I Finite State Approach to Vehicle Merging

D. E. IHiTlEY du Pont Associate Professor in Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. Assoc. Mem. ASME I Finite State Approach to Vehicle lirgiig 1 A stati space approach is taken to vehicle merging on high speed highways. The vehicles ate assumed to be traveling in equal sized "slots" which move at the group velocity. At points where two or more lanes merge, some vehicles must be moved forward or backward to other slots to accomplish the merge. The state of a group of vehicles to be merged is defined in terms of the slots occupied at any time. A finite set of admissible terminal stales, representing possible merged configurations, is easily determined. The sequence of moves required to obtain a merge is found as a shortest path in the space of all states, running from the initial state to the terminal manifold. Various costs may be applied to moves in this space, such as time consumed, or number of vehicles being moved simultaneously. Costs may also be assigned to the terminal arrangements, reflecting, for example, the size of platoons in the resulting merge. Estimates are made of required computing load and the method is compared with other approaches. Introduction H T THE merge points in future automated highways, some method of rearranging the vehicles to allow merging will be necessary. The approach one takes to this problem is influenced substantially by the operational characteristics assumed to be in effect away from the merge. If one assumes that vehicles nearing a merge could have a variety of positions and velocities, one concludes that velocities as well as positions must be controlled by a. central algorithm during merging. This led Athans [1| 2 to describe merging in terms of a first order vector differential equation and to extract merging action as an optimal control law based on a quadratic penalty. In his treatment, a state vector of dimension 2n is required, where n is the number of cars being merged. Real time communication of position and velocity information between ail the merging cars is also required. If one assumes, by contrast, that the vehicles in normal cruising are constrained by on-board autonomous control loops to travel at or near group velocity V in imaginary "slots" of length A,3 then one tends to ignore velocities and think of merging as pure rearrangement. Vehicles may have to be •Work supported in part by DOT, OHicc of High-Speed Ground Transportation, contract C-85-65. ^Numbers in brackets designate Reierenees at end of paper. "The advantages of this approach for maintenance of longitudinal control are well known [8, 9], Contributed by the Automatic Control Division of T H E AMERICAN SOCIETY OF MECHANICAL ENGINEEHS and presented at the Joint ASCE-AS±\IK Transportation Engineering Conference, Seattle, Wash., July 26-30, 1971. Manuscript received at ASME Headquarters, Marcii 13, 1972. shifted from slot to slot in order to accomplish the merge. The slots are synchronized by a signal traveling along the roadway [2, 6], A predetermined velocity program in each car is called into action to direct any shift called for by a central control station, which also monitors progress. These assumptions led Godfrey [3| to investigate sequential merging strategies for resolving conflicts as they came up at a merge. He proposed and tested several decision rules for granting right of way to one or the other of two cars arriving simultaneously and seeking to enter the same downstream slot. Inter-vehicle communication is not required. Examining these two approaches, we find that neither gives the designer direct control over how the vehicles shall move during merging and how they shall end up. Athans explicitly penalizes the possibility of collisions, wide variations in velocity, and control effort. Godfrey judges his strategies by how they avoid shifting any car too many slots. This latter cost will be shown in the following to be expressible as a terminal cost only, leaving one free to pick such path costs as the number of vehicles being shifted simultaneously. Athans' approach is continuous, in the sense that positions and velocities of vehicles are monitored continuously in time, and central control is being exerted continuously during the merge. Godfrey's approach is discrete in that the central control calls only for cars to move an integer number of slots, and makes such calls only at isolated times. On-board controllers in each car exert controls to accomplish the moves. The merging method described in the following is discrete. A finite group of vehicles to be merged is identified and described by a state vector. Terminal "merged" states are enumerated and a merging control sequence is obtained as a shortest path in the space of all possible slates, the path running from the initial state to the terminal manifold. The motivation for this approach is to be found in reference [7], Journal of Dynamic Systems, Measurement, and Control JUNE 1 9 7 2 / 147 Copyright © 1972 by ASME Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 11/02/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use MERGE POINT JSLLJSLL U-A ~4~ A -4— A -4«- A —j Fig. 3 Transition diagram for either lane in Fig. 1 Fig. 1 Two lanes of vehicles approaching merge STATE OF LANE 2 (decimal equivalent! • -POSIBLE MERGE STATE Fig. 4 Transition diagram for either lane, showing cost equal