Bio-inspired control by overlapping adaptive clusters: a vehicle platoon case study

IFAC PapersOnLine 56-2 (2023) 1460–1465 ScienceDirect Available online at www.sciencedirect.com 2405-8963 Copyright © 2023 The Authors. This is an open access article under the CC BY-NC-ND license . Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2023.10.1832 10.1016/j.ifacol.2023.10.1832 2405-8963 Copyright © 2023 The Authors. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0/ ) Bio-inspired control by overlapping adaptive clusters: a vehicle platoon case study Ovidiu Pauca ∗Anca Maxim ∗Jos´e M. Maestre ∗∗ Constantin F. Caruntu ∗ ∗“Gheorghe Asachi” Technical University of Iasi, Department of Automatic Control and Applied Informatics, Blvd. D. Mangeron 27, 700050 Iasi, Romania (e-mail:[email protected]) ∗∗ University of Seville, Department of Systems and Automation Engineering, Camino Descubrimientos, S/N.- Isla Cartuja, 41092 Seville, Spain Abstract: Clustering vehicles in a platoon by exchanging information through communication networks leads to an improvement in their performance with a moderate cooperation effort. In particular, in this paper, a bio-inspired methodology for overlapping coalitions of vehicles is proposed, which optimizes communication and control performance. The results obtained using the proposed adaptive clusters method are compared with the ones obtained by a conventional coalitional control strategy, and the comparison illustrates the improved efficiency of the proposed method. Keywords: Bio-inspired control, coalitional control, overlapping topology, vehicle platoon 1. INTRODUCTION Cooperative autonomous vehicles use control solutions based on existing vehicle-to-vehicle (V2V) and vehicle-to- infrastructure (V2I) communication to reduce fuel consumption, ensure people’s safety by avoiding collisions, improve travel comfort, and reduce the occupied space on roads by maintaining a safety distance between vehicles (Badue et al., 2019). The performance of the overall system depends on issues as: the use of the communication network, the chosen topology, the node management, and the bandwidth length. The same holds for the lifetime of batteries feeding actuators and wireless communications devices (Maestre et al., 2014). One possibility to balance the communication effort is the use of clustering methods, which dynamically reorganise the communication topology to reduce communication costs and improve performance (Maglaras and Katsaros, 2016). Additional advantages are: i) bandwidth optimisation - more clusters can use the same communication frequency with respect to geographical constraints; ii) network topology stability - clusters are formed to ensure better communication and performance ⋆The work of C.F. Caruntu was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDIUEFISCDI, project number PN-III-P1-1.1-TE-373 2019-1123, within PNCDI III. The work of O. Pauca and A. Maxim was supported by “Institutional development through increasing the innovation, development and research performance of TUIASI – COMPETE 2.0”, project funded by contract no. 27PFE /2021, financed by the Romanian government. The work of A. Maxim was also supported by “Gheorghe Asachi” Technical University of Iasi (TUIASI), through the Project “Performance and excellence in postdoctoral research 2022”. The work of J.M. Maestre was supported by grant PID2020- 119476RB-I00 funded by MCIN/AEI/ 10.13039/501100011033. so, they can be used for a long time if no other changes appear; iii) reduction of delays - the communication inside clusters can be organised more efficiently; iv) scalability - the division of the system into small groups allows the use of distributed control strategies to optimise the performances of the whole group (Ayyub et al., 2022). Based on these advantages, there are some studies in the literature focused on coalitional control. For example, (Leon Calvo and Mathar, 2018) use a coalitional approach to improve the traffic flow by grouping the vehicles in platoons. Another study based on the coalitional game theory is represented by (Youyun et al., 2019), where the performance of the network communication is improved to ensure the stability of the whole cluster. Moreover, (Barreiro-Gomez and Zhu, 2022) propose a solution where the Shapley values are used to determine the most relevant links to select the coalition. Finally, (Baldivieso- Monasterios and Trodden, 2021) propose a coalitional distributed model predictive control strategy for a coupled mass-spring-damper system where the coalitions are chosen to ensure feasible and stable control. The idea of cooperative clusters comes from nature, where cooperative formations are naturally formed. Each formation has its own shape and motion, e.g., flocking motion for birds, schools of fish, and herds for animals. These examples from nature succeed to form and move in their groups using multiple communication types. Moreover, each member of the formation has to pay attention only to its neighbours in order not to collide (Lewis et al., 2014). Three rules which govern the motion of clusters have been determined in (Reynolds, 1987): i) avoid collision with neighbours; ii) match velocity and direction of motion with Bio-inspired control by overlapping adaptive clusters: a vehicle platoon case study Ovidiu Pauca ∗Anca Maxim ∗Jos´e M. Maestre ∗∗ Constantin F. Caruntu ∗ ∗“Gheorghe Asachi” Technical University of Iasi, Department of Automatic Control and Applied Informatics, Blvd. D. Mangeron 27, 700050 Iasi, Romania (e-mail:[email protected]) ∗∗ University of Seville, Department of Systems and Automation Engineering, Camino Descubrimientos, S/N.- Isla Cartuja, 41092 Seville, Spain Abstract: Clustering vehicles in a platoon by exchanging information through communication networks leads to an improvement in their performance with a moderate cooperation effort. In particular, in this paper, a bio-inspired methodology for overlapping coalitions of vehicles is proposed, which optimizes communication and control performance. The results obtained using the proposed adaptive clusters method are compared with the ones obtained by a conventional coalitional control strategy, and the comparison illustrates the improved efficiency of the proposed method. Keywords: Bio-inspired control, coalitional control, overlapping topology, vehicle platoon 1. INTRODUCTION Cooperative autonomous vehicles use control solutions based on existing vehicle-to-vehicle (V2V) and vehicle-to- infrastructure (V2I) communication to reduce fuel consumption, ensure people’s safety by avoiding collisions, improve travel comfort, and reduce the occupied space on roads by maintaining a safety distance between vehicles (Badue et al., 2019). The performance of the overall system depends on issues as: the use of the communication network, the chosen topology, the node management, and the bandwidth length. The same holds for the lifetime of batteries feeding actuators and wireless communications devices (Maestre et al., 2014). One possibility to balance the communication effort is the use of clustering methods, which dynamically reorganise the communication topology to reduce communication costs and improve performance (Maglaras and Katsaros, 2016). Additional advantages are: i) bandwidth optimisation - more clusters can use the same communication frequency with respect to geographical constraints; ii) network topology stability - clusters are formed to ensure better communication and performance ⋆The work of C.F. Caruntu was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDIUEFISCDI, project