Distributed Model Predictive Control for Platooning of Heterogeneous Vehicles with Multiple Constraints and Communication Delays

In this paper, the vehicle platoon control problems for a group of heterogeneous vehicles are investigated, where the multiple constraints of the vehicles and the communication delays among the vehicles are taken into consideration. A distributed model predictive control (DMPC) scheme is proposed to drive the heterogeneous vehicles into the desired platoon. In this DMPC framework, the multiple constraints, including the control constraints, state constraints, and jerk constraints, are employed to describe the practical characteristics of vehicles and the communication delays are time-varying and bounded. In this framework, a group of platoon control schemes is proposed based on the DMPC techniques. Furthermore, the feasibility and stability of the proposed vehicle platoon control system are strictly analyzed. Finally, numerical simulation and experiment with TurtleBot3 mobile robots are provided to validate the effectiveness of proposed approaches.


Introduction
In recent years, since the dramatic increase of vehicles and the inferiority of human drivers, much more attention has been paid to the traffic problems (e.g., traffic congestion, road accidents, and air pollution) [1][2][3]. Vehicle platoon control is an effective way to increase the capability of roads and the fuel efficiency. It requires a leader vehicle in the platoon to follow a reference trajectory and the remaining vehicles to follow the leader vehicle with desired distances. Meanwhile, the autonomous vehicles in the platoon can reduce traffic jams and road accidents significantly. e idea of vehicle platoon can be dated back to the Eighties when Partner for Advanced Transportation Technology (PATH) in California was established [4]. On-board cameras or laser sensors are used to measure the velocity, distance, and position of the surrounding vehicles and the vehicles in the platoon can cooperate with others via vehicleto-vehicle (V2V) communication [5]. en, various control techniques have been considered in the literature, including consensus-based control, sliding mode control, and model predictive control. For instance, the consensus formation control strategy is proposed for autonomous vehicular strings with V2V communication connections in [6]. In [7], the distributed adaptive control strategies based on the integral sliding mode control (ISMC) technique are proposed to maintain a rigid formation for a string of vehicle platoon in one dimension. A model predictive control system for a hybrid electric vehicle platoon considering the route information has been presented in [8]. A DMPC algorithm is proposed for the platoon of vehicles with nonlinear dynamics. Meanwhile, the stability and string stability are analyzed in [9]. In our previous work [10], a DMPCbased control scheme is proposed for a group of nonlinear vehicles. Due to the superiorities in dealing with constraints and optimizing control performance, the DMPC becomes more and more popular in vehicle platoon control [11][12][13].
More physical requirements, such as ride comfort, fuel economy, and velocity limit, have been imposed on vehicle platoon controllers to improve the driving performance. For instance, a smooth function tanh(·) is applied to restrict the control input of the vehicles in the platoon in [14]. In [15], a (i) A delay-involved DMPC strategy is proposed for the discrete-time vehicle platoon control problem subject to multiple constraints and communication delays. Multiple constraints, including control constraints, state constraints, and jerk constraints, are considered in each DMPC-based optimization problem. Time-varying and bounded communication delays are taken into consideration in the V2V communication, which can be dealt by the proposed DMPC strategy. (ii) e feasibility and stability of proposed platoon control system are strictly analyzed. In detail, the feasibility of the proposed control strategy is proven by iteratively ensuring the multiple constraints and the terminal constraints. e stability of the vehicle platoon system is demonstrated through the Lyapunov stability theory. In addition, numerical simulation is presented to verify the theoretical results. (iii) An experiment with three TurtleBot3 mobile robots on the Robot Operating System (ROS) platform is conducted to verify the proposed DMPC algorithm.
In the experiment, these mobile robots are driven to the desired platoon by using the proposed DMPC algorithm and the experiment results validate the feasibility and effectiveness of proposed approaches in practical applications. e remainder of this paper is organized as follows. In Section 2, preliminaries and problem formulation are presented. Section 3 introduces the DMPC-based vehicle platoon algorithm with multiple constraints and communication delays. Section 4 analyzes the feasibility and stability of the proposed control system. e simulation and experiment results are presented in Section 5. Section 6 concludes this paper.
Notation: R stands for the set of real numbers. R n stands for the n-dimension real space. Given a matrix M, M > 0 (M ≥ 0) means the matrix is positive definite (positive semidefinite). M 1 ≥ M 2 means that M 1 − M 2 ≥ 0. For a given column vector v, ‖v‖ represents the Euclidean norm. e P-weighted norm is defined as , where P is a given matrix with appropriate dimension. Given matrix Q, λ(Q) and λ(Q) represent the minimum and maximum of the absolute values of the eigenvalues for Q.