to number of car-slots moved -PRESENT STATE <B © ® £> 3 5 6 9 INITIAL STATE 10 STATE OF LANE I (decimal equivalenr) 12 POSSIBLE MERGE STATE Fig. 2 State space corresponding to Fig. 1 Discrete State Model for Merging Consider two lanes of vehicles approaching a merge, as in Fig. 1. There is a conflict in the second slot upstream of the merge. Define the state of a lane as follows. Construct a binary number whose highest order bit represents the slot nearest, the merge point. This bit is unity if the slot is occupied and zero if not. Proceed upstream similarly. The resulting binary number uniquely describes the locations of vehicles and empty slots and is convenient for computer representation. For the vehicles in Fig. 1, we have state of lane 1 = 1100 = 12 w (1J state of lane 2 = 0110 = 6I0 (Subscript 10 denotes decimal notation.) Assuming that we consider only the four slots nearest the merge point in each lane, the set of allowed states for lane 1 is the same as for lane 2: (in decimal notation) 3, 5, 6, 9, 10, 12 (2) In Fig. 2 we show the state space for these two example lanes, with the present state (12, 6) indicated. The terminal states which allow merging are on the antidiagonal as shown. These states satisfy the following relation: state (si, s2) is a merge state if and only if Si + s, = 15io (3) Here, 15 is the largest value Si or s2 could have if all four cars were in a platoon in one lane, as they will be after being merged. This result obviously generalizes to any such space: For a space which represents merging N cars into a single solid platoon, a state (si, Sa) is a possible merge state if and only if s, + s , = 2-v 1 (4) Of course, it is not necessary to merge vehicles into a, solid platoon and there may indeed be good reasons not to. Platoons will be assumed, however, in what follows to simplify the discussion. An allowed transition in such a space is defined to be one in which any number of cars move one slot each, all in the same direction. Other definitions are possible, but this one will do for illustration. Then we may draw a transition diagram for each of the lanes in Fig. 1 (see Fig. 3). The numbers inside the circles correspond to states. Using this diagram we can draw 148 / J U N E 19 7 2 Fig. 5 Portion of Fig. 2 showing allowed transitions and costs based on number of car-slots moved transition lines on the state space. Aside from cluttering the figure, this would be incomplete until we added lines representing simultaneous moves in both lanes, which are allowed in our definition. Complete examples will be shown in the following. Path Costs and Termina! Costs We can do a lot of things with the foregoing transition diagrams. Assume first that we assign a cost to each transition equal to the total number of slots traversed by all the cars moving in that transition. These costs are shown in Fig. 4. If we apply these costs to Fig. 2 and allow no diagonal moves in the state space,*1 then it is clear that, all shortest paths from initial state (12, 6) to the terminal set are of cost = 2, and that there are three terminal states so reached: (9, 6), (10, 5), and (12, 3). If we allow diagonal transitions and are concerned solely with the total number of slots moved by all the cars involved in each transition as a cost,3 then each diagonal move must cost the same as the shortest nondiagonal path which accomplishes the same maneuver. The relevant section of Fig. 2 is shown in Fig. 5 to represent these facts. Clearly all shortest paths to the terminal set have the same length, illustrating the fact proved by Godfrey [3], that "total delay" in merging is independent of merging strategy. Thus we could make this cost a terminal cost, rather than a path cost if we wished. A more interesting terminal cost is found by squaring the net number of slots moved by each car during the entire merge, and summing over all cars moved. For the foregoing example, this cost is 4 for terminal state (9, 6) since one car moves 2 slots in the transition from (12, 6) to (9, 6). Similarly, this cost is 2 for terminal states (12, 3) and (10, 5) since here two cars move one slot each. This cost criterion tends to favor merges in which no vehicle is moved a long distance compared to distances 4 A diagonal move in a space such as that in Fig. 2 is one which is not parallel to either axis. A move parallel to one axis represents motion of cars in one lane only, while a diagonal move represents simultaneous motion of cars in both lanes. s Godfrey calls this the total dtlay [3]. Transactions of the ASME Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 11/02/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use LANE a X—X —s<-^< INITIAL STATE X X s x Fig. 7 Sequence of vehicle positions corresponding to shortest path from (12, 6) to (10, 5) in Fig. 6 [2l-TERMINAL COST Fig. 6 State space w i t h costs assigned to penalize m o v i n g p l a t o o n s and favor diagonal moves moved by the others. This is one of the criteria Godfrey uses to judge his merging strategies. Another useful terminal cost comes into play if we allow more than enough room in which to place the cars. Then we may distribute empty slots as we wish. Usually it is desirable to avoid merging vehicles into long platoons, so we could penalize terminal states according to the iength of platoons in them. Terminal costs may be used exclusively to pick a terminal state, as Godfrey