number PN-III-P1-1.1-TE-373 2019-1123, within PNCDI III. The work of O. Pauca and A. Maxim was supported by “Institutional development through increasing the innovation, development and research performance of TUIASI – COMPETE 2.0”, project funded by contract no. 27PFE /2021, financed by the Romanian government. The work of A. Maxim was also supported by “Gheorghe Asachi” Technical University of Iasi (TUIASI), through the Project “Performance and excellence in postdoctoral research 2022”. The work of J.M. Maestre was supported by grant PID2020- 119476RB-I00 funded by MCIN/AEI/ 10.13039/501100011033. so, they can be used for a long time if no other changes appear; iii) reduction of delays - the communication inside clusters can be organised more efficiently; iv) scalability - the division of the system into small groups allows the use of distributed control strategies to optimise the performances of the whole group (Ayyub et al., 2022). Based on these advantages, there are some studies in the literature focused on coalitional control. For example, (Leon Calvo and Mathar, 2018) use a coalitional approach to improve the traffic flow by grouping the vehicles in platoons. Another study based on the coalitional game theory is represented by (Youyun et al., 2019), where the performance of the network communication is improved to ensure the stability of the whole cluster. Moreover, (Barreiro-Gomez and Zhu, 2022) propose a solution where the Shapley values are used to determine the most relevant links to select the coalition. Finally, (Baldivieso- Monasterios and Trodden, 2021) propose a coalitional distributed model predictive control strategy for a coupled mass-spring-damper system where the coalitions are chosen to ensure feasible and stable control. The idea of cooperative clusters comes from nature, where cooperative formations are naturally formed. Each formation has its own shape and motion, e.g., flocking motion for birds, schools of fish, and herds for animals. These examples from nature succeed to form and move in their groups using multiple communication types. Moreover, each member of the formation has to pay attention only to its neighbours in order not to collide (Lewis et al., 2014). Three rules which govern the motion of clusters have been determined in (Reynolds, 1987): i) avoid collision with neighbours; ii) match velocity and direction of motion with Bio-inspired control by overlapping adaptive clusters: a vehicle platoon case study Ovidiu Pauca ∗Anca Maxim ∗Jos´e M. Maestre ∗∗ Constantin F. Caruntu ∗ ∗“Gheorghe Asachi” Technical University of Iasi, Department of Automatic Control and Applied Informatics, Blvd. D. Mangeron 27, 700050 Iasi, Romania (e-mail:[email protected]) ∗∗ University of Seville, Department of Systems and Automation Engineering, Camino Descubrimientos, S/N.- Isla Cartuja, 41092 Seville, Spain Abstract: Clustering vehicles in a platoon by exchanging information through communication networks leads to an improvement in their performance with a moderate cooperation effort. In particular, in this paper, a bio-inspired methodology for overlapping coalitions of vehicles is proposed, which optimizes communication and control performance. The results obtained using the proposed adaptive clusters method are compared with the ones obtained by a conventional coalitional control strategy, and the comparison illustrates the improved efficiency of the proposed method. Keywords: Bio-inspired control, coalitional control, overlapping topology, vehicle platoon 1. INTRODUCTION Cooperative autonomous vehicles use control solutions based on existing vehicle-to-vehicle (V2V) and vehicle-to- infrastructure (V2I) communication to reduce fuel consumption, ensure people’s safety by avoiding collisions, improve travel comfort, and reduce the occupied space on roads by maintaining a safety distance between vehicles (Badue et al., 2019). The performance of the overall system depends on issues as: the use of the communication network, the chosen topology, the node management, and the bandwidth length. The same holds for the lifetime of batteries feeding actuators and wireless communications devices (Maestre et al., 2014). One possibility to balance the communication effort is the use of clustering methods, which dynamically reorganise the communication topology to reduce communication costs and improve performance (Maglaras and Katsaros, 2016). Additional advantages are: i) bandwidth optimisation - more clusters can use the same communication frequency with respect to geographical constraints; ii) network topology stability - clusters are formed to ensure better communication and performance ⋆The work of C.F. Caruntu was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDIUEFISCDI, project number PN-III-P1-1.1-TE-373 2019-1123, within PNCDI III. The work of O. Pauca and A. Maxim was supported by “Institutional development through increasing the innovation, development and research performance of TUIASI – COMPETE 2.0”, project funded by contract no. 27PFE /2021, financed by the Romanian government. The work of A. Maxim was also supported by “Gheorghe Asachi” Technical University of Iasi (TUIASI), through the Project “Performance and excellence in postdoctoral research 2022”. The work of J.M. Maestre was supported by grant PID2020- 119476RB-I00 funded by MCIN/AEI/ 10.13039/501100011033. so, they can be used for a long time if no other changes appear; iii) reduction of delays - the communication inside clusters can be organised more efficiently; iv) scalability - the division of the system into small groups allows the use of distributed control strategies to optimise the performances of the whole group (Ayyub et al., 2022). Based on these advantages, there are some studies in the literature focused on coalitional control. For example, (Leon Calvo and Mathar, 2018) use a coalitional approach to improve the traffic flow by grouping the vehicles in platoons. Another study based on the coalitional game theory is represented by (Youyun et al., 2019), where the performance of the network communication is improved to ensure the stability of the whole cluster. Moreover, (Barreiro-Gomez and Zhu, 2022) propose a solution where the Shapley values are used to determine the most relevant links to select the coalition. Finally, (Baldivieso- Monasterios and Trodden, 2021) propose a coalitional distributed model predictive control strategy for a coupled mass-spring-damper system where the coalitions are chosen to ensure feasible and stable control. The idea of cooperative clusters comes from nature, where cooperative formations are naturally formed. Each formation has its own shape and motion, e.g., flocking motion for birds, schools of fish, and herds for animals. These examples from nature succeed to form and move in their groups using multiple communication types. Moreover, each member of the formation has to pay attention only to its neighbours in order not to collide (Lewis et al., 2014). Three rules which govern the motion of clusters have been determined in (Reynolds, 1987): i) avoid collision with neighbours; ii) match velocity and direction of motion with Bio-inspired control by overlapping adaptive clusters: a vehicle platoon case study Ovidiu Pauca ∗Anca Maxim ∗Jos´e M. Maestre ∗∗ Constantin F. Caruntu ∗ ∗“Gheorghe Asachi” Technical University of Iasi, Department of Automatic Control and Applied Informatics, Blvd. D. Mangeron 27, 700050 Iasi, Romania (e-mail:[email protected]) ∗∗ University of Seville, Department of Systems and Automation Engineering, Camino Descubrimientos, S/N.