is either kt + DT or kt + T Therefore we can compare the performance obtained under either one of these two terminal times select the one with better performance to determine the optimal arrival time for Problem 1 In both cases the travel time is now fixed hence the only objective is to minimize the energy consumption Thus we have the following problem formulation: Problem 3: Fixed Terminal Time Optimal Control Problem subject to min ut tp t u t dt x t p = l 38 t p = kt + DT or kt + T 39 v min v t v max 4 u min u t 41 1 Arrival Time t p = kt +DT : In this case it is clear that that the vehicle must use less time than the one specified by t p in Problem higher acceleration Define a function { v t p + 1 t p l for t p vmax v h v = v max t p 1 v max v l for t p > vmax v Observe that the terminal time t p = kt + DT is possible if only if h v The main result for this case is given in the following theorem Theorem 4: Let x t v t u t be an optimal solution to Problem 3 with t p = kt + DT Then the optimal control u t satisfies u t = arg min u u t t τ u + ut v v t p + τ t u t where τ is the first time on the optimal path when v τ = v max if τ < t p ; τ = t p otherwise Proof: Similar to the proof of Theorem 1 we will use the direct adjoining approach [] to solve the fixed terminal time optimal control problem The Hamiltonian H v u λ Lagrangian L v u λ µ η are defined as Hu v λ = u + λ 1 v + λ u Lu v λ µ η = H + µ u + η 1 v min v + η v v max respectively where λt = [λ 1 t λ t] T ηt = [η 1 t η t] T µ t µ t [u t ] = η 1 t η t η 1 t [v min v t] + η t [v t v max ] = Note that as in the the proof of Theorem 1 u t for all t therefore the constraint u t u min is relaxed which implies that u t = arg min u + λ u 4 ut u t = min { λ } t λ t From the proof of Theorem 1 we know that µ t is a redundant variable λ 1 is a constant Let us first assume that l > v min t p Note that the case of l = v min t p cannot occur when t p = kt + DT however it may occur when t p = kt + T this case will be discussed later Again we can prove the fact that v t = v min happens only at t = t but without using the transversality condition as we did in the free terminal

7 7 time optimal control problem The property that λ t has no discontinuities also still holds The costate λ t satisfies λ t = λ 1 + η 1 t η t 43 Similarly we can show that η 1 t = 43 reduces to λ t = λ 1 44 for t [t τ λ τ = By solving the differential equation 44 we get λ t = λ 1 t τ 45 Again since the Hamiltonian is not an explicit function of time by the condition H t = H t p we have u t + λ 1 v t λ 1 t τ u t = λ 1 v t p 46 where the fact that λ t p = u t p = has been used From 46 we can obtain λ 1 = u t v t p + t τ u t p v t 47 For t [τ t p ] we can just let η t = λ 1 If v t p < v max then τ = t p in 46 The proof is completed by substituting λ 1 in 47 into 45 then λ into 4 Given the terminal time kt + DT the road length l the value of v can be classified into one of five cases as shown in Table II Note that if Case i is infeasible for some v the given parameters we can treat Ji u as infinity The TABLE II OPTIMAL SOLUTION CLASSIFICATION FOR PROBLEM 3 WITH t p = kt + DT Optimal Control Performance Case I u = umax u t = J u 1 Case II u t = v t p = v max J u Case III u t = v t p < v max J u 3 Case IV u < umax v t p = v max J u 4 Case V u < umax v t p < v max J u 5 performances associated with each case in Table II as well as the detailed calculations are given in Appendix D After obtaining the performance for each cases with t p = kt +DT we select the one with the smallest energy consumption that is kt +DT Ju = min {J1 u J5 u } with the corresponding optimal acceleration profile 1 The dash in Φ means that the variable t cannot reach the upper bound therefore that case is inapplicable here Similar explanations apply to other Φs defined in Table I Arrival Time t p = kt + T : In this case the vehicle must use less acceleration than in the free terminal time case Depending on the initial speed v there are three cases to consider First if l = v kt + T t then the vehicle can cruise through the intersection with the constant speed v without any acceleration Case VI in Table III The energy consumption in this case is If on the other h J u 6 = l > v kt + T t then the problem can be solved using the result of the case t p = kt + DT analyzed above Finally if l < v kt + T t then the vehicle must decelerate to reach the traffic light while in its green state Therefore the control input is only subject to the constraint u min u t The main result in this case is