Coordinated Cooperative Distributed Decision-Making Using Synchronization of Local Plans

Centralized decision-making for a Networked Control System (NCS) suffers from a high computational burden on the planning agent. Distributed agents, which compute cooperative decision-making, increase computational performance. In cooperative decision-making, agents locally plan for a subset of all agents. Due to only local system knowledge of the agents, these local plans are inconsistent with the local plans of other agents. This inconsistency leads to the infeasibility of plans. This article introduces an algorithm for synchronizing local plans for cooperative distributed decision-making. The algorithm enables parallel decision-making and achieves feasible decisions. The algorithm consists of two iterative steps: local planning and global synchronization. In the local planning step, the agents parallelly compute local decisions, referred to as plans. Subsequently, consistency of the local plans across agents is achieved using global synchronization. The globally synchronized plans act as reference decisions for the local planning step in the next iteration. In each iteration, the local planning guarantees locally feasible plans, while the synchronization guarantees globally consistent plans in that iteration. The parallel algorithm converges to globally feasible decisions if the coupling topology is feasible. We introduce requirements for the coupling topology to achieve convergence to globally feasible decisions and present the algorithm using a model predictive control example. Our evaluations with car-like robots show that globally feasible decisions are achieved.


A. Motivation
I N AN NCS, multiple networked agents have a common control task. An example of an NCS is a network of robots, which plan their trajectories to reach a goal position and avoid collisions. According to the definitions of [1], communicating agents make their decisions in a centralized or distributed manner. In centralized decision-making, one agent makes the decisions for all agents based on global system knowledge. Centralized decision-making may impose a high computational burden on the central agent. Therefore, the decision-making can be distributed to dispense the resource usage in the NCS. The actual gain in control quality and communication costs depend on the distribution method.
A difficulty of distributed decision-making compared to centralized decision-making is that there is no agent with global system knowledge. All agents make their decisions based on local system knowledge, which leads to inconsistencies in their local decisions. The agents share their local decisions to achieve global system knowledge. In cooperative decision-making, each agent makes decisions for itself and all its coupled neighbors. The coupled neighbors of an agent depend on the coupling topology of the NCS. We use the term plans to refer to local decisions. Fig. 1 shows an example of agents' local system knowledge with the local system view of agent v A in Fig. 1(a) and the local system view of agent v B in Fig. 1(b). Agent v A is coupled with agents v B , v C , and v D , while agent v B is coupled with agents v A , v C , v E , and v F . Agent v A considers its neighbors and itself, i.e., and agent v B considers its neighbors and itself, i.e., where V (A) and V (B) denote the sets of the neighbors of agent v A and v B , respectively, i.e., v E , v F / ∈ V (A) and v D / ∈ V (B) . The decision-making of agents v A and v B includes different sets of agents and; hence, the plans of agents v A and v B for each other are inconsistent, e.g., the plan of agent v A for v B and the plan of agent v B for itself differ [2]. In cooperative decision-making, agents have to communicate to achieve consistent plans. We call the property of consistent plans planning consistency.
Section I-B presents related work that addresses the planning consistency problem in cooperative distributed decisionmaking. Subsequently, Section I-C states the contribution of this article and Section I-D introduces the organization of this article. On each side, the black colored agent considers the blue colored agents in its decision-making and ignores the gray colored agents. This leads to inconsistent plans, since the agents include different sets of agents in their decision-making.