- Isla Cartuja, 41092 Seville, Spain Abstract: Clustering vehicles in a platoon by exchanging information through communication networks leads to an improvement in their performance with a moderate cooperation effort. In particular, in this paper, a bio-inspired methodology for overlapping coalitions of vehicles is proposed, which optimizes communication and control performance. The results obtained using the proposed adaptive clusters method are compared with the ones obtained by a conventional coalitional control strategy, and the comparison illustrates the improved efficiency of the proposed method. Keywords: Bio-inspired control, coalitional control, overlapping topology, vehicle platoon 1. INTRODUCTION Cooperative autonomous vehicles use control solutions based on existing vehicle-to-vehicle (V2V) and vehicle-to- infrastructure (V2I) communication to reduce fuel consumption, ensure people’s safety by avoiding collisions, improve travel comfort, and reduce the occupied space on roads by maintaining a safety distance between vehicles (Badue et al., 2019). The performance of the overall system depends on issues as: the use of the communication network, the chosen topology, the node management, and the bandwidth length. The same holds for the lifetime of batteries feeding actuators and wireless communications devices (Maestre et al., 2014). One possibility to balance the communication effort is the use of clustering methods, which dynamically reorganise the communication topology to reduce communication costs and improve performance (Maglaras and Katsaros, 2016). Additional advantages are: i) bandwidth optimisation - more clusters can use the same communication frequency with respect to geographical constraints; ii) network topology stability - clusters are formed to ensure better communication and performance ⋆The work of C.F. Caruntu was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDIUEFISCDI, project number PN-III-P1-1.1-TE-373 2019-1123, within PNCDI III. The work of O. Pauca and A. Maxim was supported by “Institutional development through increasing the innovation, development and research performance of TUIASI – COMPETE 2.0”, project funded by contract no. 27PFE /2021, financed by the Romanian government. The work of A. Maxim was also supported by “Gheorghe Asachi” Technical University of Iasi (TUIASI), through the Project “Performance and excellence in postdoctoral research 2022”. The work of J.M. Maestre was supported by grant PID2020- 119476RB-I00 funded by MCIN/AEI/ 10.13039/501100011033. so, they can be used for a long time if no other changes appear; iii) reduction of delays - the communication inside clusters can be organised more efficiently; iv) scalability - the division of the system into small groups allows the use of distributed control strategies to optimise the performances of the whole group (Ayyub et al., 2022). Based on these advantages, there are some studies in the literature focused on coalitional control. For example, (Leon Calvo and Mathar, 2018) use a coalitional approach to improve the traffic flow by grouping the vehicles in platoons. Another study based on the coalitional game theory is represented by (Youyun et al., 2019), where the performance of the network communication is improved to ensure the stability of the whole cluster. Moreover, (Barreiro-Gomez and Zhu, 2022) propose a solution where the Shapley values are used to determine the most relevant links to select the coalition. Finally, (Baldivieso- Monasterios and Trodden, 2021) propose a coalitional distributed model predictive control strategy for a coupled mass-spring-damper system where the coalitions are chosen to ensure feasible and stable control. The idea of cooperative clusters comes from nature, where cooperative formations are naturally formed. Each formation has its own shape and motion, e.g., flocking motion for birds, schools of fish, and herds for animals. These examples from nature succeed to form and move in their groups using multiple communication types. Moreover, each member of the formation has to pay attention only to its neighbours in order not to collide (Lewis et al., 2014). Three rules which govern the motion of clusters have been determined in (Reynolds, 1987): i) avoid collision with neighbours; ii) match velocity and direction of motion with Bio-inspired control by overlapping adaptive clusters: a vehicle platoon case study Ovidiu Pauca ∗Anca Maxim ∗Jos´e M. Maestre ∗∗ Constantin F. Caruntu ∗ ∗“Gheorghe Asachi” Technical University of Iasi, Department of Automatic Control and Applied Informatics, Blvd. D. Mangeron 27, 700050 Iasi, Romania (e-mail:[email protected]) ∗∗ University of Seville, Department of Systems and Automation Engineering, Camino Descubrimientos, S/N.- Isla Cartuja, 41092 Seville, Spain Abstract: Clustering vehicles in a platoon by exchanging information through communication networks leads to an improvement in their performance with a moderate cooperation effort. In particular, in this paper, a bio-inspired methodology for overlapping coalitions of vehicles is proposed, which optimizes communication and control performance. The results obtained using the proposed adaptive clusters method are compared with the ones obtained by a conventional coalitional control strategy, and the comparison illustrates the improved efficiency of the proposed method. Keywords: Bio-inspired control, coalitional control, overlapping topology, vehicle platoon 1. INTRODUCTION Cooperative autonomous vehicles use control solutions based on existing vehicle-to-vehicle (V2V) and vehicle-to- infrastructure (V2I) communication to reduce fuel consumption, ensure people’s safety by avoiding collisions, improve travel comfort, and reduce the occupied space on roads by maintaining a safety distance between vehicles (Badue et al., 2019). The performance of the overall system depends on issues as: the use of the communication network, the chosen topology, the node management, and the bandwidth length. The same holds for the lifetime of batteries feeding actuators and wireless communications devices (Maestre et al., 2014). One possibility to balance the communication effort is the use of clustering methods, which dynamically reorganise the communication topology to reduce communication costs and improve performance (Maglaras and Katsaros, 2016). Additional advantages are: i) bandwidth optimisation - more clusters can use the same communication frequency with respect to geographical constraints; ii) network topology stability - clusters are formed to ensure better communication and performance ⋆The work of C.F. Caruntu was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDIUEFISCDI, project number PN-III-P1-1.1-TE-373 2019-1123, within PNCDI III. The work of O. Pauca and A. Maxim was supported by “Institutional development through increasing the innovation, development and research performance of TUIASI – COMPETE 2.0”, project funded by contract no. 27PFE /2021, financed by the Romanian government. The work of A. Maxim was also supported by “Gheorghe Asachi” Technical University of Iasi (TUIASI), through the Project “Performance and excellence in postdoctoral research 2022”. The work of J.M. Maestre was supported by grant PID2020- 119476RB-I00 funded by MCIN/AEI/ 10.13039/501100011033. so, they can be used for a long time if no other changes appear; iii) reduction of delays - the communication inside clusters can be organised more efficiently; iv) scalability - the division of the system into