given in the following theorem Theorem 5: Let x t v t u t be an optimal solution to Problem 3 with t p = KT + T Then the optimal solution u t satisfies u u t τ t u t = arg min u u + min ut v t p v τ t u t where τ is the first time on the optimal path when v τ = v max if τ < t p ; τ = t p otherwise Proof: The Hamiltonian H u v λ the Lagrangian L u v λ µ η are defined as H u v λ = u + λ 1 v + λ u L u v λ µ η = H u v λ + µ u min u + η 1 v min v + η v v max respectively where λt = [λ 1 t λ t] T ηt = [η 1 t η t] T µ t µ t [u min u t] = η 1 t η t η 1 t [v min v t] + η t [v t v max ] = As before we do not include the constraint u t since we have already established in Lemma that u t According to Pontryagin s minimum principle the optimal control u t must satisfy u t = arg min H u v t u t λ t min ut which allows us to express u t in terms of the costate λ t that is { u t = max u min λ } t 48

8 8 with λ t The Lagrange multiplier µ t is redundant as before The costate λ 1 is a constant The co-state λ t satisfies λ t = L v = λ 1 + η 1 t η t First it is easy to see that v v min Let τ be the first time that v τ = v min then u t = for t τ Again since the Hamiltonian is not an explicit function of time by the condition we have H τ = H τ + u τ + λ τ u τ = 49 According to 48 we either have u τ = u min or u τ = λτ When u τ = u min the above equality becomes u min + λ τ u min = which contradicts the minimum principle 48; when u τ = λτ 49 becomes u τ u τ = Therefore only λ τ = u τ = is possible that is to say λ u have no discontinuities at τ At the terminal time t p the following costate boundary condition holds: λ t p = γ1 v [v min v] v=v t p + γ v [v v max] v=v t p that is λ t p = γ1 + γ γ 1 γ γ 1 [v min v t p ]+γ [v t p v max ] = At t p we know that v t p v max Thus γ = Likewise it is easy to obtain γ 1 = Therefore we have λ t p = Since the Hamiltonian is not an explicit function of time the condition dh t = dt implies that [u t + λ t] u t + [η 1 t η t] u t = Since the first term is always zero as before the above condition becomes [η 1 t η t] u t = When v = v max we have η 1 t = that is Recall that η t u t = λ t = L v = λ 1 + η 1 t η t Since λ 1 > then λ t must decrease Therefore u t < η t = for all t For t [t τ η 1 t = Therefore λ t = λ 1 for t [t τ For t [τ t p λ t = λ 1 + η 1 t = Solving the above differential equation we obtain for t [t τ By the condition we have that is λ t = λ 1 τ t 5 H t = H t p u t + λ 1 v + λ 1 τ t u t = λ 1 v t p λ 1 = u t v t p v τ t u t The proof is completed by substituting λ 1 into 5 then λ into 48 TABLE III OPTIMAL SOLUTION CLASSIFICATION FOR PROBLEM 3 WITH t p = kt + T Optimal Control Performance Case VI u t = v t = v J u 6 Case VII u t = u min v t p = v min J u 7 Case VIII u t = u min v t p > v min J u 8 Case IX u t < u min v t p = v min J u 9 Case X u t < u min v t p < v min J u 1 The classification of all possible solutions with t p = kt +T is shown in Table III The performances associated with each case in this table as well as the detailed calculations are given in Appendix E After obtaining the energy consumption from J6 u through J1 u we can select kt +T Ju = min {J6 u J1} u where Ji u can be treated as infinity if Case i is infeasible Finally we can compare the two performances obtained that is J kt +DT kt +DT = kt + DT + J J kt +T = kt + T + J u kt +T u determine the optimal performance to be the one with a smaller value

9 9 Speed m/s Acceleration m/s Speed m/s Acceleration m/s Distance m Distance m Fig Case I in Table I with v = Fig 3 Case III in Table I with v = IV NUMERICAL EXAMPLES We have simulated the system defined by the vehicle dynamics 1 associated constraints optimal control problem parameters with values given as follows The minimum maximum speeds are 78 m/s m/s The maximum acceleration deceleration are set to 5 m/s 9 m/s respectively The weights in 3 are set using ρ = 9549 that is = 133 = In this case the values 1 u max = 563 v m + v M v M = 566 are almost the same Thus if we romly generate the initial speed v from a uniform distribution on the interval [v min v max ] different initial speeds fall roughly equally into the two different cases in the first row in Table I The total cycle time for the traffic light is 6 s with different patterns We first test the optimal controller on a road of length m Figure depicts the case when the initial speed is relatively slow The vehicle starts with full acceleration when the speed limit is reached it switches to no acceleration The vehicle arrives at the traffic light within the first