Collaborative Platooning of Automated Vehicles Using Variable Time-Gaps Aria HasanzadeZonuzy∗, Sina Arefizadehy, Alireza Talebpoury, Srinivas Shakkottai∗ and Swaroop Darbhaz ∗Dept. of Electrical and Computer Engineering, Texas A&M University yDept. of Civil Engineering, Texas A&M University zDept. of Mechanical Engineering, Texas A&M University Email:fazonuzy, sinaarefizadeh, atalebpour, sshakkot, [email protected] Abstract— Connected automated vehicles (CAVs) could po- tentially be coordinated to safely attain the maximum traffic flow on roadways under dynamic traffic patterns, such as those engendered by the merger of two strings of vehicles due a lane drop. Strings of vehicles have to be shaped correctly in terms of the inter-vehicular time-gap and velocity to ensure that such operation is feasible. However, controllers that can achieve such traffic shaping over the multiple dimensions of target time-gap and velocity over a region of space are unknown. The objective of this work is to design such a controller, and to show that we can design candidate time-gap and velocity profiles such that it can stabilize the string of vehicles in attaining the target profiles. Our analysis is based on studying the system in the spacial rather than the time domain, which enables us to study stability as in terms of minimizing errors from the target profile and across vehicles as a function of location. Finally, we conduct numeral simulations in the context of shaping two platoons for Fig. 1: The Eulerian view of two successive vehicles. merger, which we use to illustrate how to select time-gap and velocity profiles for maximizing flow and maintaining safety. Since the events of interest to us such as traffic merger I. INTRODUCTION requires that platoons be shaped at a particular location in space, we use the so-called Eulerian viewpoint of the Traffic shaping in terms of achieving desired time-gaps platoon in which we specify the state by observing the and velocities over platoons of vehicles is needed to handle system from a fixed location in space s; (as opposed to variable traffic flows on highways caused by mergers at the Lagrangian viewpoint which does so by moving with highway entrances, departures at exits or prevailing road con- the vehicle of interest and specifying state as a function of ditions such as a lane drop. If uncontrolled, such events could time t.) Thus, we use a fixed Cartesian coordinate system to lead to shockwave formation and breakdown. Connected indicate the position of each vehicle at each time. Assume automated vehicles (CAVs) have the potential to maintain that all vehicles are traveling at velocity v; and that vehicle maximum traffic flow on roadways under such dynamically i − 1 has just passed location s: Consider the scenario in changing traffic patterns. However, even in the CAV setting, which the distance between the vehicles i−1 and i is d: The arXiv:1710.02186v1 [cs.SY] 5 Oct 2017 significant coordination is needed to ensure that traffic is time-gap between the two vehicles, denoted as τ is the time shaped in a manner that allows the seamless merger of the required for vehicle i to just pass location s: The distance vehicle platoons in a safe manner. traveled by the vehicle i during this time interval is d + l: Determining the safe operating points in the system re- A. Safe Traffic Flow quires a kinematic model of the vehicles. For simplicity Consider the scenario of a platoon of vehicles in which of exposition we use a so-called second order model of the length of each vehicle is l units, and the maximum vehicular dynamics under which each vehicle can directly deceleration possible by a vehicle is amin: As illustrated control its acceleration (within limits). Such a model does in Figure 1, a safe operating point of this platoon would be not account for the lag between the application of a control to ensure that for any vehicle i; if the vehicle ahead of it, input and the desired acceleration actually being realized. i − 1 were to stop instantaneously, then vehicle i must be However, it is relatively straightforward to extend the results able to come to a stop without hitting vehicle i − 1: This is that we present in this paper to a third order model that a conservative approach, but provides a model under which incorporates lag. Furthermore, since lag is a random variable collision-free operation can be deterministically guaranteed that depends on individual vehicle dynamics and roadway as long as the safety criterion is not violated. conditions, the best option is often to develop controllers for 6 5.8 5.6 5.4 5.2 Traffic Shaping Region 6.5 5 6 6 4.8 Highway – Mainstream Platoon Merged Platoon 5.5 5.8 5 4.6 5.6 Platoon 5.4 4.5 4.4 5.2 – Substream 4 1600 5 Road 4.2 C Feeder 1550 Time-gap (sec) 3.5 4.8 1400 1450 1500 1550 1600 4.6 1500 3 Fig. 3: Traffic shaping for smooth platoon merger. A 4.4 1450 2.5 4.2 1400 4 2 1350 B 20 Upstream 1.5 Traffic Shaping Region Downstream 0 5 10 15 20 25 18 Velocity (m/sec) Velocity Profile 16 Even Vehicles Odd Vehicles Fig. 2: Safety Region in terms of Time-Gap and Velocity. 