small groups allows the use of distributed control strategies to optimise the performances of the whole group (Ayyub et al., 2022). Based on these advantages, there are some studies in the literature focused on coalitional control. For example, (Leon Calvo and Mathar, 2018) use a coalitional approach to improve the traffic flow by grouping the vehicles in platoons. Another study based on the coalitional game theory is represented by (Youyun et al., 2019), where the performance of the network communication is improved to ensure the stability of the whole cluster. Moreover, (Barreiro-Gomez and Zhu, 2022) propose a solution where the Shapley values are used to determine the most relevant links to select the coalition. Finally, (Baldivieso- Monasterios and Trodden, 2021) propose a coalitional distributed model predictive control strategy for a coupled mass-spring-damper system where the coalitions are chosen to ensure feasible and stable control. The idea of cooperative clusters comes from nature, where cooperative formations are naturally formed. Each formation has its own shape and motion, e.g., flocking motion for birds, schools of fish, and herds for animals. These examples from nature succeed to form and move in their groups using multiple communication types. Moreover, each member of the formation has to pay attention only to its neighbours in order not to collide (Lewis et al., 2014). Three rules which govern the motion of clusters have been determined in (Reynolds, 1987): i) avoid collision with neighbours; ii) match velocity and direction of motion with Bio-inspired control by overlapping adaptive clusters: a vehicle platoon case study Ovidiu Pauca ∗Anca Maxim ∗Jos´e M. Maestre ∗∗ Constantin F. Caruntu ∗ ∗“Gheorghe Asachi” Technical University of Iasi, Department of Automatic Control and Applied Informatics, Blvd. D. Mangeron 27, 700050 Iasi, Romania (e-mail:pauca.[email protected]o) ∗∗ University of Seville, Department of Systems and Automation Engineering, Camino Descubrimientos, S/N.- Isla Cartuja, 41092 Seville, Spain Abstract: Clustering vehicles in a platoon by exchanging information through communication networks leads to an improvement in their performance with a moderate cooperation effort. In particular, in this paper, a bio-inspired methodology for overlapping coalitions of vehicles is proposed, which optimizes communication and control performance. The results obtained using the proposed adaptive clusters method are compared with the ones obtained by a conventional coalitional control strategy, and the comparison illustrates the improved efficiency of the proposed method. Keywords: Bio-inspired control, coalitional control, overlapping topology, vehicle platoon 1. INTRODUCTION Cooperative autonomous vehicles use control solutions based on existing vehicle-to-vehicle (V2V) and vehicle-to- infrastructure (V2I) communication to reduce fuel consumption, ensure people’s safety by avoiding collisions, improve travel comfort, and reduce the occupied space on roads by maintaining a safety distance between vehicles (Badue et al., 2019). The performance of the overall system depends on issues as: the use of the communication network, the chosen topology, the node management, and the bandwidth length. The same holds for the lifetime of batteries feeding actuators and wireless communications devices (Maestre et al., 2014). One possibility to balance the communication effort is the use of clustering methods, which dynamically reorganise the communication topology to reduce communication costs and improve performance (Maglaras and Katsaros, 2016). Additional advantages are: i) bandwidth optimisation - more clusters can use the same communication frequency with respect to geographical constraints; ii) network topology stability - clusters are formed to ensure better communication and performance ⋆The work of C.F. Caruntu was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDIUEFISCDI, project number PN-III-P1-1.1-TE-373 2019-1123, within PNCDI III. The work of O. Pauca and A. Maxim was supported by “Institutional development through increasing the innovation, development and research performance of TUIASI – COMPETE 2.0”, project funded by contract no. 27PFE /2021, financed by the Romanian government. The work of A. Maxim was also supported by “Gheorghe Asachi” Technical University of Iasi (TUIASI), through the Project “Performance and excellence in postdoctoral research 2022”. The work of J.M. Maestre was supported by grant PID2020- 119476RB-I00 funded by MCIN/AEI/ 10.13039/501100011033. so, they can be used for a long time if no other changes appear; iii) reduction of delays - the communication inside clusters can be organised more efficiently; iv) scalability - the division of the system into small groups allows the use of distributed control strategies to optimise the performances of the whole group (Ayyub et al., 2022). Based on these advantages, there are some studies in the literature focused on coalitional control. For example, (Leon Calvo and Mathar, 2018) use a coalitional approach to improve the traffic flow by grouping the vehicles in platoons. Another study based on the coalitional game theory is represented by (Youyun et al., 2019), where the performance of the network communication is improved to ensure the stability of the whole cluster. Moreover, (Barreiro-Gomez and Zhu, 2022) propose a solution where the Shapley values are used to determine the most relevant links to select the coalition. Finally, (Baldivieso- Monasterios and Trodden, 2021) propose a coalitional distributed model predictive control strategy for a coupled mass-spring-damper system where the coalitions are chosen to ensure feasible and stable control. The idea of cooperative clusters comes from nature, where cooperative formations are naturally formed. Each formation has its own shape and motion, e.g., flocking motion for birds, schools of fish, and herds for animals. These examples from nature succeed to form and move in their groups using multiple communication types. Moreover, each member of the formation has to pay attention only to its neighbours in order not to collide (Lewis et al., 2014). Three rules which govern the motion of clusters have been determined in (Reynolds, 1987): i) avoid collision with neighbours; ii) match velocity and direction of motion with Bio-inspired control by overlapping adaptive clusters: a vehicle platoon case study Ovidiu Pauca ∗Anca Maxim ∗Jos´e M. Maestre ∗∗ Constantin F. Caruntu ∗ ∗“Gheorghe Asachi” Technical University of Iasi, Department of Automatic Control and Applied Informatics, Blvd. D. Mangeron 27, 700050 Iasi, Romania (e-mail:[email protected]) ∗∗ University of Seville, Department of Systems and Automation Engineering, Camino Descubrimientos, S/N.