does, or they may be combined with path costs to pick both a path and a terminal state, as Athans does, or terminal costs may be used to pick a terminal state while path costs are used to find a desirable path to that slate. In the first instance one does not care what maneuvers the cars go through while coming to a merge state. In the second instance one must choose a weight to combine the two kinds of costs before a p a t h and terminal state can be chosen. In the third instance one may pick the terminal state for purely configurational reasons and pick a path to that state for reasons based only on maneuvering. Since these two considerations, maneuvering and final configuration, do not bear on each other, the third method is a natural one. It is available to us using the methods discussed here. Possible maneuvering considerations include: 1 Avoid moving platoons—instead move one car at a time. This might improve reliability. Less monitoring is needed and slight differences in the vehicles' responses will not matter very much; 2 Achieve the merge state in the fewest number of transitions—this means move along diagonals in state space which tends to minimize the time needed to merge; •3 Remove the conflict nearest the merge first. This is also a reliability consideration. To achieve the first of these, one method is to make the cost of each state transition equal to the square of the total number of slots moved by all the cars during that transition. To achieve the third, we note that assigning the high order bit to the slot nearest the merge point guarantees that conflict states near the merge point will be concentrated in one easily identified corner of state space. For the space of Fig. 2, these head-end conflict states satisfy ,S'i and s-> > 9 simultaneously 0'') All transitions in this region can then be assigned extra, high costs. The result is that all shortest paths will leave this region in as few transitions as possible, with subsequent maneuvers being determined largely by other costs. If should be noted that resolution of head-end conflicts is the only consideration discussed which requires simultaneous examination of maneuvers in both lanes. In other cases, the lanes may be considered separately. To do this, we use a condition like (4) to enumerate the possible merge states. Then for each pair (initial state, terminal state) we consider paths in two Fig. 8 Transition diagram with costs to favor moving platoons transition diagrams, one diagram for each lane. For example, take initial state (12, 0) and terminal state (10, .">). Then we find the minimum cost path from 12 to 10 in one transition diagram and from 6 to o in the other, add these costs to the terminal cost on (10, 5) and compare with similar results using the other terminal states. The lowest: overall cost; wins and designates both the terminal state and each lane's move sequence. Since the lanes are being dealt with simultaneously, maneuvering consideration (2) in the foregoing is achieved automatically. This sequential solution procedure uses much less computer memory than the overall optimal path approach employing the full state space and usually gives optimal or near-optimal results. As an example, assume that the cars are arranged as in Fig. 1, and that we wish to avoid moving platoons and avoid moving any cars excessively long distances. To counteract the time lost, by not moving platoons, let us favor diagonal moves. Then we could assign the costs as in Fig. 6. Note in particular the high cost of moving, for example, from (!), 6) to (9, 3) in one transition (by moving a two car platoon) compared with the cost of accomplishing the same net transition in two steps in which one ear is moved at each step. It is clear that (10, 5) wins with total cost 2 + \ / 2 - The sequence of vehicle positions is shown in Fig. 7. By contrast, suppose that we wished, in the interest of short, merging time, to favor moving platoons but, in the interest of computer memory requirements, decided to work only with transition diagrams, and used terminal penalties as before. Then we could assign costs as in Fig. S. Note in particular the low cost of moving from state 6 to state 3 in one transition (via a two car platoon) compared with the cost of doing so in two transitions which move one car each. Now we find that (12, 3) wins with a total cost of 2 + V 2 . Had we used the full space we would have discovered that (10, 5) gives the same cost. The example shows that good results, though not necessarily the best, can be obtained by considering the much more compact transition diagrams. Implementation Strategy The proposed merging method operates by looking upstream and planning ahead the required moves. This can be implemented in the following way. Look upstream from the last set of merged cars until a conflict is detected. Count upstream from there, totaling cars in both lanes and distance in slots. When the number of cars equals the number of slots (call the number A") then these K cars can be conceptually isolated and merged into a solid platoon merely by rearranging them within their region of highway, the region being A' slots long. If there are U cars in this stretch of lane 1 and L, cars in the opposite stretch of lane 2 (/,, + L% = K) then the transition diagram for lane 1 contains K\/((Li)\(K - U)\) states and that for lane 2 has K\/((U)KK — £•>)!) states. These Journal of Dynamic Systems, Measurement, and Control J u N E I 9 7 2 / 149 Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 11/02/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use (START) Fig. 9 group THIS PATTERN REPEATS INDEFINITELY Ifi THIS ' J ^ C T I O N M a r k o v process r e p r e s e n t i n g counting back to find a m e r g e numbers are maximum when L, = L« = K/2, so that K is a measure of the computing load required to merge. That is, in states, more that case the space would have \K\/{(K/'i)\)'!