Vehicle Modeling with Multiple Constraints.
Consider a longitudinal vehicle platoon of N a + 1 vehicles, which contains a leading vehicle (noted as leader, indexed by 0) and N a following vehicles (noted as followers, indexed from 1 to N a ). e dynamics for the ith vehicle can be described as follows: where p i (t), v i (t), and m i are the position, velocity, and mass of the ith vehicle, respectively. F i (t) is the force generated by the vehicle engine with the derivative as follows: where ς i and c i (t) denote time constant and throttle input of the vehicle. F g i (t) � − m i g sin(θ i (t)) represents the force due to gravity, where g is the acceleration of gravity and θ i (t) is the road slope. F aero is the aerodynamic resistance with ρ, A i , and c di being the air density, cross-sectional area of the vehicle, and air drag coefficient. F drag i (t) � − c r m i g cos(θ i (t)) represents the rolling resistance where c r is the rolling coefficient.
Combining the differentiation to (1) with (3), it results in erefore, the third-order dynamics for the ith vehicle is formulated as follows: In order to deal with the nonlinear model, we linearize it by employing precise feedback linearization [30]: According to (5) and (7), we have After discretizing the system with sampling period T, we obtain where In order to improve the driving performance, multiple constraints are considered for each vehicle in the platoon as follows: (1) Control constraints: u min ≤ u i ≤ u max , where u min < 0 and u max > 0 are bounds of control input for each vehicle. For the convenience of subsequent description, we define the compact set U i to represent the control constraints. ese constraints can be expressed as a set of linear inequalities: where Journal of Advanced Transportation 3 Remark 1. In practice, there exist some relationships among these constraints. Particularly, for the relationships between the vehicle's velocity and acceleration, define a i and a i as the boundaries of the acceleration a i . en, for a certain velocity v i , we have that where a i (v i ) and a i (v i ) are both the functions of the velocity v i . Since v i is always bounded by [v min , v max ], we can obtain that where e error model for the ith follower can be described as follows: . d 0 is the desired distance between two adjacent vehicles. It can be obtained that where x i,des (k) and u i,des (k) are the desired state and control trajectories for the ith follower according to the leader. For the error model, the set of linear inequalities representing the multiple constraints in (11) should be modified as follows: where e objectives of vehicle platoon control are to track the leader and maintain the desired distance between two consecutive vehicles subject to multiple constraints and communication delays. at is, 4 Journal of Advanced Transportation

Communication
Topology. e communication topology among vehicles can be characterized as a weighted . . , N a , meaning that the ith vehicle can send its information to followers in the set.

Assumption 2.
ere is a spanning tree in graph G, in which the root node of this spanning tree is the leader vehicle.
Assumption 3. Suppose that the leader can plan its desired trajectory in advance and send this trajectory to followers through communication topology.
Remark 2. Assumption 3 is necessary due to that the DMPC algorithm needs to make optimization by using the desired states in the receding horizon.