- Isla Cartuja, 41092 Seville, Spain Abstract: Clustering vehicles in a platoon by exchanging information through communication networks leads to an improvement in their performance with a moderate cooperation effort. In particular, in this paper, a bio-inspired methodology for overlapping coalitions of vehicles is proposed, which optimizes communication and control performance. The results obtained using the proposed adaptive clusters method are compared with the ones obtained by a conventional coalitional control strategy, and the comparison illustrates the improved efficiency of the proposed method. Keywords: Bio-inspired control, coalitional control, overlapping topology, vehicle platoon 1. INTRODUCTION Cooperative autonomous vehicles use control solutions based on existing vehicle-to-vehicle (V2V) and vehicle-to- infrastructure (V2I) communication to reduce fuel consumption, ensure people’s safety by avoiding collisions, improve travel comfort, and reduce the occupied space on roads by maintaining a safety distance between vehicles (Badue et al., 2019). The performance of the overall system depends on issues as: the use of the communication network, the chosen topology, the node management, and the bandwidth length. The same holds for the lifetime of batteries feeding actuators and wireless communications devices (Maestre et al., 2014). One possibility to balance the communication effort is the use of clustering methods, which dynamically reorganise the communication topology to reduce communication costs and improve performance (Maglaras and Katsaros, 2016). Additional advantages are: i) bandwidth optimisation - more clusters can use the same communication frequency with respect to geographical constraints; ii) network topology stability - clusters are formed to ensure better communication and performance ⋆The work of C.F. Caruntu was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDIUEFISCDI, project number PN-III-P1-1.1-TE-373 2019-1123, within PNCDI III. The work of O. Pauca and A. Maxim was supported by “Institutional development through increasing the innovation, development and research performance of TUIASI – COMPETE 2.0”, project funded by contract no. 27PFE /2021, financed by the Romanian government. The work of A. Maxim was also supported by “Gheorghe Asachi” Technical University of Iasi (TUIASI), through the Project “Performance and excellence in postdoctoral research 2022”. The work of J.M. Maestre was supported by grant PID2020- 119476RB-I00 funded by MCIN/AEI/ 10.13039/501100011033. so, they can be used for a long time if no other changes appear; iii) reduction of delays - the communication inside clusters can be organised more efficiently; iv) scalability - the division of the system into small groups allows the use of distributed control strategies to optimise the performances of the whole group (Ayyub et al., 2022). Based on these advantages, there are some studies in the literature focused on coalitional control. For example, (Leon Calvo and Mathar, 2018) use a coalitional approach to improve the traffic flow by grouping the vehicles in platoons. Another study based on the coalitional game theory is represented by (Youyun et al., 2019), where the performance of the network communication is improved to ensure the stability of the whole cluster. Moreover, (Barreiro-Gomez and Zhu, 2022) propose a solution where the Shapley values are used to determine the most relevant links to select the coalition. Finally, (Baldivieso- Monasterios and Trodden, 2021) propose a coalitional distributed model predictive control strategy for a coupled mass-spring-damper system where the coalitions are chosen to ensure feasible and stable control. The idea of cooperative clusters comes from nature, where cooperative formations are naturally formed. Each formation has its own shape and motion, e.g., flocking motion for birds, schools of fish, and herds for animals. These examples from nature succeed to form and move in their groups using multiple communication types. Moreover, each member of the formation has to pay attention only to its neighbours in order not to collide (Lewis et al., 2014). Three rules which govern the motion of clusters have been determined in (Reynolds, 1987): i) avoid collision with neighbours; ii) match velocity and direction of motion with Bio-inspired control by overlapping adaptive clusters: a vehicle platoon case study Ovidiu Pauca ∗Anca Maxim ∗Jos´e M. Maestre ∗∗ Constantin F. Caruntu ∗ ∗“Gheorghe Asachi” Technical University of Iasi, Department of Automatic Control and Applied Informatics, Blvd. D. Mangeron 27, 700050 Iasi, Romania (e-mail:[email protected]) ∗∗ University of Seville, Department of Systems and Automation Engineering, Camino Descubrimientos, S/N.- Isla Cartuja, 41092 Seville, Spain Abstract: Clustering vehicles in a platoon by exchanging information through communication networks leads to an improvement in their performance with a moderate cooperation effort. In particular, in this paper, a bio-inspired methodology for overlapping coalitions of vehicles is proposed, which optimizes communication and control performance. The results obtained using the proposed adaptive clusters method are compared with the ones obtained by a conventional coalitional control strategy, and the comparison illustrates the improved efficiency of the proposed method. Keywords: Bio-inspired control, coalitional control, overlapping topology, vehicle platoon 1. INTRODUCTION Cooperative autonomous vehicles use control solutions based on existing vehicle-to-vehicle (V2V) and vehicle-to- infrastructure (V2I) communication to reduce fuel consumption, ensure people’s safety by avoiding collisions, improve travel comfort, and reduce the occupied space on roads by maintaining a safety distance between vehicles (Badue et al., 2019). The performance of the overall system depends on issues as: the use of the communication network, the chosen topology, the node management, and the bandwidth length. The same holds for the lifetime of batteries feeding actuators and wireless communications devices (Maestre et al., 2014). One possibility to balance the communication effort is the use of clustering methods, which dynamically reorganise the communication topology to reduce communication costs and improve performance (Maglaras and Katsaros, 2016). Additional advantages are: i) bandwidth optimisation - more clusters can use the same communication frequency with respect to geographical constraints; ii) network topology stability - clusters are formed to ensure better communication and performance ⋆The work of C.F. Caruntu was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDIUEFISCDI, project number PN-III-P1-1.1-TE-373 2019-1123, within PNCDI III. The work of O. Pauca and A. Maxim was supported by “Institutional development through increasing the innovation, development and research performance of TUIASI – COMPETE 2.0”, project funded by contract no. 27PFE /2021, financed by the Romanian government. The work of A. Maxim was also supported by “Gheorghe Asachi” Technical University of Iasi (TUIASI), through the Project “Performance and excellence in postdoctoral research 2022”. The work of J.M. Maestre was supported by grant PID2020- 119476RB-I00 funded by MCIN/AEI/ 10.13039/501100011033. so, they can be used for a long time if no other changes appear; iii) reduction of delays - the communication inside clusters can be organised more efficiently; iv) scalability - the division of the system into small groups allows the use of distributed control strategies to optimise the performances of the whole group (Ayyub et al., 2022). Based on these advantages, there are some studies in the literature focused on coalitional control. For example, (Leon Calvo and Mathar, 2018) use a coalitional approach to improve the traffic flow by grouping the vehicles in platoons. Another study based on the coalitional game theory is represented by (Youyun et al., 2019), where the performance of the network communication is improved to ensure the stability of the whole cluster. Moreover, (Barreiro-Gomez and Zhu, 2022) propose a solution where the Shapley values are used to determine the most relevant links to select the coalition. Finally, (Baldivieso- Monasterios and Trodden, 2021) propose a coalitional distributed model predictive control strategy