B. Related Work
Various research has addressed planning consistency in cooperative distributed decision-making. We classify cooperative distributed decision-making approaches into sequential and parallel approaches. Examples of sequential approaches are the works in [3] and [4]. The agents make local decisions while each agent considers the goals of coupled agents in its decision-making. The sequential order of computations achieves planning consistency. Decisions of the prior agents are known to the following agents. Another sequential approach is presented in [5]. The first agent makes multiple options for future decisions and communicates them to the second agent. The second agent makes its own decisions for each of the options. Subsequently, the second agent selects the decisions that are best for both agents. The agents swap their roles in each time step. The authors state that this approach can be extended to more than two agents. An example of a parallel approach is the work in [6]. The authors address the planning consistency by coupling agents with all agents that have a path to each other in the coupling topology. Planning consistency is achieved by considering the complete graph of active couplings, which results in the central problem, or by considering complete sub-graphs for disjoint coupling graphs. Another idea to address the planning consistency is to adapt the coupling weights in the coupling graph. The coupling weights model the uncertainty of neighbors depending on the influence of their neighbors' decisions. The work in [7] proposes to choose the weighting factors depending on the vertex degrees in the coupling graph. Nevertheless, they did not evaluate this idea. In [8], agents have initial decisions and make proposals to adapt the decisions in favor of their local costs. The other agents need to acknowledge the changes in the initial decisions. However, this method depends on initial decisions and becomes intractable for large-scale systems. An extension to this approach uses fuzzy logic as a negotiation scheme for cooperative Distributed Model Predictive Control (DMPC) [9]. Nevertheless, fuzzy logic requires application knowledge and manual modifications to the process. Other approaches make use of game theory, e.g., of min-max games [10] and coalitional games [11], [12], [13]. These approaches, however, focus on coalitional approaches rather than cooperative approaches. Coalitional approaches cluster agents in the coupling topology, leading to potentially high node degrees and high computation times.
Several authors apply cooperative decision-making and propose practice-driven approaches. In [14] external soft constraints extend the optimization problem. These additional constraints increase the distance between vehicles to avoid collisions even if the plans are not planning consistent. The authors of [15] use a hierarchical approach for vehicles at intersections to reduce the computation complexity. Another example of networked and autonomous vehicles is the work in [16]. The vehicles drive on fixed paths in an unsignalized intersection scenario. In this approach, the vehicles within a fixed distance of the intersection form a complete coupling graph and use consensus to achieve consistent speed profiles for collision avoidance. Consensus is also used in [17] to achieve a consistent direction of movement for unmanned aerial vehicles. The authors of [18] use cooperative decision-making in power networks for energy scheduling with shared energy sources and storage. They use an iterative and sequential approach to converge to a global solution. In [19], the energy schedule of a power network is optimized using the distributed optimization method of [20]. The work in [21] uses a fuzzy Q-learning approach for controlling a chemical production line. Another approach of fuzzy logic in a chemical plant is proposed in [22].
To the best of our knowledge, there is no sufficient work addressing the planning consistency problem in cooperative decision-making for agents that make their decisions in parallel. Recent methods depend on coupling topologies with a high number of links or initial decisions. Other methods are application-specific and are limited to a specific scenario.
Another method with goals similar to DMPC is distributed optimization. In distributed optimization, Lagrangian duality approaches of the optimization problem are often used [23], [24]. However, Lagrangian duality methods suffer from slow convergence [25]. An application of Lagrangian duality is the Alternating Direction Method of Multipliers (ADMM). ADMM is mostly used for convex and smooth optimization [26], since it converges only in special cases of non-convex and non-smooth optimization problems [27]. Another approach of distributed optimization is the Lagrangian decomposition. As ADMM, Lagrangian decomposition is mostly used for convex optimizations, e.g., in [28], since it requires strong duality, which is guaranteed in convex optimization problems, but rarely holds in non-convex optimization problems [29]. The Lagrangian framework [30] with penalty parameters [31] solve this restriction, but requires modifications of the optimization problem and induces non-separability of the objective. Several approaches tries to achieve separability based on augmented Lagrangian approaches [32], [33]. A recent augmented Lagrangian approach solves non-convex distributed optimization problems [34]. Nevertheless, the constraints are still convex equations. To the best of our knowledge, distributed non-convex optimization using Lagrangian duality approaches is not yet solved. DMPC follows a different approach by distributing the control problem rather than distributing the optimization problem. The different approaches lead to different methods for analyzing distributed computations. Both approaches may be combined, e.g., the works in [35], [36] implement cooperative DMPC using ADMM methods. Nevertheless, this paper focus on DMPC rather than distributed optimization.

C. Contribution of This Article
This article introduces an algorithm for cooperative distributed decision-making that achieves planning consistent decisions, without depending on pre-computations or initial decisions. We utilize the idea of synchronization of systems and apply it to local plans. The synchronization of local plans guarantees global consistency. To this end, we extend the synchronization method analyzed in, e.g., [1] to systems with multiple synchronization vectors, i.e., one synchronization vector for the decision of each agent. Nevertheless, only the coupled neighbors of an agent v i and v i itself participate in the planning and synchronization for v i [3], [7]. We propose an iterative algorithm, which alternates two steps: local planning and global synchronization. In each iteration, the agents parallelly compute local plans, which are globally synchronized across the agents. Using the synchronized plans as local reference decisions in the next iteration, the algorithm converges to globally feasible decisions if the coupling topology is feasible. This article presents theoretical analyses of convergence and global feasibility of this algorithm and demonstrates the complete pipeline from uncoupled agents to globally feasible solutions. All parts of this pipeline are purely distributed. We evaluate our algorithm using DMPC and apply it to the CPM Lab, an open-source platform for networked and autonomous vehicles.

D. Organization of This Article
This article is structured as follows. Section II presents our cooperative decision-making algorithm using a DMPC approach and Section III-A analyzes the feasibility and convergence of this approach. Subsequently, Section IV evaluates the algorithm in trajectory planning scenarios for networked car-like robots. Finally, Section V concludes this article.

