Truck platoon control considering heterogeneous vehicles

: This paper presents control algorithms enabling autonomous heterogeneous trucks to 1 drive in platoons. Heterogeneous trucks imply that the hardware information (e.g., truck length, 2 break, accelerator, or engine) of a truck may be distinct from that of another truck. We deﬁne a 3 platoon as a collection of trucks where a manually driven truck (leader truck) is followed by several 4 automatically controlled following trucks. The proposed approach is to make every autonomous truck 5 keep following the leader’s trajectory while maintaining a designated distance from its predecessor 6 truck. As far as we know, this paper is unique in developing both lateral maneuver and speed control 7 considering a platoon of heterogeneous trucks. The efﬁciency of the proposed approach is veriﬁed 8 using simulations. 9

If the front wheel is located at distance L i from the rear wheel along the orientation of the truck p i , then the Ackermann Steering model [26] is In Ackermann steering model, each wheel has its own pivot point and the system is constrained in 82 such a way that all wheels of the car drive on circles with a common center point, avoiding skid [26]. 83 We acknowledge that in the case where slip sideways are allowed, more complicated truck models in 84 [27][28][29] are required. However, as we consider slip sideways, we need to set various parameters, such 85 as the truck mass or inertia.

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Considering heterogeneous trucks, each truck has distinct motion model from other trucks.

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Setting complicated models for each individual truck is time-consuming, and a truck's model may 88 not represent the true dynamics of the truck as the truck runs a longer distance. For instance, the 89 functionality of a truck's hardware (such as engine, accelerator, or braking system) gets worse as time 90 goes on. Also, various road (or tire) situations may make effects on slip sideways. Accurate motion 91 modeling in various environments may be impossible in practice.

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(1) can handle heterogeneous trucks by allowing L i = L j for j = i. L i = L j implies that the length 93 of the i-th truck is distinct from that of the j-th truck. Thus, this paper uses (1) as the motion model of 94 each heterogeneous truck.

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A car cannot arbitrarily change its speed within one time step. Thus, let a i A denote the maximum acceleration of a truck p i . Also, let a i D denote the maximum deceleration of p i . Here, a i A and a i D are determined by hardware information of p i , such as engine, accelerator, or braking system of p i . This implies that (2) presents the feasible range of s i k considering the maximum acceleration and deceleration of p i . The 96 maximum speed of p i is s i M , i.e. s i k ≤ s i M for all k.

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This is due to the fact that the hardware information (such as engine or braking system) of p i may be 99 distinct from that of p j .

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A car cannot change its heading abruptly between adjacent time steps, due to the car's geometry such as the maximum steering angle of the car. Letψ i M present the maximum yaw rate of p i . We have Let δ i M denote the maximum steering angle of a truck p i . Then,ψ i M is derived aṡ This implies that as the length of a truck p i increases, the maximum yaw rate of p i decreases. Also, as 101 the maximum steering angle of p i increases, the maximum yaw rate of p i increases. Since we consider 102 heterogeneous trucks, We describe how to control the speed of each truck for longitudinal control of the truck. The 119 leader is controlled using ACC so that it maintains a constant speed. Also, the speed of a following 120 autonomous truck is controlled to maintain a desired distance (reference platooning distance) from its 121 predecessor truck.

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The reference platooning distance at time step k for a truck p i is denoted as d i k . This implies that 123 the desired distance between p i and p i−1 at time step k is d i k .

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The main source of time delay between trucks is from the parasitic time delay induced by the 125 mechanical systems and the communication delay generated by the communication and computing 126 devices. We assume that the time delay in communication between the leader and a truck p i is bounded 127 by a certain constant, say t d .

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The travel distance of p i within t d seconds can be estimated as t d * s i k . In order to avoid collision due to delayed communication, the reference distance d i k is set as a bigger value than t d * s i k . Also, we set the lower bound for d i k for the safety of trucks. Let min D > 0 present the lower bound for d i k . We use Here, g ≥ 1 and min D are positive constants. As the speed of a truck converges to s 1 , d i k converges to 129 g * t d * s 1 + min D , which is a positive constant. This separation change approach in (5) was inspired 130 by the constant time headway spacing policy, which was also used in [21].

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Suppose that p i visited a waypoint W n−1 and is heading towards the next waypoint W n . The 132 curvature at W n−1 is K n−1 , and the curvature at W n is K n . Since p i and p i−1 are adjacent trucks, assume 133 that both p i and p i−1 move along an arc path with radius, say R = 2 K n +K n−1 . See Figure 1. The arc path 134 connecting p i and p i−1 is depicted with a bold arc in this figure.

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The distance between p i and p i−1 at time step k is Recall that the desired distance between p i and p i−1 at time step k is d i k . The arc angle associated At time step k, p i−1 moves along the arc path with speed s i−1 k within one sampling interval. The arc angle associated to the travel distance s i−1 k T is Note that p i−1 is the predecessor of p i , thus it moves away from p i .