for a coupled mass-spring-damper system where the coalitions are chosen to ensure feasible and stable control. The idea of cooperative clusters comes from nature, where cooperative formations are naturally formed. Each formation has its own shape and motion, e.g., flocking motion for birds, schools of fish, and herds for animals. These examples from nature succeed to form and move in their groups using multiple communication types. Moreover, each member of the formation has to pay attention only to its neighbours in order not to collide (Lewis et al., 2014). Three rules which govern the motion of clusters have been determined in (Reynolds, 1987): i) avoid collision with neighbours; ii) match velocity and direction of motion with your neighbours; iii) stay close to your neighbours. The advantages of an adaptive cluster compared to a single agent are that the group can ensure robustness when an agent from the group fails, reconfigurability and target localisation. A cooperative cluster has four main properties: group objective, agents (members), information topology, and control algorithms. The group objectives may include flocking, escorting, vehicle formation maintenance, cooperative load transport, and optimal sensing (Bai et al., 2011). One of the most studied clustering methods for vehicles is vehicle platooning (Aghababa and Saif, 2020). Among the advantages of this approach, we can mention the increase of safety, the reduction of costs and pollution by decreasing the amount of used fuel (Fang et al., 2021). This paper proposes a coalitional control strategy to maintain constant distances between the vehicles of a platoon. Each vehicle is controlled by an individual controller and can share and receive information (i.e., sensors measurements, internal states values) through dynamic communication links which can be enabled or disabled. The communication topology is chosen online to improve the performance of the platoon and to reduce the number of active communication links. To design the corresponding controllers for the vehicles, two solutions based on the minimisation of a cost function are studied and compared with a conventional coalitional control strategy based on linear matrix inequalities (LMIs). Inspired by groups from nature, the vehicles have to find the best way to cooperate to reach the imposed target with minimal costs. The main novelty of the paper is the use of overlapping coalitions, which have not been used before in the context of coalitional control. In particular, overlapping coalitions allow a given vehicle (agent) to be member of multiple coalitions at the same time. The proposed methodology fuzzifes the discrete set of topologies to improve performance because there are more degrees of freedom to compute the controller, in comparison to conventional coalitional control design methods. 2. PROBLEM SETTING Following the coalitional control framework of (Chanfreut et al., 2021), we consider a system formed by Msubsystems, S={1,2, .., M}, each of them with dynamics: xi(k+ 1) = Aiixi(k)+Biiui(k)+di, di=j∈S−{i}Aijxj(k)+Bijuj(k),(1) where xi∈Rnand ui∈Rmrepresent the state and the input of subsystem i∈S, respectively, and direpresents the disturbance that affects subsystem i. Starting from (1), the equation that describes the dynamics of the overall system is given by: xS(k+ 1) = ASxS(k)+BSuS(k),(2) where xS∈RnSand uS∈RmSaggregate all states and inputs of all sub-systems i∈S. The communication network between subsystems is described by a graph (S,L), where Sis the set of subsystems and Lis the set of edges representing communication directed links between agents: L⊆LS={{i, j}|{i, j}⊆S,i=j}, T={Λ|Λ⊆L}.(3) The directed links between agents can be enabled or disabled by the controller and we define the network topology Λ ∈Tas the set of active links at a given time. Also, coalitions are defined as the partitions of the set of agents due to the network configuration. Since the partitions are not disjoint (i.e., with overlapping clusters, a given agent can belong to more than one coalition). To clarify this, let us focus on the overall feedback gain, which defines the control law of the overall system: uS(k+ 1) = KΛxS(k),(4) where KΛis the overall feedback gain control matrix, corresponding to topology Λ ∈T. In particular, the objective is to steer the state of the overall system (2) to the origin by minimizing a global cost function: J(uS,Λ,k)= ∞  j=0 xT S(k+j)QSxS(k+j)+ +uT S(k+j)RSuS(k+j)+cξ(Λ(k+j)), (5) where QS∈RnS×nSand RS∈RmS×mSare positive defined weighing matrices, c∈Ris the weight of the communication links enabled in topology Λ, and ξ(Λ) is the communication cost of topology Λ, which is defined as: ξ(Λ)=(no∗(KΛ)/n −M)/2,(6) where no∗(KΛ) represents the number of the non-zero elements of the control matrix KΛ,nis the number of the states of each subsystem. Since the minimization of global cost function (5) is difficult in real-time, the infinity sum is replaced by the following finite sum: J(Λ) = N  j=0 xT S(k+j)QSxS(k+j)+ +uT S(k+j)RSuS(k+j)+cNξ(Λ(k)), (7) where N∈Z+represents the number of time samples for which system Sis evaluated. This cost function depends only on the topology Λ ∈Tbecause the command uS is computed using (4) and the parameters of KΛare determined in advance. To determine the topology for the next Ntime instants, the cost function (7) is minimised using Λ ∈T. Finally, the whole control method is described in Algorithm 1. Algorithm 1: Control method if mod(k, N)!=0 then Compute and apply the command using (4); else (1) All subsystems share their current states; (2) Starting from the current values of the state of the overall system xS(k): (a) For all Λ ∈T, compute the cost function J(Λ) using (7) and model (2); (b) The smallest value for J(Λ) determines the topology Λ, which is used by the system for the next Ntime samples. end Ovidiu Pauca et al. / IFAC PapersOnLine 56-2 (2023) 1460–1465 1461 Copyright © 2023 The Authors. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0/ ) your neighbours; iii) stay close to your neighbours. The advantages of an adaptive cluster compared to a single agent are that the group can ensure robustness when an agent from the group fails, reconfigurability and target localisation. A cooperative cluster has four main properties: group objective, agents (members), information topology, and control algorithms. The group objectives may include flocking, escorting, vehicle formation maintenance, cooperative load transport, and optimal sensing (Bai et al., 2011). One of the most studied clustering methods for vehicles is vehicle platooning (Aghababa and Saif, 2020). Among the advantages of this approach, we can mention the increase of safety, the reduction of costs and pollution by decreasing the amount of used fuel (Fang et al., 2021). This paper proposes a coalitional control strategy to maintain constant distances between the vehicles of a platoon. Each vehicle is controlled by an individual controller and can share and receive information (i.e., sensors measurements, internal states values) through dynamic communication links which can be enabled or disabled. The communication topology is chosen online to improve the performance of the platoon and to reduce the number of active communication links. To design the corresponding controllers for the vehicles, two solutions based on the minimisation of a cost function are studied and compared with a conventional coalitional control strategy based on linear matrix inequalities (LMIs). Inspired by groups from nature, the vehicles have to find the best way to cooperate to reach the imposed target with minimal costs. The main novelty of the paper is the use of overlapping coalitions, which have not been used before in the context of coalitional control. In particular, overlapping coalitions allow a given vehicle (agent) to be member of multiple coalitions at the same time. The proposed methodology fuzzifes the discrete set of topologies to improve performance because there are more degrees of freedom to compute the controller, in comparison to conventional coalitional control design methods. 