A. Algorithm Overview
The agents periodically make their decisions. In each time step, the agents alternate local planning and global synchronization steps to iteratively converge to a globally feasible solution. We sketch our algorithm as follows: 1) Make local plans in parallel 2) Synchronize local plans with neighbors 3) Go to 1), if synchronized plans are not locally feasible 4) Apply feasible inputs 5) Communicate new states The agents make their local plans (step 1) in parallel over a directed and time-varying network. The local plans consider only local system knowledge. Our algorithm globally synchronizes the local plans (step 2) and iteratively converges to globally consistent plans. If the synchronized plans are locally infeasible, the agents replan using the global consistent plans as reference decision in the next iteration (step 3). By alternating local planning and global synchronization, the agent's plans converge to globally feasible decisions. The agents then apply their globally feasible decisions (step 4) and communicate their new states (step 5) to continue with the next time step.
Algorithm 1 shows the pseudo code of our algorithm. Each agent v i executes Algorithm 1 in each time step. Consider an NCS consisting of N dynamically decoupled agents V = {v 1 , . . . , v N }. A weighted and directed coupling topology graph G = (V, E) models the NCS, where E is the set of coupling links. Let A denote the weighted adjacency matrix of G and α (i,j) are the element of the ith row and jth column of A. The set of in-neighbors for an agent v i is defined as Decisions are planned over the planning horizon H p and we use (·) to refer to the full planning horizon. The inputs of Algorithm 1 are the references r (j) (·), ∀j : v j ∈ V (i) ∪ {v i }, which lead the agents to their goal states, objective functions J (j) , ∀j : v j ∈ V (i) ∪ {v i }, and set of coupled neighbors V (i) . The reference is limited to the planning horizon H p and; hence, is updated in each time step. Let x (i) (·), u (i) (·), and Δu (i) (·) denote the decisions, control inputs, and variations in control inputs of agent v i , respectively. Let x        − . If the globally synchronized plans are not locally feasible (line 3), the agents plan again using the globally synchronized plans as local references. Section II-B introduces our assumptions on the NCS and the following subsections explain the local planning in Section II-C and the global synchronization in Section II-D in more detail. Section II-E introduces extensions to Algorithm 1.

B. Assumptions
This section states our assumptions about the NCS.
Control (CMPC) can find a feasible solution. We assume that CMPC generates feasible solutions.

2) Communication Possible Between Coupled Agents:
The coupling graph determines the communication links between the agents. If agents are coupled, we assume communication to be possible. However, the coupling topology may consider physical limitations on communication by using coupling links only when the corresponding agents are able to communicate with each other.

3) States and References Known to Neighbors:
The decision-making uses local planning for neighboring agents. Thus, each agent has to know the goals and current states of the neighboring agents. Uncertainties of the agents' states are beyond the scope of this article but may be considered in the coupling constraints of the optimization problem (18).

4) Dynamics Known to Neighbors:
All agents require knowledge about the dynamics of their coupled neighbors. In the case of homogeneous dynamics, this is easily achieved. In the case of heterogeneous dynamics, each agent has to communicate its model to its neighbors before they can start the decisionmaking. Uncertainties of the agents' dynamics are beyond the scope of this article but may be considered in the dynamic constraints (8) and (13) and the coupling constraints (18) of the optimization problem.

5) Knowledge of Coupling Graph:
All agents have to know their neighbors and the corresponding weights in the coupling graph. Whenever the coupling graph changes, affected agents require an update of their neighbors. This is the case if the agents generate the coupling graph locally. When an external device generates the coupling graph, it has to inform the agents about updates to the coupling graph.

C. Optimization of Plans
At every time step t, each agent v i solves a local optimization problem in order to generate local plans for itself and its coupled neighbors. The agents share their local plans to achieve global system knowledge. The objective function consists of multiple parts. The overall objective function is given as One objective is to stay on the reference, i.e., where l(·) denotes the costs for the deviation between the decision and the reference. The deviation between the decision and the reference in the last step of the planning horizon H p has higher costs for the same deviation than in other planning steps, i.e., where l H p (·) denotes the costs for the deviation between decision and reference at the end of the planning horizon H p . We evaluate the costs for deviations between the decision and reference at the last step of the planning horizon H p higher than deviations at other steps. This ensures that each agent approaches its final goal states and provides stability of the planning algorithm [37], [38]. A further objective is to keep the control input variations low, i.e., (4) where l u (·) denotes the costs for control input variations. For cooperation, the objective function also considers coupling objectives to neighboring agents, i.e., (·) denotes the coupling objective between agents v i and v j . The objective function also considers the coupling objectives from neighboring agents to one another, i.e., The objective function for agent v i is minimized aŝ where Δu The optimization constraints for agent v i are formulated as follows. The solution space is constrained to the dynamics by where represents the dynamics of agent v i . Furthermore, feasible states are ensured by and where X (i) is the set of feasible states of agent v i and X H p is the set of feasible states of agent v i at the time step H p . We constraint the changes of the control input as and where U (i) is the set of feasible control inputs of agent v i and ΔU (i) is the set of feasible control input variations of agent v i . We constrain the decisions for agent v j in the optimization of agent v i in the same way as the decisions of agent v i as where (8) and represent the constraints on feasible states and control input variations, similar to (9)- (12). Additionally, multiple agents have coupling constraints, represented by for the coupling constraints between v i and its neighbors and for the coupling constraints between the neighbors of v i , where c ) denotes the coupling constraints between agents v i and v j . This optimization problem returns ΔÛ (i) (·), but agent v i applies only Each agent v i considers only hypothetical decisions of its coupled neighbors.
In order to avoid conflicts with the actual plans of neighboring agents, the DMPC algorithm has to satisfy the planning consistency property as stated in Definition 1 for each time step t.
Definition 1 (Planning consistency): Planning consistency is the property that the plan x If the planning consistency is not satisfied, conflicts between decisions may occur, e.g., if agent v i guarantees conflict freeness between decisions x In this case, the planned decisions are locally feasible in each agent, but not globally feasible within the NCS. In order to achieve planning consistency, we introduce synchronization in Section II-D.