Delay-Involved DMPC-Based Vehicle
Platoon Algorithm is section introduces the formulation of DMPC for the vehicle platoon subject to multiple constraints and communication delays.
For the error model in (15), the pair e matrix P i > 0 is the unique positive definite solution of the Lyapunov matrix equation: where Q i and R i are positive definite matrices. en, there e detailed proof is shown in the Appendix.
Remark 3. For the error model, the multiple constraints in (17) represent the satisfaction of (11). e feedback control Assumption 4. Suppose that the communication delays are bounded, i.e., τ � n * T, n * ∈ R and 0 ≤ n * ≤ N − 1, where N is the length of predictive horizon used in DMPC. And the vehicle equipment has a storage function.
In order to describe the communication delays, the time domain is divided by the time instants k, k � 0, 1, . . .. Assume that at time k, all the followers generate the control signals simultaneously and send their information to other followers that are connected with them. At the time k + 1, each follower measures its system state. However, the communication delays occur in the process of transmitted information among the vehicle communication topology. As a result, each follower may not be able to receive its neighbors' information at the time k + 1. Remark 4. k is the time that can generate the control signal through the DMPC-based optimization problem. Note that due to the communication delays, the next time for the optimization problem is not k + 1, but k + 1 + n k . Meanwhile, the last time for the optimization problem is not k − 1 similarly. Define the last time for the optimization problem as k and the communication delays as n k ; then it can be obtained that k � k + 1 + n k .
Remark 5. At the synchronization time when all the followers can receive their neighbors' information, followers can also receive the trajectory plan for twice time length of the receding horizon from the leader. en, the leader makes new plan for another twice time length of the receding horizon.
For the ith follower at time k, the vehicle platoon with multiple constraints under DMPC scheme can be described in the following optimization problem.

Problem 1.
e optimization problem is given by Journal of Advanced Transportation 5 subject to (for p � 0, 1, . . . , N − 1) where denotes the unknown variables to be optimized; is the set of neighbor followers of the ith follower; ε i is the constant determined in Lemma 1; x a i (k) and x a j (k) are the assumed state trajectories with is denoted as the collection of the ith follower's neighbors. e assumed control trajectories are generated as follows: where u * i (k + p | k) and u * j (k + p | k) are the optimal control trajectories at time k; Remark 6. In (21a), x i (k + p|k) − x i,des (k + p) and u i (k + p|k) − u i,des (k + p) represent the state errors and input errors from the desired equilibrium. x i (k + p|k) − x a i (k + p|k) means that vehicle i tries to maintain its assumed trajectory. x i (k + p|k) − x a j (k + p|k) − d i,j is the error between vehicle i and the assumed trajectory of its neighbor j.
At time k + p, p � 0, . . . , n k , the control input u i (k + p) � u * i (k + p|k) is applied and then the optimization problem at k + 1 + n k is constructed. For more details, the control process of the ith follower is shown in Figure 1. In this figure, the control inputs for time k to k + n k are from the optimal control trajectory at time k, and the next optimization problem is constructed at time k + 1 + n k .
For each follower, when the follower states are outside the terminal set Ω i (ε i ), the control input signal is applied according to the optimization problem; when the follower states enter the terminal set Ω i (ε i ), the stabilizing state feedback law u i (k) � K i x i (k) + u i,des (k) is applied. e delay-involved DMPC algorithm is detailed in Algorithm 1.

Feasibility Analysis.
In order to prove the iterative feasibility by the induction principle, Problem 1 needs to be feasible at the initial time instant k � 0, i.e., there exists a control trajectory driving the initial sate into the terminal set while satisfying all the constraints. is requirement can be fulfilled by choosing an appropriate prediction horizon N.
Assumption 5. At time k � 0 with the initial state x i (0), there exists a prediction horizon N such that Problem 1 has a solution.