green light cycle When the initial speed is relatively large the vehicle should not start with full acceleration This is the case shown in Fig 3 In the last two figures the traffic light starts at a green state The following two figures show the case when the traffic light starts at a red state It can be inferred from the first two plots that the arrival time obtained from the free terminal time optimal control problem should be within the red light interval Figure 4 shows a case when the initial speed is slow The optimal arrival time obtained from the free terminal time optimal control is 1186 seconds However the traffic light in the first 4 seconds is red The optimal time for the vehicle to arrive at the intersection is 4 seconds The vehicle has adequate time to accelerate therefore it does not start with Distance m Speed m/s Acceleration m/s Fig 4 Case V in Table II with v = 4634 full acceleration it is unnecessary to accelerate to the maximum speed Figure 5 exhibits a different traffic light pattern where the traffic light in the first seconds is red Due to a relatively large initial speed the vehicle has to decelerate to cross the intersection when the traffic light is green In the following we test the optimal controller on a road of length 3 m Due to this length the optimal arrival time usually does not fall within the first green light cycle sometimes it is impossible for the vehicle to arrive at the traffic light within this cycle For the case in Fig 6 the optimal arrival time calculated from the free terminal time optimal control problem is seconds Unfortunately this arrival time belongs to a red light interval Therefore full acceleration is used to reach the speed limit cross the intersection at 1 seconds when the traffic light is green Figure 7 shows the case when the vehicle has a relatively fast initial speed compared to Fig 6 Therefore the vehicle does not start with full acceleration to reach the speed limit catch the green light at 1 seconds

10 1 Speed m/s Acceleration m/s Speed m/s Acceleration m/s Distance m Distance m Fig 5 Case X in Table III with v = Fig 7 Case IV in Table II with v = Speed m/s Acceleration m/s Speed m/s Deceleration m/s Distance m 1 Distance m Fig 6 Case II in Table II with v = Fig 8 Case X in Table III with v = For the last case in Fig 8 the initial speed is very large The best option is to decelerate the vehicle to cross the intersection at 1 seconds when the traffic light is green Exploring the time-energy tradeoff In order to compare the performance between i autonomous vehicles under the optimal control developed ii a human driver we arbitrarily define the following rules as the driving behavior of a human driver: Full acceleration when the traffic light is green; No acceleration/deceleration when the traffic light is red We calculate the performance of both autonomous vehicles human drivers for the different scenarios encountered from Fig to Fig 8 summarize the results in Table IV The improvement is more than 1% for the case in Fig 4 The performance improvement is calculated as the performance difference between the human driver autonomous vehicle divided by the performance of the human driver It is particularly challenging for a human driver to make a decision when he/she faces a steady red traffic light Also note that the weighting parameter ρ is chosen to be in favor of travel time rather than energy efficiency Therefore the performance improvement would be larger when we decrease the weighting parameter ρ which provides a trade-off between energy consumption travel time Figure 9 shows the travel time the energy consumption when we vary the parameter ρ from to 1 The initial speed is chosen as v = By exploring the trade-off curve one may select am appropriate weight parameter ρ depending on a particular application of interest For instance if energy efficiency is a major concern Fig 9 suggests to not select a large value for ρ since the energy consumption grows rapidly In this case the human driver approaches the intersection at red light with the speed We assume that the human driver is able to stop before the traffic light immediately In addition we did not consider the energy consumptions of sudden braking restarting the vehicle 3 In this case the human driver approaches the intersection with the maximum speed at red light We assume that the human driver is able to stop before the traffic light immediately from the maximum speed In addition we did not consider the energy consumptions of sudden braking restarting the vehicle