D. Synchronization of Plans
The local plansx (i) − , ∀i : v i ∈ V resulting from the optimization performed by agent v i are locally feasible, but may not be planning consistent, i.e.,x We synchronize the plans using distributed local average. Our idea is inspired by consensus in [39]. The synchronization assumes homogeneous dynamics for all local plans for one agentx For agents with heterogeneous dynamics, the optimization considers the different dynamics, see (8) and (13). Different plans for the same agent use the same dynamics, e.g.,x . The information flow during the synchronization phase of Algorithm 1 is modelled as a synchronization graph, which is introduced in Definition 2.
Definition 2 (Synchronization graph): A synchronization graph represents the information flow during the synchronization phase of Algorithm 1.
to v j to be considered in the synchronization process. The synchronization graph is defined as The relation between the optimization adjacency matrix A and the synchronization adjacency matrix A s is The synchronization graph consists of reduced synchronization graphs for each agent. Definition 3 introduces the reduced synchronization graphs.
Definition 3 (Reduced synchronization graph): Reduced synchronization graphs are sub-graphs of synchronization graphs. The reduced synchronization graph G (i) s for agent v i consists of the vertices of agent v i and all its neighbors and all edges between these vertices. It is defined as G Agent v i synchronizes the plan of agent v j as where α (k,i) is the reversed weighting factor resulting from (20). The agents communicate the locally averaged plans x (−) − (·) and iteratively perform local average steps and communicate the intermediate results to their neighbors until the plans are synchronized, i.e., x Synchronization is achieved, if the requirements of Theorem 1 are satisfied. Theorem 1 uses the terms spanning tree and rooted trees, which is introduced in Definition 4.
Definition 4 (Spanning tree and rooted tree): In an undirected In spanning trees, all vertices have a path to one another.
In a directed graph is a tree that contains all vertices of G d and only directed edges that points to the root vertex (in-tree), or from the root vertex (out-tree). In an in-tree, all vertices have a path to the root vertex. In an out-tree, the root vertex has a path to all other vertices.
Theorem 1: The synchronization converges to a solution if and only if each reduced synchronization graph G (i) s contains a spanning tree in the undirected case, and an out-tree in the directed case.
Proof: According to [39], the distributed local average converges to a solution if and only if at least one vertex has a path to all other vertices. This is the case if an undirected graph contains a spanning tree, or if a directed graph contains an out-tree. We synchronize the plans of each agent using distributed local average. To achieve global synchronization, all plans must be synchronized. Hence, for each plan, a spanning tree or out-tree is required. The reduced synchronization graph G (i) s represents the synchronization topology for the decision x (i) (·). Therefore, the synchronization converges to a solution if and only if all reduced synchronization graphs G (i) s contain spanning trees or out-trees.
The following example demonstrates the synchronization process.
Example 1: Fig. 2(b) shows an example coupling graph, which contains an out-tree. Agent v A plans for agents v A and v B . The agents v A , v C , and v D plan for agent v A . Hence, agent v A should synchronize its plan with agents v C and v D . Note that agent v B is irrelevant for the synchronization of the plan of agent v A . Therefore, the synchronization graph for v A can be reduced to agents v A , v C , and v D and their couplings to one another. The blue colored vertices and edges in Fig. 2  v D , as shown in green in Fig. 2(b), complete an out-tree with s . Note that any additional edge that completes an out-tree with any root is sufficient for convergence of the synchronization.
We achieve an out-tree in all reduced coupling graphs G (i) s , ∀i : v i ∈ V by utilizing the directed edges in G (i) s in both directions. In the example in Fig. 2(b), the agent pairs v A and v C , and v A and v D communicate with each other and; hence, the green colored edges are used to achieve an out-tree in G (A) s .

E. Coupling Weight Adaptation
Algorithm 1 only converges for feasible coupling weights α (i,j) . For infeasible coupling weights, the synchronization of local plans results in the reference plans, i.e., ∀i : v i ∈ V : x (i) (·) = r (i) (·). We achieve convergence by adapting the coupling weights during computation of Algorithm 1. We adapt the coupling weights according to the current local plans x (i) j (·) and the synchronized plans In order to reduce the number of iterations required by Algorithm 1, we adapt the coupling weights before the synchronized plans result in the reference plans. To this end, we classify two problems, which indicate that the convergence may not be achieved for the current coupling weights: 1) Convergence Problem: The convergence problem indicates that the synchronized plan x (i) (·) for an agent v i is approximately the reference r (i) (·) ≈ x (i) (·), but not locally feasible. This may be caused due to local system knowledge. In the case of the convergence problem, we adapt the coupling weights to favor convergence to a local plan x (i) j (·), v j ∈ V (i) ∪ v i with higher deviation from the reference than x (i) (·). If an agent v i detects the convergence problem in x (i) (·), i.e., ∃j ∈ V (i) : , it decreases the coupling weight α (i,j) .
2) Discord Problem: The discord problem indicates that the local plans x (i) − (·) for an agent v i are far apart of one another. The discord problem occurs, if the local plans x , and x (i) k (·), respectively. In the case of the discord problem, we adapt the coupling weights to favor convergence to the direction of the majority of local plans. In the case of no majority, we do not adapt any coupling weights. If an agent v i detects the discord problem to the set of agents V