Theorem 1. Suppose that Assumptions 4 and 5 hold. e proposed distributed DMPC-based vehicle platoon scheme with multiple constraints and communication delays is iteratively feasible.
e key point of eorem 1 is to show that u i (k + 1 + n k + p | k + 1 + n k ), p � 0, . . . , N − 1 is a feasible control trajectory at time k + 1 + n k satisfying all constraints. e proof is provided in the Appendix. When the follower states enter the terminal set, the local state feedback control u i (k) � K i x i (k) + u i,des (k) is applied, which guarantees the multiple constraints.
Finally, the feasibility of the proposed approach is guaranteed.

Stability Analysis.
e stability analysis is divided into two parts. When the follower states are outside the terminal set, the sum of the optimal control objective functions of all the followers will prove to be an appropriate Lyapunov function; when the follower states enter the terminal set, the local Lyapunov function in Lemma 1 can be used.
When the follower states are outside the terminal set, the feasible control trajectory is generated as (A.3) at time k + 1 + n k . It can be obtained that Apply u i * (k + 1 | k) Apply u i Generate assumed state trajectories x a -i (· | k + 1 + n k ) and x i a (· | k + 1 + n k ) Generate control signals u i * (· | k + 1 + n k ) Send information x i * (· | k + 1 + n k ) out Apply u i * (k + 1 + n k | k + 1 + n k ) k k + 1 k + 1 + n i k k + 1 + n k k + N Generate the optimal control trajectory u * i (k + p | k), p � 0, . . . , N − 1 by solving Problem 1; (4) Generate the state trajectory x * i (k + p | k), p � 0, . . . , N and send it out; (5) Apply the control input u i (k + p) � u * i (k + p | k), p � 0, . . . , n k and receive all the state trajectories of its neighbors at k + 1 + n i k ; (6) Generate the assumed state trajectories x a i (k + 1 + n k + p | k + 1 + n k ) and x a − i (k + 1 + n k + p | k + 1 + n k ), p � 0, . . . , N − 1 for the next optimization problem at k + 1 + n k . (7) else (8) Apply the control input as u i (k) � K i x i (k) + u i,des (k); (9) end if (10) end while ALGORITHM 1: Delay-involved DMPC-based vehicle platoon algorithm.

Journal of Advanced Transportation 7
Define ; then it can be obtained that 8 Journal of Advanced Transportation e detailed proof is presented in the Appendix.
In order to calculate the upper bound of Δ i , we have In addition, we have a ij n k p�0 j∈N i

Theorem 2. For all followers, suppose that Assumptions 1-5 hold. If
is guaranteed and the communication delays are bounded as τ ≤ (N − 1)T. e sum of all followers' objective function is strictly monotonically decreasing. en, the states of each follower outside the terminal set Ω i (ε i ) will enter the set finally with the delay-involved DMPC-based vehicle platoon algorithm.
We will prove that the optimal value of the cost function can be qualified as a Lyapunov function such that the followers' states will enter the terminal set. e detailed proof is presented in the Appendix.
When the follower states enter the terminal set, the stability of the vehicle platoon system is proven by the local Lyapunov function: According to Lemma 1, we have