-f than for other values of Li and LL. The number of states indicates the amount of computer memory required to store the space and the amount of time required to find the shortest path. See [4] for an efficient pathfinding algorithm. To obtain an estimate of this computing load, let us assume that cars are placed in each lane by independent Bernoulli processes with occurrence probability ]>i for lane I and p-> for lane 2. That is, there is a -car in a slot in lane i with probability Pi independent of all other slots in either lane, and so on. Clearly we require f>\ + p-. < 1 in order for merging to be possible on the average. Define = Pi'l- - p -o,2 A p -0.4 o P .0.6 x p -o.8 J ALL FOR p.-p. _1 1 5_ F i g . 10(a) Pi + P-i «i D P r o b a l i t y m a s s f u n c t i o n f o r K f o r v a r i o u s v a l u e s of p P'i'l = Ih'fh 22 u S u' a lie • e a a •\f ' a , a (1 <a w L Js s V a S e V, P - a e e el a s c ) 1 I , e | e e j k ,0 s a e h,. 20 1 8 6 • i e •H »e e e e e is-, i Aa s a < 14 s s a a j 'e n e we s ai a j i " — e n e >eg g e - i we a M S e i g> •- a e 12 it si i i s t n !1 a a i --s e esp a e T ] Wf go es a e / — > s % a s oe ' a s e ess w c — e we i i i" ue i a I we t i t i s>a e = e i' s c e ai k a -. e a'e je a i i id A — i -. ne i a co i J i e e n ea i A" a -IJ u e, a i / A w » a j ' as pa an e e s s i g fl w gij h e i e-. I <• esi ii ' i A" e x*i i / w e uc 1 e ie-.ui s depend s i< g ' a i l wejk i a '> 0 1 0.2 C3 04 0.5 OS 0.7 0.8 09 1.0 P, +P 2 sepiua p 1 s; ' -awa s • i j I J I ge •>< ig s \\s A" a i :i o /v as iF i g . 10{o) M e a n K a n d s t a n d a r d d e w i a t i o n <rK v e r s u s Pi + PFhus i a ' o e a i s a ie-s he a - u e u< se MI , I — p, ^ ' i s A" v\ i 1 p ai I e ess JI i • js -ft e ' >t in sc A e n n hi ge e i g in e no e ' s i< i ' e a.a-'a e n >e s w e A is •, a ( I e a previously proposed methods but has some advantages ver oe bo i we-' ' s wa e a se i e i< k auea< i" e i i u each. As with the others, it will work well if traffic deiisoy is procedure. not too high. It is easy to vary this procedure to include interspersed empty Some advantages of this approach are: solts in the merged group. If we wish in empty slots (in > 0), 1 A group of vehicles to be merged may be identified as far we keep counting umil « = in. This obviously will increase A_\ in advance of the merge point as desired. This provides the The terminal states may be identified in a similar manner ami system with valuable look-ahead characteristics which can use those with the spaces not desirably distributed may be eliminated computer time advantageously and coordinate with overall by giving them very large terminal cost. network scheduling; 2 Regardless of the number of cars to be merged, the diSummary mension of the state space is two; '•> Merging strategies may be chosen to satisfy a varieU of requirements. Desirable features of the final merged conallows a designer considerable freedom in choosing how the vefiguration can be sought independently of desirable features of hicles are to be maneuvered and in what configurations they the merging maneuvers; will emerge. This method shares some of the features of twro 1 i r< 150 / J U N E 19 7 2 Transactions of the ASME Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 11/02/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use 4 The state representation is chosen in such a way that vehicles nearest the merge point may bo maneuvered into merge positions before the others, an important feature if time is short. References 1 Athans, M., "A Unified Approach to the Vehicle Merging problem," Transportation Research, Vol. 3, No. 1, Apr. 1909, pp. 123-134. 2 Baumann, D.M.B., et. al., Project Melran, M.I.T. Press. 3 Godfrey, M. B., "Merging in Automated Transportation Systems," >SeD thesis, M.I.T., Department of Mechanical Engineering, 1908. 4 Hart, P., el al., "A Formal Basis for the Heuristic Determination of Minimum Cost P a t h s , " IEEE Trans. Si/sl. Sci. and Ci/h., VSSC-4, July, 190S, pp. 100-107. 5 Howard, U. A., Dynamic Programming and Markov Processes, Technology Press and Wiley, New York, 1960. 0 "Study of Synchronous Longitudinal Guidance as Applied to Intercity Automated Highway Networks," T R W Systems Group Report, No. 06818-W666-RO-00, Sept, 1969. 7 Whitney, I). E., "State Space Models of Remote Manipulation Tasks," 'IEEE Trans. Automatic Control, Vol. AC-14, No. 0, Dec. 1969, pp. 617-623. 8 Wilkie, 1). F., "A Moving Cell Control Scheme for Automated Transportation Systems," Transportation Science, Vol. 4, No. 4, Nov. 1970, pp. 347-364. it Whitney, I). E., and M. Tomizuka, "Normal and Emergency Control of a String of Vehicles by Fixed-Reference Sampled-! )at a Control," in press. Journal of Dynamic Systems, Measurement, and Control JUNE 1972 / Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 11/02/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use 151