A. Feasibility Analysis
For feasibility analysis, we define local feasibility and global feasibility of decisions.

Definition 5 (Local feasibility): Decisions are locally feasible in agent
Definition 6 (Global feasibility): Decisions are globally feasible in all agents, x (−) (·) ∈ F, if the decisions fulfill the constraints in (8)- (19) in all agents and the decisions are planning consistent, i.e., In order to achieve global feasibility of decisions, Theorem 2 states that the decisions have to be locally feasible and planning consistent, and the coupling topology is required to have a distance of maximum two between any pair of conflicting agents. Definition 7 introduces the distance between two vertices in a graph.
Theorem 2: Let C ⊂ V × V denote the sets of conflicts between agents. Agents, which implement a cooperative distributed decision-making algorithm, achieve globally feasible decisions if and only if C1 all decisions are locally feasible, i.e., x C2 the decisions are planning consistent, i.e., x C3 the distance in the coupling graph between conflicting agents is maximum 2, i.e., The conflicting agents have a distance in the coupling graph of at most 2 if and only if C3a conflicting agents are neighbors, i.e., , or C3b conflicting agents have a common neighbor, i.e., r Case 1: Assume locally infeasible decisions: r Case 2: Assume planning inconsistent decisions: r Case 3: Assume a pair of conflicting agents are not coupled and they do not have a common neighbor: For a set of agents in which each pair of agents conflict with each other, the coupling topology results in a topology with a diameter of maximum 2. Coupling topologies with a diameter of two enable Algorithm 1 to make feasible decisions for all agents, since it is most restrictive. Definition 8 introduces the diameter of a graph.

Definition 8 (Diameter):
The diameter d of a graph G is defined as A diameter d means that all vertices are connected with a maximum distance of d.

B. Diameter Extension
We extend the synchronization process to also forward information about the plans of neighboring agents in each iteration. Hence, the plans are not only provided to the neighbors, but also to agents that have a distance of two in the coupling topology. These agents use the forwarded plans as non-cooperative constraints in their planning to avoid conflicts. To this end, we introduce the set of indirectly known agents V for each agent v i that consists of all agents that have a distance of two to v i in the coupling graph G. The additional constraints are If the plans are locally feasible in each agent, it holds that no conflicts occur between v i and v j nor between v i and v o for any o . Algorithm 1 does not depend on the forwarding of synchronized plans, see Theorem 2. Nevertheless, the forwarding of synchronized plans to the neighbors of the neighbors improves the convergence and enables to detect and resolve conflicts with agents of one additional step in the coupling graph. The forwarded plans are planning consistent, since the synchronization is repeated until no plans change in the last step of the synchronization in (21). This relaxes the condition C3 of Theorem 2 to distances of maximum three between conflicting agents by adding C3c : for conflicting agents v i and v j . Thus, coupling topologies with diameter three are feasible.

Corollary 1 shows that Synchronized Cooperative Distributed Model Predictive Control (SC-DMPC) requires only one iteration for convex systems.
Corollary 1: If the solution space of the optimization problem is convex, SC-DMPC requires only one iteration.
Proof: In convex solution spaces, the following statement holds:x Thus, Algorithm 1 terminates in line 3 after the first iteration, if the conditions of Theorem 2 are fulfilled. Due to in general non-convex solution space of the optimization problem in (7)- (19), (30) does not hold. Therefore, if the synchronized plans are not locally feasible, i.e., ∃i − (·)) between the resulting local plans and the synchronized plans is minimized. Algorithm 1 will converge to feasible decisions, i.e., where n (·) denotes the nth iteration of Algorithm 1. However, the rate of convergence and; therefore, the number of iterations m required for convergence depends on the coupling topology, as Theorem 1 and 2 show. However, there are coupling topologies that are insufficient for SC-DMPC, as stated in Theorem 2. Section III-E summarizes the requirements on the coupling topology. Nevertheless, according to Corollary 2, if a solution exists for centralized planning, there exists a coupling topology for which Algorithm 1 will find a solution in the trivial case of a complete and unweighted coupling topology, which corresponds to centralized planning. We investigated guarantees on less trivial coupling topologies in [40]. Corollary 2: If a solution of centralized decision-making exists, there exists a coupling topology for synchronized cooperative decision-making to find a globally feasible solution.
Proof: Cooperative decision-making coincides with centralized decision-making for a fully connected coupling topology, i.e., a complete coupling graph with In this case, each agent solves the central optimization problem for all agents, considering their objectives and constraints. Therefore, all agents generate the same plans, which are globally feasible if a solution of a centralized decision-maker exists. Since no synchronization is required, one optimization step is enough to generate these decisions.

D. Communication Requirements
In each iteration of SC-DMPC, the agents communicate their local plans to their neighbors. The synchronization step is iterative and requires each agent to update the local plans of itself and its neighbors by considering its neighbors' local plans. For each reduced synchronization graph G the synchronization requires communication with only direct neighbors. An agent v i is able to locally compute the output of the synchronization of its neighbors V (i) without additional communication, if it knows the trajectories of the neighbors of its neighbors {v k |v j ∈ V i ∧ v k ∈ V j }. Hence, the agents do not require any communication during the synchronization step when using broadcast or two-hop communication. Thus, SC-DMPC requires only one communication step per iteration.