2. PROBLEM SETTING Following the coalitional control framework of (Chanfreut et al., 2021), we consider a system formed by Msubsystems, S={1,2, .., M}, each of them with dynamics: xi(k+ 1) = Aiixi(k)+Biiui(k)+di, di=j∈S−{i}Aijxj(k)+Bijuj(k),(1) where xi∈Rnand ui∈Rmrepresent the state and the input of subsystem i∈S, respectively, and direpresents the disturbance that affects subsystem i. Starting from (1), the equation that describes the dynamics of the overall system is given by: xS(k+ 1) = ASxS(k)+BSuS(k),(2) where xS∈RnSand uS∈RmSaggregate all states and inputs of all sub-systems i∈S. The communication network between subsystems is described by a graph (S,L), where Sis the set of subsystems and Lis the set of edges representing communication directed links between agents: L⊆LS={{i, j}|{i, j}⊆S,i=j}, T={Λ|Λ⊆L}.(3) The directed links between agents can be enabled or disabled by the controller and we define the network topology Λ ∈Tas the set of active links at a given time. Also, coalitions are defined as the partitions of the set of agents due to the network configuration. Since the partitions are not disjoint (i.e., with overlapping clusters, a given agent can belong to more than one coalition). To clarify this, let us focus on the overall feedback gain, which defines the control law of the overall system: uS(k+ 1) = KΛxS(k),(4) where KΛis the overall feedback gain control matrix, corresponding to topology Λ ∈T. In particular, the objective is to steer the state of the overall system (2) to the origin by minimizing a global cost function: J(uS,Λ,k)= ∞  j=0 xT S(k+j)QSxS(k+j)+ +uT S(k+j)RSuS(k+j)+cξ(Λ(k+j)), (5) where QS∈RnS×nSand RS∈RmS×mSare positive defined weighing matrices, c∈Ris the weight of the communication links enabled in topology Λ, and ξ(Λ) is the communication cost of topology Λ, which is defined as: ξ(Λ)=(no∗(KΛ)/n −M)/2,(6) where no∗(KΛ) represents the number of the non-zero elements of the control matrix KΛ,nis the number of the states of each subsystem. Since the minimization of global cost function (5) is difficult in real-time, the infinity sum is replaced by the following finite sum: J(Λ) = N  j=0 xT S(k+j)QSxS(k+j)+ +uT S(k+j)RSuS(k+j)+cNξ(Λ(k)), (7) where N∈Z+represents the number of time samples for which system Sis evaluated. This cost function depends only on the topology Λ ∈Tbecause the command uS is computed using (4) and the parameters of KΛare determined in advance. To determine the topology for the next Ntime instants, the cost function (7) is minimised using Λ ∈T. Finally, the whole control method is described in Algorithm 1. Algorithm 1: Control method if mod(k, N)!=0 then Compute and apply the command using (4); else (1) All subsystems share their current states; (2) Starting from the current values of the state of the overall system xS(k): (a) For all Λ ∈T, compute the cost function J(Λ) using (7) and model (2); (b) The smallest value for J(Λ) determines the topology Λ, which is used by the system for the next Ntime samples. end 1462 Ovidiu Pauca et al. / IFAC PapersOnLine 56-2 (2023) 1460–1465 Remark 1. All the links are considered enabled when the system has to switch the communication topology, so that all subsystems can share their inputs and states. 3. ANALYSIS OF LINKS RELEVANCE USING GAME THEORY Game theory can be used to find out which links are more relevant. This can be done using a characteristic function, f(Λ), which determines the benefits obtained using a certain coalition Λ. For this reason, the characteristic function is computed using the cost defined by (7), i.e., f(Λ) = J(Λ) (Maestre et al., 2014). In order to have a link game where the communication links represent the players, the f(Λ0), with Λ0={∅} (i.e., the subsystems do not communicate) has to be 0. To fulfil it, the characteristic function is redefined as: f(Λ) = f(Λ) −f(Λ0).(8) The characteristic function evaluates the topologies, but the Shapley value must be used to obtain an evaluation of each link l∈Λ: σl(Λ,u)=  Ω⊆Λ−{l} ξ(Ω)!(ξ(Λ) −ξ(Ω) −1)! ξ(Λ)! ˜σ(9) ˜σ(Ω) = w(Ω ∪{l})−w(Ω),(10) where w(Ω) = N  j=0 [xT S(k+j)QSxS(k+j)+ +uT S(k+j)RSuS(k+j)]. (11) The Shapley value efficiently distributes the communication and performance costs between links for a given state xSand can be used to determine the most relevant links. 4. FEEDBACK DESIGN METHODS In this section, some design alternatives to compute the control feedback gains KΛare assessed. All methods begin with the gain matrix from the decentralized case and finish with the last one, which is the centralized case. Each algorithm has as an input variable a large set of initial states for the overall system (denoted S0) to remove the bias caused by a specific system configuration. The expected output of each algorithm is the control feedback gain matrix for each communication topology. 4.1 LMI-based methodology The first method used is based on a LMI-based optimisation problem, in which the control gain matrix is determined for each topology, solving a set of linear inequalities. The advantage of this method is that it requires low computational power to compute these matrices. For more details regarding this method, the interested readers are referred to (Chanfreut et al., 2021). 4.2 Independent state-feedback gain matrix for each communication topology (IK) The second method was first introduced by (Maxim et al., 2022) and, for each topology, the feedback matrix KΛis determined by minimising a cost function, which is one step in Algorithm 2. The advantage of this approach is that it can be used to compute the control gain matrix for overlapping topologies. The platoon has more degrees of freedom to choose the best topology to improve its performance. Algorithm 2: IK Starting from a large set of initial states S0of the overall system xS(k), for each initial state xS(k) do the following: for all Λ∈Tdo Find the feedback matrix KΛminimising the cost function: J(KΛ)=kstop j=0 [xT S(k+j)QSxS(k+j)+ +uT S(k+j)RSuS(k+j)] ; The state xS(k+j) is obtained using (2), j≥1; The command uT S(k+j) is computed using (4). end 4.3 Modular K matrix suitable for all possible communication topologies (ModK) This subsection presents a method that has as result a modular matrix K, from which the state feedback gain matrix for each topology can be extracted. The method reduces the required computational power. Thus, the components of this matrix Kare determined in an incremental approach for each topology, considering also the feedback gain matrix obtained for previous topologies, based on Algorithm 3. Algorithm 3: ModK Starting from a large set of initial states S0of the overall system xS(k), for each initial state xS(k) do the following: Find the matrix Ksuch that; for all Λ∈Tdo Defining matrix KΛconsidering the common elements from matrix K, which have been already computed in the previous steps; Find the unknown elements of feedback matrix KΛ(which are specific for topology Λ) by minimising the cost function: J(KΛ)=kstop j=0 [xT S(k+j)QSxS(k+j)+ +uT S(k+j)RSuS(k+j)] ; Add in matrix Kthe new elements corresponding to the matrix KΛ; xS(k+j) is obtained using (2), j≥1; The command uT S(k+j) is computed using (4). end The methods represented by Algorithms 2 and 3 begin to compute the feedback gain matrix starting from the decentralised coalition and increase the complexity of the coalitions until the centralised coalition when all links are enabled. Again, this method gives the possibility for a platoon to have more degrees of freedom to choose the best topology by using overlapping topologies. The methods described in this section are used to design the feedback gain matrix of each communication topology for a vehicle platoon in a case study that will be described in the next section. 