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At each time step k, the i-th truck sets its reference speed, say r i k , as follows.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 7 June 2020 doi:10.20944/preprints202006.0066.v1 In the case where both p i and p i−1 move along a straight path, R in (6) is ∞. (9) is not well-defined in this case. In the case where both p i and p i−1 move along a straight path, the following equation is used to set the reference speed r i k .
Recall that the leader is controlled using ACC so that it maintains a constant speed. While the leader moves using ACC, we set the bound for r i k as This implies that if r i k using (9) or (10) is above γ * s i−1 k , then we set (12) implies that the reference speed of p i cannot be much faster than the speed of p i−1 . The effect 137 of changing γ is analyzed in Section 4.

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The reference speed r i k is used to change the speed of the truck at each time step. In the case where r i k > s i k−1 + a i A * T, the reference speed r i k cannot be reached within one time step. Thus, we use Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 7 June 2020 doi:10.20944/preprints202006.0066.v1 In the case where r i k < s i k−1 − a i D * T, r i k cannot be reached within one time step. Thus, we use Otherwise, the reference speed r i k can be reached within one time step. Thus, we can set 3.2. Lateral control 139 We describe how to control the lateral maneuver of each truck. Consider the case where p i at time step k heads towards a waypoint whose coordinate is ( At each time step k, we control the yaw of p i so that it converges to the reference yaw angle Ψ i k .

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Recall Ψ i k is set to make the truck head towards the next waypoint. Let e ψ = Ψ i k − ψ i k for convenience.
Recall thatψ i M = s i k T * L i tan(δ i M ) denotes the maximum yaw rate of a truck p i . If e ψ >ψ i M * T, then we set the steering angle as The third equation of (1) is (18) and (19) lead to Here, sign(e ψ ) denotes the sign of e ψ . This implies that we change the yaw of the truck with maximum 142 yaw rateψ i M .

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If e ψ ≤ψ i M * T, then we set the steering angle as (21) and (19) lead to This implies that the yaw angle of the truck converges to the reference yaw angle. We need to handle the case where a truck is too close to its predecessor truck. In other words, 146 we handle the case where D i−1,i k < sa f eD. Recall that the distance between p i and p i−1 at time step To avoid collision, it is desirable to set sa f eD so that is satisfied. However, as sa f eD increases, the separation between neighboring trucks increases. Thus, 151 the effect of fuel reduction under truck platoon decreases. Thus, there is a trade-off between fuel 152 reduction and vehicle safety.

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We present MATLAB simulations to verify our control laws. The initial speed of every 166 heterogeneous truck is zero. We used (1) to model the motion of every truck.

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Our simulation settings are as follows. As a method to simulate localization error in the waypoints, Localization based on multiple sensors is not within the scope of this paper and is presented in various papers, such as [31][32][33]. In the case where there are sensing or communication infrastructure along a road, we can use the infrastructure to localize a truck. In order to localize a truck, many estimation methods were utilized, such as time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA), and received signal strength [34][35][36][37][38][39][40][41][42][43][44][45].

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The leader increases its speed with the maximum acceleration a 1 A until the speed reaches 80 km/h.
Then, the leader maintains its speed as 80 km/h using ACC. After the leader moves for 16 minutes, 186 the scenario ends.  192 Figure 4 shows the speed of each heterogeneous truck with respect to time(sec). We use γ = 1.01.

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The speed of every truck converges to the speed of the leader (truck 1) as time goes on. Due to the 194 noise in the system, the truck speed is also noisy.
195 Figure 5 shows the separation between every neighboring truck. Initially, the separation between 196 neighboring trucks increases. However, the separation converges to the desired separation as time 197 goes on. Next, we analyze the effect of changing γ. Figure 6 shows the speed of each heterogeneous truck 200 with respect to time(sec). The only difference from Figure 4 is that we use γ = 1.001 instead of γ = 1.01.

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See that the deviation of every truck's speed decreased, compared to the case where γ = 1.01 in Figure   202 4. Due to the noise in the system, the truck speed is also noisy. 203 Figure 7 shows the separation between every neighboring truck. Initially, the separation between 204 neighboring trucks increases. See that it takes more time in making the separation converge to the 205 desired separation, compared to the case where γ = 1.01 in Figure 5.

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This paper presents control algorithms enabling heterogeneous trucks to drive in platoons. We 220 make all heterogeneous trucks keep following waypoints along the leader's trajectory. Every truck 221 visits the leader's waypoints sequentially while maintaining a designated distance from its predecessor 222 truck. As far as we know, this paper is unique in developing both lateral maneuver and speed control 223 considering a platoon of heterogeneous trucks. The efficiency of the proposed approach is verified 224 using simulations. As our future works, we will verify the effectiveness of the proposed approach 225 using experiments with real heterogeneous trucks.  We use γ = 1.01. No Gaussian noise is used. See that the deviation of every truck's speed decreased, compared to the case where Gaussian noise is used. A truck speed overshoot (marked with arrow in Figure 8) happens at the moment when the truck changes its lane. See Figure 3 for the moment when the truck changes its lane.