E. Coupling Topology Requirements
The following paragraphs discuss coupling topology requirements: 1) Physical Requirements: Decision conflicts depend on the physics of the application. To guarantee globally feasible decisions, Section III-A states that the coupling topology has to contain a path of length three or smaller for each pair of conflicting agents.
2) Maximum Vertex Degree: A higher vertex degree results in higher optimization times. To achieve low computation times in each iteration, the vertex degree should be as small as possible. Since the agents plan in parallel, even node degrees are desired to minimize waiting times.
3) Time-Varying Coupling Topologies: For the convergence of Algorithm 1 to globally feasible decisions, the synchronization requires the reduced synchronization graph to contain a spanning tree or out-tree. According to [41], in the case of time-variant coupling topologies, it is sufficient for distributed local average to converge if there is a coupling topology contains a spanning tree or out-tree that is applied frequently enough. If no coupling graph contains a spanning tree or out-tree, the convergence of distributed local average is achieved if the union graph of the coupling graphs contains a spanning tree or out-tree and if the agents are connected frequently enough [42]. Since we do not expect the network to vary during a SC-DMPC iteration, we require each coupling topology to contain a spanning tree or out-tree in each reduced synchronization graph. As stated in Sections II-D and III-E, this is always the case in Algorithm 1.

IV. EVALUATION
This section presents our evaluation results. We evaluate the SC-DMPC method in a networked trajectory planning simulation of car-like robots. The car-like robots plan trajectories and share information over a communication network. The car-like robots' dynamics follow the kinematic bicycle model according to [43]. We now present our evaluation setup in Section IV-A.

A. Evaluation Setup
We evaluate our SC-DMPC algorithm in the CPM Lab [44], an open-source test platform for networked and autonomous vehicles. It provides up to 20 car-like robots for networked trajectory planning. The computations run physically distributed on an Ubuntu 18.04 system per car-like robot. Each computation unit has two cores at 1.6 GHz each and 16 GB of RAM. The architecture for experiments in NCS guarantees deterministic and reproducible experiments [45]. Our evaluation uses MAT-LAB 2020a and the optimization tool of IBM CPLEX 12.10. We solve the optimization problem using the mixed integer modelling of [46]. We evaluate SC-DMPC in two different scenarios: N -Circle and N -Parallel. Fig. 3 shows a sketch of both evaluation scenarios for four car-like robots. In the N -Circle scenario, we place the car-like robots uniformly distributed in a circle. The car-like robots' reference trajectories point with constant velocity through the center of the circle. In order to avoid collisions in the center, the car-like robots have to adjust their directions and deviate from their references. The car-like robots in the parallel scenario drive with constant velocity in a parallel formation. Once they reach the obstacle, they have to deviate from their references. The reference is to drive straight to the other side of the obstacle. To avoid collisions, the car-like robots have to adjust their directions. We place the car-like robot v 1 in the middle of the formation, i.e., the car-like robots v 1 , v 2 , and v 3 are blocked by the obstacle for all N . We place the car-like robots with even IDs to the left of v 1 and the car-like robots with odd IDs to the right of v 1 . The car-like robot IDs are in ascending order from the middle to the outer car-like robots. All car-like robots have the same lateral distance to their physical neighbors. The scenarios are referred to as N -Parallel and N -Circle, where N denotes the number of car-like robots. We define the states and references as follows: where x (i) j (t) and y (i) j (t) are the planned coordinates in x-and y-direction of agent v j for agent v i and x (i) (t) and y (i) (t) are the reference coordinates in x-and y-direction for agent v i at time step t, respectively. We define the cost functions as follows: where Q (i) (k) is the weighting matrix of the tracking error, where Q (i) (H p ) is the weighting matrix of the terminal tracking error, where R (i) (k) is the weighting matrix of the steering angle variation. The coupling constraint is defined as where A v j (t + k) is the area occupied by v j in time step t + k. In our application of car-like robots, no coupling objective

B. Coupling Topologies
We derive coupling topologies using our method of [40]. Our method uses game theory to generate coupling topologies with only local knowledge of the topology. In the game, each agent is a player with the option to activate or deactivate coupling links to other agents. Fig. 4 sketches the procedure to derive unidirectional coupling topologies. In the first step, each agent locally takes decisions without coupling constraints and communicates the decisions to the other agents. Thereafter, the agents compute the utility for the possible coupling links and enable couplings with high utility and close couplings with low utility. The agents use the resulting coupling topology in the SC-DMPC and repeat the coupling topology generation in each time step. Each agent compute its utility ut (i) of couplings to agent v i as where ut keep denote the part utilities for conflict avoidance, buffer to conflicts, number of links, and keeping couplings active, respectively. The part utilities are computed as follows: The utility for conflict avoidance is where k con|ij ∈ [1, H p ] ⊂ N represents the time step of a conflict between the decisions of the agents v i and v j . c con is a weighting factor.
The utility for buffer to conflict is where d buf defines an appropriate distance of a conflict, while d ij defines the distance from the closest conflict. c buf is a weighting factor. The utility for the number of links is with weighting factor c link . The utility for keeping couplings active is where r con defines the maximum reward for keeping a coupling and k ij represents the time since when the coupling between the agents v i and v j is enabled.