11 11 TABLE IV PERFORMANCE COMPARISON BETWEEN HUMAN DRIVER HD AND AUTONOMOUS VEHICLE AV Travel Time HD AV Improvement Fig % Fig % Fig % Fig NA Fig % Fig % Fig % 3 484% 485% from to 1 Fig 9 Trade-off between travel time energy consumption as ρ approaches 1 On the other h a small ρ is likely not a better option since we can see that energy consumption does not significantly increase with ρ increasing as long as ρ < 7 approximately In fact when ρ increases from to 7 the travel time is significantly reduced by 484% whereas the energy consumption increases by only 485% It is noteworthy that both curves show different trends around the circled area shown in Fig 9: this is mainly because the optimal control has included the full acceleration part when the parameter ρ is large V CONCLUSIONS This paper provided the optimal acceleration/deceleration profile for autonomous vehicles approaching an intersection based on the traffic light information which could be obtained from an intelligent infrastructure via VI communication The solution for the above problem had the key feature of avoiding idling at a red light Comparing with similar problems solved by numerical calculations we provided a real-time analytical Energy Cost solution The proposed algorithm offered better efficiency in terms of travel time energy consumption which has been verified through extensive simulations The simulation results showed that the algorithm achieved substantial performance improvement compared with vehicles with heuristic human driver behavior There are a few avenues available for extending this work In particular there is a need to consider a practical scenario where interferences from other road users present A possible way of doing this is to predict the driving behavior of vehicles ahead It is also desirable to develop a more general algorithm by taking into account traffic light information at multiple intersections APPENDIX A PROOF OF LEMMA 1 Let us first consider the case of constant control input By solving the differential equation it is straightforward to get the expression for v t 1 As a byproduct we have v t = v + t t u By solving the differential equation 1 it follows that x t 1 = x + t1 t [v + t t u] dt = x + v t 1 t + 1 u t 1 t The energy consumption J u is then easy to obtain Next let us consider the case that u t = u t 1 t for t [t t 1 ] Solving the differential equation we obtain v t 1 = v + t1 t [u t 1 t] dt = v + 1 u t 1 t As a byproduct we have v t = v + 1 u t t Solving the differential equation 1 yields t1 x t 1 = x + [v + 1 ] u t t dt t = x + v t 1 t u t 1 t 3 The energy consumption is then calculated as J u = t1 t u t 1 t dt = 1 3 u t 1 t 3 APPENDIX B PROOF OF LEMMA We will prove the result by a contradiction argument Let us assume that u t t p are the optimal control the optimal arrival time of Problem respectively In addition we assume that there exists an interval [t 1 t ] such that u t < Next we construct another control input u t such that u t = u t for t < t 1 u t = for t [t 1 t ] It is then straightforward to get v t 1 = vt 1 x t 1 = xt 1

12 1 We now invoke the comparison lemma [] which compares the solutions of the differential inequality vt ft v with the solution of the differential equation ut = ft u asserts that If v u then vt ut By applying the comparison principle to the dynamics of vt it follows that v t < vt 51 for t > t 1 until v t = v max By applying the comparison principle again to the dynamics of xt it follows that x t < xt for t > t 1 Then according to the terminal condition x t p < xt p = l we conclude that t p > t p therefore we have t p t < t p t 5 Let τ be the time when vτ = v max we assume that τ > t without loss of generality The remaining control input of ut is thus constructed as { u u t = t for t t < min {τ t p } for t min {τ t p } By using the inequality 51 at t = τ we have v τ < vτ = v max Recalling that u t < u t = for t [t 1 t ] it follows that tp t u t dt = < t1 t u t dt + t1 + t u t dt + min{τtp} t t p t u t dt min{τtp} t t t 1 u t dt u t dt u t dt The above inequality together with 5 contradicts the optimality of u t t p in 9 completes the proof by contradiction Therefore we conclude tha u t for all t [t t p] APPENDIX C CALCULATIONS FOR TABLE I Let us assume that A Case I: v t p = vmax Let us first find the time duration δ such that u t decreases from to while the speed increases from v to the maximum speed v max under the optimal control u t = 53 v max Integrating 53 on both sides yields = δ v max which can be simplified as