Numerical Simulation.
A simulation with 6 vehicles is provided to verify the proposed approach. Index the vehicles as 0,1, . . ., 5, where 0 denotes the leader and 1, . . ., 5 are the followers. e communication topology is that each follower can communicate with its predecessor and the leader except follower 1, which can only communicate with the leader. For heterogeneous vehicles, we choose different ς i as follows: ς 0 � ς 1 � 0.5, ς 2 � 0.6, ς 3 � 0.4, ς 4 � 0.2, and ς 5 � 0.8. And the multiple constraints are bounded by e sampling period is given as T � 0.1 s. e predictive horizon is N � 20, and the terminal set is determined as with the bound τ ≤ (N − 1)T during the simulation time.
Based on these parameters, the simulation results are as follows.
e states (including position, velocity, and acceleration) and the related errors between the real and desired values for followers are illustrated in Figures 2-4. e jerks of followers are shown in Figure 5 and the control inputs are presented in Figure 6.
From Figures 2-4, it can be obtained that all states converge to the desired values and the related errors converge to 0. Meanwhile, the velocity constraints (v i (k) ∈ [0, 8]) and acceleration constraints (a i (k) ∈ [− 6, 6]) are guaranteed. In detail, Figure 2 depicts that the convergence time of the position states is about 3.6 s. e velocities of followers converge to the leader's velocity (5m/s) at about 3.8 s in Figure 3 and the accelerations converge to 0 at about 0.4 s according to Figure 4. From Figure 5, the jerk constraints (Δa i (k) ∈ [− 3, 3]) are satisfied apparently. As shown in Figure 6, the control input curves reveal that the control inputs converge to 0 and the input constraints 25,25]) are satisfied. In addition, the feasibility of the proposed strategy is guaranteed. According to the simulation results, the proposed delay-involved DMPCbased vehicle platoon algorithm can drive all vehicles with multiple constraints and communication delays to the desired platoon effectively.

Experiment with TurtleBot3 Mobile Robots.
e Tur-tleBot3 is a ROS standard platform robot, which can use enhanced 360°LiDAR, 9-Axis Inertial Measurement Unit, and precise encoder for research and development (see Figure 7). e hardware specifications of the mobile robots are shown in Table 1.         e experiment results are shown in Figures 9-11, where the tb3_0, tb3_1, and tb3_2 denote the leader, follower 1, and follower 2 in the platoon, respectively. Figures 9 and 10 show the positions and velocities of the TurtleBot3 mobile robots. ey illustrate that the mobile robots can be driven to the desired platoon using the proposed algorithm. In Figure 10, although the velocities have a little oscillation, they can converge to the leader's velocity 0.05 m/s at about 20 s. Figure 11 reveals that the distances between each follower and the leader satisfy the desired spacing (0.25 m), which shows the stability of the TurtleBot3 platoon. In summary, the experiment results validate the feasibility and effectiveness of the proposed approaches.
e oscillation of velocity curves might be caused by the measuring errors of the speed sensors or external disturbances. To solve this problem, we will apply a Kalman filter in the experiment to acquire a better performance in our future work.

Conclusion
e platoon control problem for vehicles with multiple constraints and communication delays has been studied by using the dual-mode DMPC strategy in this paper. e heterogeneous decoupled vehicle platoon with multiple constraints constructs an optimization problem for each vehicle. In addition, a delay-involved DMPC algorithm is proposed to deal with the time-varying and bounded communication delays by using the waiting mechanism. e iterative feasibility of the proposed scheme is proven and the stability conditions are provided. Finally, numerical simulation and experiment are presented to verify the feasibility and effectiveness of proposed approaches.

Proof of Lemma 1
Proof. According to the definition of Ω i (ε i ), there exists ε i such that the set is not nonempty and satisfies (17). en, it is necessary to prove that Ω i (ε i ) is an invariant set with the control law u i (k) � K i x i (k): Due to P i > 0 and (20), it can be obtained that en, for any x i (k) ∈ Ω i (ε i ), we have x i (k + 1) ∈ Ω i (ε i ) with the control law u i (k) � K i x i (k).

Proof of eorem 1
Proof. e proof is derived by induction. First, according to Assumption 5, the constrained optimization Problem 1 is feasible at time k � 0 for each follower. Second, we assume that Problem 1 is feasible at time k, k ≥ 1. ird, we need to prove that there exists a feasible solution to Problem 1 at time k + 1 + n k .
According to Lemma 1, it can be obtained that where x i * (k + N | k) � x i * (k + N | k) − x i,des (k + N). It results in that when p � N − 1 − n k , . . . , N − 1, we have K i x i (k + 1 + n k + p | k + 1 + n k ) + u i,des (k + 1 + n k + p)  Gaps between follower and leader (m) Figure 11: Gaps between followers and leader in the experiment.