C. Methodology
We evaluate the deviation between the decisions and the references, computation time, and the number of iterations of the SC-DMPC method in both scenarios with different numbers of car-like robots. We also compare SC-DMPC with CMPC. Table I lists our evaluation parameters. We set the maximum computation time to 400 ms. After that time, the car-like robots turn into an invariant state, which is braking to zero velocity. In the following, we introduce our evaluation metrics.

1) Deviation Between Decisions and References:
We evaluate the deviation between decisions and references as the cumulative distance of the final trajectories to the original reference trajectories of each time step. The synchronized trajectories in each iteration do not influence the deviation between decisions and references in this evaluation. The distance metric is the euclidean distance between the sample points. The deviation between decisions and references metric D is given as where N t denotes the number of simulation steps. We compare the deviation between decisions and references of SC-DMPC and CMPC as a suboptimality ratio given by where D SC−DM P C is the deviation between decisions and references of the SC-DMPC and D CMP C is that of CMPC.

2) Computation Time:
The computation time is measured for the optimization step only. The computation time for the synchronization step is omitted, since it is negligible in comparison with the optimization step. We separately evaluate the number of iterations of the synchronization. For CMPC, there is only one optimization. Therefore, the computation time for the optimization step is the overall computation time In SC-DMPC, there are multiple optimizations per time step, i.e., one for each car-like robot and iteration. Since the car-like robots optimize their trajectories in parallel in each iteration, we measure the sum of all maximum optimization times per iteration as overall computation time as follows: where n is the number of iterations and t

D. Deviation Between Decisions and References
Fig . 5 shows the deviation between decisions and references according to (45) in the N -Circle scenario. For two and three car-like robots, the trajectories have about the same cumulative deviation using SC-DMPC and CMPC. For four car-like robots,  the deviation between decisions and references using SC-DMPC is about 4% higher than the deviation using CMPC. In the case of five car-like robots, the deviation between decisions and references is smaller than for four car-like robots. This may be caused by the additional coupling per car-like robot in the coupling graph. Fig. 6 shows the deviation between decisions and references in the N -Parallel scenario. For two car-like robots, the deviation between decisions and references is the same for SC-DMPC and CMPC. For three car-like robots, the deviation between decisions and references using SC-DMPC is about 3% higher than the deviation using CMPC. More car-like robots do not lead to an increase of the suboptimality ratio of SC-DMPC over CMPC in all cases. A higher number of coupling links has a positive effect on the suboptimality ratio, while more car-like robots may have a negative effect on the suboptimality ratio.
The centralized computing vehicle in CMPC considers all vehicles in its trajectory planning. Thus, CMPC results in the global optimum according to the objective function. In contrast to CMPC, each vehicle in SC-DMPC considers only a subset of vehicles in its trajectory planning. SC-DMPC converges to globally feasible solutions. Nevertheless, the globally feasible solutions may differ from the global optimum. The trajectories generated by CMPC deviate less from the reference trajectories than the trajectories generated by SC-DMPC. Fig. 7 shows the median and maximum computation times of SC-DMPC and CMPC in the N -Circle scenario on a logarithmic scale. The computation time of both approaches are similar for two car-like robots. This is because the coupling topology becomes complete. For more car-like robots, the computation time of CMPC increases faster than the computation time of SC-DMPC to over 360 ms in the median and over 10 s in the maximum for five car-like robots. The computation time of SC-DMPC increases to about 140 ms in the median and about 3 s in the maximum. In this scenario, we achieved a speedup of more than three using four car-like robots. In the 5-Circle scenario, the computation times were not applicable in real-time. Fig. 8 shows the computation time of the N -Parallel scenario on a logarithmic scale. For two car-like robots, the median and maximum computation times of SC-DMPC and CMPC are similar. For higher numbers of car-like robots, the SC-DMPC shows a better scalability than CMPC. In the 10-Parallel scenario, SC-DMPC achieved a speedup of about three. SC-DMPC was able to compute all N -Parallel scenarios in real-time, while CMPC was able to compute the N -Parallel scenarios for N ≤ 6. For more car-like robots, CMPC exceeded the time-bound of 400 ms.

E. Computation Time
In CMPC, the centralized computing vehicle considers all vehicles in its trajectory planning. Contrary, the planning vehicles in SC-DMPC consider less vehicles. Thus, the optimization time of SC-DMPC is lower than the optimization time of CMPC. In applications with a fixed deadline for the trajectory planning, SC-DMPC is able to consider more vehicles than CMPC.