δ = v max According to Lemma 1 we know that v max = v + u maxv max which can be written as v = 1 u max v max with the assumption that 1 u max v max v min For the same amount of time the distance that the vehicle travels is d = vmax 3 u3 maxvmax ρ u ρ t According to Theorem 1 the optimal control can be parameterized in terms of the speed vt as u t = u t = u t = ρt v max if v t 1 u max ρu v max if 1 u max ρu v max v t v max if v t = v max There are different cases depending on the relationship between the initial speed v the road length l Remind that the analysis is under the assumption that 1 u max v max v min 1 Case I1 v 1 u max ρu v max : The first column in Table I In this case the vehicle will first accelerate to v t 1 = 1 u max ρu v max using the maximum acceleration Then it will travel a distance d to reach v max At time t 1 we have x t 1 = v t 1 t + 1 t 1 t It is easy to figure out that 1 u max ρu v max v t 1 t = To achieve the maximum speed v max the road length l must satisfy l xt 1 + d = v max v + vmax 1 6 u3 maxvmax ρ u ρ t

13 13 Case I 1 u max ρu v max < v v max : The third column in Table I In this case the vehicle will not start with full acceleration we have u t = τ t v max where τ is the time when v τ = v max According to Lemma 1 we can obtain x τ = v τ t + τ t 3 6 v max v max = v + 4 v max τ t 54 We can calculate from 54 to get τ t = v max v v max 55 By using 55 a necessary condition for v t to reach the maximum speed v max is l v v max v v max v max v v max v v max 56 B Case II: v tp < vmax 1 Case II1 v < 1 u max ρu v max : The second column in Table I In this case the road length l < v max v + vmax 1 6 u3 maxvmax ρ u ρ t is not long enough for the vehicle to reach the maximum speed Let us assume that the speed when the acceleration starts to decrease at time t 1 is v According to Lemma 1 it takes the time t 1 t = v v for the vehicle to reach the speed v by using the maximum acceleration x t 1 = v v The speed v increases to v t p by using a linearly decreasing optimal control from to It is easy to get that u t = v t p Therefore the time for to decrease to is δ = v t p According to Lemma 1 we can obtain v t p = v + u max v t p which is v t v p = 1 ρu u max By the road length constraint we are able to calculate v from the equality l = v v + v 1 ρu u + 4 max 3 v ρ t u 3 max 1 ρu u max that is l + v v = ρu t max ρt + 8 u 4 ρ 3 1 ρu t max u ρ t Case II v > 1 u max ρu v max : The fourth column in Table I In this case the road length l < v v max v v max v max v v max v v max is not large enough for the vehicle to reach the speed limit the maximum acceleration will not be used either According to Theorem 1 the optimal control can be parameterized as u t = ρ t t p t v t p According to Lemma 1 we have v t p = v + ρ t t p t 4 v 57 t p l = v t p t + 6 v t 3 t p t 58 p By solving the equation 57 we can obtain v t p = v + v + ρt t p t By substituting for v t p we are able to obtain t p from 58 APPENDIX D DETAILED CALCULATIONS FOR TABLE II There are different cases depending on the initial speed v the time duration kt + DT the road length l A Case I: u = u t = This case corresponds to hv = The vehicle accelerates fully until it arrives at the traffic light or the maximum speed is reached According to Lemma 1 the vehicle reaches the maximum speed by spending time δ = v max v Depending on the values of kt +DT δ we have different energy consumptions { J1 u umax v max v if vmax v u = max < kt + DT u max kt + DT t if vmax v kt + DT For all other cases h v > u t for some t

14 14 B Case II: u t = v t p = v max The time t 1 is when the acceleration starts to decrease that is 1 u max τ t 1 = 59 v v max + τ t From 59 we can obtain According to Lemma 1 τ = v max v t 1 + t v t 1 = v + t 1 t x t 1 = v t 1 t + 1 t 1 t x τ = x t 1 +v t 1 τ t u max τ t v v max + τ t Therefore we have l = t p τ v max + x τ 6 We can solve the equation 6 to get t 1 The energy consumption can be expressed as J u = t 1 t u max + 1 u 4 max τ t [v v max + τ t ] C Case III: u t = v t p < v max In this case τ = t p First we need to find the time t 1 such that the acceleration starts to decrease that is 1 u max t p t 1 v v = t p + t p t By solving the above equation for v t p we can obtain v t p = v + t p + t 1 t 61 According to Lemma 1 the speed the distance of the vehicle at t 1 are v t 1 = v + t 1 t x t 1 = v t 1 t + 1 t 1 t respectively From the road length constraint l = x t 1 + v t 1 t p t t p t 1 6 we are able to calculate t 1 The energy consumption for this case can be expressed as J3 u = t p + t 