14 12 10 Velocity (m/sec) the second order model, and then study their performance 8 using a third order model (with random lags) numerically. 6 Let dmin be the smallest distance between the two vehicles 4 1400 1420 1440 1460 1480 1500 1520 1540 1560 1580 1600 such that we satisfy our safety requirement at velocity v: Location (m) Then denoting the absolute value of the maximum possible Fig. 4: Traffic shaping a platoon into sub-platoons of size deceleration of the vehicle by amin, the minimum distance two. for safe operation is v2 dmin = : (1) platoon corresponds to the point B = (v2; τ2). Suppose that 2amin the entering platoon from the feeder is already at v2: In Assuming vehicle i has constant velocity v, the kinematic order to merge the two platoons, we need to first shape the relation between v; τ and dmin is mainstream into sub-platoons such that the velocity of all the vehicles is reduced to v ; and the time-gaps between vehicles vτ = d + l; 2 min in each sub-platoon are selected appropriately to ensure that where τ is minimum time-gap required for safety. Then from the merged platoon operates at point B = (v2; τ2). (1) we have the relation between time-gap and velocity for Figure 4 provides an illustration of the above idea in which safe operation as all vehicles of the mainstream platoon initially operate at v l point A = (v1; τ1): The graph shows the desired velocity of τ = + : (2) every vehicle as a function of its location, and below that 2a v min we provide a snapshot of how the vehicles are positioned B. Need for Traffic Shaping in space at a particular time instant. As the vehicles of the Figure 2 provides an illustration of the relationship derived mainstream platoon proceed along the highway (left to right above. The length of each vehicle, l is chosen as 6 m (and in the figure), they are shaped into sub-platoons, each of size conservatively represents both the actual vehicle length plus two. The lead vehicle of each sub-platoon (whose indices are the standstill spacing desired) and amin is chosen to be 4 even numbers) eventually operates at point C = (v2; τ3); m/s2 in this example. The x-axis represents the velocity v. while the following vehicle in each sub-platoon (whose The y-axis represents minimum time-gap τ achievable for indices are odd numbers) operates at B = (v2; τ2): Notice each velocity. There is no control law that can safely operate that by shaping the mainstream platoon in this manner, we (guarantee no collisions) outside of this convex region, i.e. have created the necessary spacings such that the substream the safe region is above the curve. However, there could be platoon can be merged into the gaps created, with all vehicles a control law that guarantees safe operation for points inside in the merged platoon operating at point B = (v2; τ2): the safety region. The procedure for merging shaped platoons is relatively Now, suppose that we have two platoons with different straightforward, and is not the focus of the paper. Our main flow rates as in Figure 3. We refer to the existing platoon goal is the design of a controller that can undertake traffic on the highway as the mainstream platoon, and the entering shaping of a single platoon. platoon from the feeder road as the substream platoon. We The shaping of a given platoon to attain a desired operating wish to merge these two asymmetric platoons, in which one point requires the specification of the time-gap and velocity vehicle from the substream enters for each pair of vehicles in as a function of location, which we refer to as a to as the mainstream. Assume that the initial (upstream) operating a time-gap and velocity profile. As in the example above, point of the mainstream is at A = (v1; τ1): Now, the these desired profiles could potentially be different for each minimum time-gap that can be supported in the merged vehicle, and need to be selected carefully so that they can be realized using a simple controller. In our example, even controller is based on feedback linearization [12], and the vehicles move from operating point A to C , while odd controller itself can be considered as a generalization of the vehicles move from A to B.