F. Number of Iterations
We also evaluate the number of iterations for the SC-DMPC approach. Fig. 9 shows the number of iterations for the N -Circle scenario. It shows the median and maximum number of iterations for optimization and the median and maximum number of iterations for synchronization. The median number of optimization and synchronization is one for two to five car-like robots. The maximum number of optimizations is one for two car-like robots, since the coupling topology becomes complete. The number of optimizations increase with increasing number of car-like robots. For five car-like robots, SC-DMPC required five optimizations. The maximum number of synchronization steps is 25 for three car-like robots and two for four car-like robots. This is because each car-like robot had an additional coupling link in the coupling topology. The maximum number of optimizations is less for four car-like robots than for three car-like robots. The maximum computation time, nevertheless, was lower for three car-like robots, see Fig. 7. Fig. 10 shows the number of iterations required in the N -Parallel scenario. For up to five car-like robots, only one iteration of optimization is required and the trajectories are instantly synchronized. For higher numbers of car-like robots, at most two optimizations per time step are required. In the same setup, SC-DMPC requires at most eight synchronization steps, while in the mean SC-DMPC requires only one synchronization step.
SC-DMPC requires more iterations in the N -Circle scenario than in the N -Parallel scenario. This is due to the number of different trajectories for each vehicle. In the N -Cirlce scenario, the number of different trajectories for each vehicle is higher than in the N -Parallel scenario. In the N -Parallel scenario, the vehicles' trajectories differ in the direction of vehicle 1. The two possibilities are to drive to the left of the obstacle or to the right. Depending on the coupling topology, the vehicles favour different directions. The driving direction of vehicle 1 determines the driving directions of the other vehicles. Thus, there are two sets of trajectories in the N -Parallel scenario. In the N -Circle scenario, each vehice has the option to drive to the left or to the right of the collision area. Depending on the coupling topology, different vehicles favour different directions for each agent. The diversity of trajectories is higher in the N -Circle scenario, resulting in higher numbers of iterations of SC-DMPC.

G. Communication Demand
In each iteration of SC-DMPC, all vehicles have to communicate their planned trajectories. The number of communication steps equals the number of optimizations in Figs. 9 and 10. Hence, in the N -Circle scenario SC-DMPC requires at maximum five communication steps and one communication step in the mean for five car-like robots. In the parallel scenario, SC-DMPC requires at maximum two communication steps and one communication step in the mean for 10 car-like robots.
In CMPC, the centralized planning vehicle receives the data from all other vehicles. After the trajectory planning, the central vehicle communicates the results to all vehicles. Thus, for N vehicles, CMPC requires the central vehicle to communicate N messages to N vehicles. SC-DMPC reduces the communication demand compared to CMPC.

H. Comparison to the State-of-the-Art
In cooperative DMPC, each agent considers its coupled neighbors. Using a coupling topology with a similar number of couplings per agent, each iteration k requires about the same computation time t comp . For N agents, sequential approaches like [3], [4] require N iterations. The computation time of these approaches is t seq = N × t comp . The computation time of iterative approaches is t iter = I × t comp for I ∈ N iterations. The computation time of iterative approaches outperforms the computation time of sequential approaches if I < N. For I = N , the computation times of sequential and iterative approaches are the same. Figs 9 and 10 show the maximum number of iterations for SC-DMPC. In the circle scenario, SC-DMPC has about the same computation time as sequential approaches for N = 3. In all other cases, the number of iterations I of SC-DMPC is lower than the number of agents N . Thus, SC-DMPC outperforms sequential approaches. SC-DMPC and sequential approaches both require one communication step per iteration. Nevertheless, the communication demand per communication step is less for sequential approaches than for SC-DMPC, since only one agent needs to communicate per time step. In SC-DMPC, all agents communicate in each communication step.
Iterative approaches may lead to CMPC in the worst-case [6], do not give guarantees to reach a feasible solution [7], or depend on system-dependent knowledge [9], [16], [17], [19], [35]. In iterative approaches, all agents need to communicate in each iteration. Thus, the communication demand depends on the number of iterations.
The authors of these papers do not give any information about generating coupling topologies or coupling weights, which both highly affect the computation time and communication demand. This work presents the complete pipeline from uncoupled agents to globally feasible solutions. All parts of this pipeline are purely distributed and do not require any central agent nor any initial solutions. The evaluation of this work goes beyond simulations by demonstrating the effectiveness of this approach in real-world experiments.

V. CONCLUSION
This article presents an algorithm for cooperative decisionmaking in the complete pipeline from uncoupled agents to globally feasible solutions. The algorithm is purely distributed and parallel. Multiple agents generate a directed and time-variant coupling topology. Subsequently, our algorithm alternates local planning and global synchronization steps to iteratively converge to a globally feasible solution. In each iteration, the agents re-examine the coupling weights and adapt them in order to accelerate the convergence to globally feasible solutions.
The theoretical analysis shows that the algorithm requires the coupling topology to fulfill some properties to converge to globally feasible decisions. First, agents that conflict with each other must have a distance in the coupling topology of maximum three. Thus, they have to be coupled with each other or have a common neighbor. Second, in order to achieve global synchronization, the reduced coupling graph of each agent has to contain a spanning tree or out-tree. Our method guarantees to contain a spanning tree in each reduced coupling graph. Third, we show that for the trivial case of a complete coupling topology, our algorithm is guaranteed to find a globally feasible solution. We will investigate the guarantees depending on less trivial coupling topologies in future research.
Our evaluations on real-world car-like robots show that the computation time of our algorithm is more than three times faster than in CMPC approaches in the N -Circle and the N -Parallel scenarios. The reference deviation between decisions and references of our algorithm is 4% higher compared to CMPC.
Future work will contain further methods for generating coupling topologies and weights to guarantee convergence, globally feasible solutions, and low computation times.