1 3t 3 D Case IV u < v t p = v max In this case the vehicle reaches the maximum speed at τ According to Lemma 1 we have v max = v + 1 u t τ t 4 v v max + τ t u t 63 Solving the above equation for u t yields u t = v max v τ t With the expression of u t Lemma 1 we can obtain l = 1 3 v + v max τ t + t p τ v max 64 We can calculate τ from 64 as τ = 3l + v max + v t 3t p v max v v max The energy consumption in this case is expressed as J4 u = 4 v max v 3 τ t E Case V: u < v t p < v max In this case τ = t f According to Lemma 1 the final speed is v t p = v + 1 u t t p t 4 v v t p + t p t u t 65 From 65 we can get u t = v t p v t p t Using the expression of u t Lemma 1 we can obtain l = v t p t + 3 v t p v t p t 66 Solving the equation 66 we have v t p = 3 l v t p t + v t p t The energy consumption in this case can be expressed as J u 5 = 3 [l v t p t ] t p t 3 APPENDIX E DETAILED CALCULATIONS FOR TABLE III A Case VII: u t = u min v t p = v min In this case the vehicle starts with full deceleration u min then at time t 1 the deceleration linearly increases until it reaches zero at t = τ Therefore at time t = t 1 we have that is u min = 1 u min τ t 1 v + τ t u min v min v v min = t τ t 1 u min 67

15 15 According to Lemma 1 the speed travel distance of the vehicle at time t 1 are v t 1 = v + u min t 1 t x t 1 = v t 1 t + 1 u min t 1 t respectively At time τ we have x τ = x t 1 +v t 1 τ t u min τ t v + τ t u min v min To satisfy the road length constraint we must have We can solve 67 to obtain l = x τ + kt + T τ v min 68 τ = t t 1 + v min v u min 68 to get t 1 The energy consumption in this case can be expressed as J7 u = t 1 t u min + 1 u 4 min τ t [v + τ t u min v min ] B Case VIII: u t = u min v t p > v min In this case τ = t f The vehicle starts with full deceleration u min at time t 1 the deceleration starts to increase Similarly we have v v t p = t t p t 1 u min 69 According to Lemma 1 we know that v t 1 = v + u min t 1 t x t 1 = v t 1 t + 1 u min t 1 t Solving 69 we can get v t p = v + t 1 + t p t u min Using the expression of v t p we can obtain t 1 by solving the following equation l = x t 1 + v t 1 t p t u min t p t 1 7 The energy consumption in this case can be expressed as J8 u = u min t p + t 1 3t 3 C Case IX: u t < u min v t p = v min In this case the vehicle starts with linearly increasing deceleration until it reaches the minimum speed v min According to Lemma 1 we have v min = v + 1 u t τ t 4 v + τ t u 71 t v min Solving u t in 71 yields u t = v min v τ t According to Lemma 1 the expression of u t the distance of the vehicle at time τ is given as x τ = 1 3 v min + v τ t Then we can solve τ from the following equation that is l = x τ + KT + T τ v min 7 τ = 3l + v min + v t 3t p v min v v min The energy consumption in this case can be expressed as J9 u = 4 v min v 3 τ t D Case X: u t < u min v t p < v min In this case τ = t p The optimal control only contains the linear increasing deceleration process According to Lemma 1 we have v t p = v + 1 u t t p t 4 v + t p t u t v t p 73 l = v t p t + 1 u t t p t 3 6 v + t p t u t v t p 74 We can solve u t v t p from to obtain u t = v t p v t p v v t p = 3 l v t p t + v t p t The energy consumption in this case is J u 1 = 3 [l v t p t ] t p t 3 REFERENCES [1] D Schrank B Eisele T Lomax J Bak 15 urban mobility scorecard Texas A&M Transportation Institute INRIX Tech Rep 15 [] L Li D Wen D Yao A survey of traffic control with vehicular communications IEEE Trans Intell Transport Syst vol 15 no 1 pp [3] E G Gilbert Vehicle cruise: Improved fuel economy by periodic control Automatica vol 1 no pp [4] J Hooker Optimal driving for single-vehicle fuel economy Transportation Research Part A: General vol no 3 pp [5] E Hellstrom J Aslund L Nielsen Design of an efficient algorithm for fuel-optimal look-ahead control Control Engineering Practice vol 18 no 11 pp [6] S E Li H Peng K Li J Wang Minimum fuel control strategy in automated car-following scenarios IEEE Transactions on Vehicular Technology vol 61 no 3 pp [7] J L Fleck C G Cassras Y Geng Adaptive quasi-dynamic traffic light control IEEE Trans Control Syst Technol vol 4 no 3 pp [8] V Milanes J Perez E Onieva C Gonzalez Controller for urban intersections based on wireless communications fuzzy logic IEEE Trans Intell Transport Syst vol 11 no 1 pp [9] J Alonso V Milans J Prez E Onieva C Gonzlez T de Pedro Autonomous vehicle control systems for safe crossroads Transportation Research Part C: Emerging Technologies vol 19 no 6 pp

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