5. CASE STUDY AND RESULTS This section illustrates the equations that describe the longitudinal dynamics of a vehicle platoon (see Fig. 1) and the results obtained using the proposed coalitional control strategy. Fig. 1. Vehicle platoon The paper considers a platoon that is formed by a leader and four followers. The leader vehicle, which is denoted by V0(and does not appear in Fig. 1), is considered moving with a constant velocity (i.e., its acceleration is null). Thus, the proposed control strategies will be used only for the follower vehicles, denoted in Fig. 1 with Vi,i={1,2,3,4}. Model (12), from the work by (Chanfreut et al., 2020), illustrates the dynamics of a follower vehicle:    ˙edi(t) ¨evi(t) ¨ai(t) ¨ai−1(t)   =   01 −tk0 00 −11 00−1/τ 0 000−1/τ       edi(t) evi(t) ai(t) ai−1(t)   +    00 00 1/τ 0 01/τ   ui(t) ui−1(t), (12) where edi(t) is the error between the distance dimaintained by vehicle ito vehicle i−1 and the target distance, defined as dri =d0+tkvi(t), d0representing the safety distance, evi(t) represents the error between the velocities of vehicles i−1 and i,tk=0.7sis the time headway, and τ=0.1sis the time constant. The parameters used for computing the cost function (7) are QS=0.1In,RS=0.1In, the link weight is set as c=0.002, N= 10 time samples, kstop = 90 time samples, and the sampling period is Ts=0.1s. The large set of initial states S0used by Algorithms 2 and 3 represents 1000 random values between −6m≤edi(k)≤6m, −4m/s ≤evi(k)≤4m/s,−10m/s2≤ai(k)≤10m/s2. The communication network is formed by six links, L= {I,II,III,IV,V,VI}, as depicted in Fig. 1, which allow bidirectional communication between all vehicles. One of the novelties of this paper is the introduction of overlapping topologies, which allow better optimisation of network communication. For example, if the links Iand II are active, then the formed topology includes the vehicles {V1,V 2,V 3}and the communication between these vehicles is provided by links I,II and IV. If, at some point, there exists a low coupling between vehicles V1and V3, then these two vehicles do not need to exchange data, i.e., the communication link IV is disabled. An overlapping topology will allow only communication between vehicle V1and V2(link Iis enabled) and between V2and V3 (link II is enabled), resulting the following overlapping topology, {{V1,V 2},{V2,V 3}}, in which V2is the common vehicle. The possible topologies for a platoon formed by four vehicles are presented in table 1. Remark 2. Note that links I,II and III are used to design the coalitions, whereas links IV,Vand VIare only used to ensure direct communication between their corresponding vehicles. For this reason, links IV,Vand VI cannot be used to form a coalition (e.g., the coalition between the vehicles V1and V3, via link IV, is not allowed). In order to illustrate the difference between overlapping typologies and the method used to compute the feedback matrix, an example is used: let us consider the topologies Λ1={V1,V 2},Λ 9={V2,V 3},Λ 6= {V1,V 2,V 3}and Λ2={{V1,V 2},{V2,V 3}}. The methods based on LMI and IK return a feedback gain matrix for each topology, except for Λ2, which is an overlapping topology and only IK can determine a control matrix for it: KΛ1=    KΛ1 11 KΛ1 12 00 KΛ1 21 KΛ1 22 00 00KΛ1 33 0 000KΛ1 44     ,KΛ9=     KΛ9 11 000 0KΛ9 22 KΛ9 23 0 0KΛ9 32 KΛ9 33 0 000KΛ9 44     ,KΛ6=    KΛ6 11 KΛ6 12 KΛ6 13 0 KΛ6 21 KΛ6 22 KΛ6 23 0 KΛ6 31 KΛ6 32 KΛ6 33 0 000KΛ6 44     , KΛ2=    KΛ2 11 KΛ2 12 00 KΛ2 21 KΛ2 22 KΛ2 23 0 0KΛ2 32 KΛ2 33 0 000KΛ2 44     , where KΛz ij ∈R1×n. For each coalition, these two methods compute a new control matrix compared to the method ModK, which computes a single matrix for all topologies and from it all feedback gain matrices are extracted for each topology, including the overlapping topologies: KΛ1=    K11 K12 00 K21 K22 00 00K33 0 000K44   ,KΛ9=   K11 000 0K22 K23 0 0K32 K33 0 000K44   , KΛ6=    K11 K12 K13 0 K21 K22 K23 0 K31 K32 K33 0 000K44   ,KΛ2=   K11 K12 00 K21 K22 K23 0 0K32 K33 0 000K44   , where Kij ∈R1×n. The communication cost also represents a difference between the overlapping coalitions and the classical coalitional method. For example, the coalitions Λ2= {{V1,V 2},{V2,V 3}} and Λ6={V1,V 2,V 3}use the same links {I,II}, but the communication cost is different because in coalition Λ2vehicles V1and V3do not communicate (i.e., link IV is disabled), compared to coalition Λ6, in which link IV is enabled. Table 1 illustrates all communication links and costs required by each coalition. This work proposes two new methods, IK and ModK, to compute the controllers for vehicles, i.e., the feedback gain matrix. The results were compared with the conventional coalitional control based on the LMI approach. The Ovidiu Pauca et al. / IFAC PapersOnLine 56-2 (2023) 1460–1465 1463 gain matrix of each communication topology for a vehicle platoon in a case study that will be described in the next section. 5. CASE STUDY AND RESULTS This section illustrates the equations that describe the longitudinal dynamics of a vehicle platoon (see Fig. 1) and the results obtained using the proposed coalitional control strategy. Fig. 1. Vehicle platoon The paper considers a platoon that is formed by a leader and four followers. The leader vehicle, which is denoted by V0(and does not appear in Fig. 1), is considered moving with a constant velocity (i.e., its acceleration is null). Thus, the proposed control strategies will be used only for the follower vehicles, denoted in Fig. 1 with Vi,i={1,2,3,4}. Model (12), from the work by (Chanfreut et al., 2020), illustrates the dynamics of a follower vehicle:    ˙edi(t) ¨evi(t) ¨ai(t) ¨ai−1(t)   =   01 −tk0 00 −11 00−1/τ 0 000−1/τ       edi(t) evi(t) ai(t) ai−1(t)   +    00 00 1/τ 0 01/τ   ui(t) ui−1(t), (12) where edi(t) is the error between the distance dimaintained by vehicle ito vehicle i−1 and the target distance, defined as dri =d0+tkvi(t), d0representing the safety distance, evi(t) represents the error between the velocities of vehicles i−1 and i,tk=0.7sis the time headway, and τ=0.1sis the time constant. The parameters used for computing the cost function (7) are QS=0.1In,RS=0.1In, the link weight is set as c=0.002, N= 10 time samples, kstop = 90 time samples, and the sampling period is Ts=0.1s. The large set of initial states S0used by Algorithms 2 and 3 represents 1000 random values between −6m≤edi(k)≤6m, −4m/s ≤evi(k)≤4m/s,−10m/s2≤ai(k)≤10m/s2. The communication network is formed by six links, L= {I,II,III,IV,V,VI}, as depicted in Fig. 1, which allow bidirectional communication between all vehicles. One of the novelties of this paper is the introduction of overlapping topologies, which allow better optimisation of network communication. For example, if the links Iand II are active, then the formed topology includes the vehicles {V1,V 2,V 3}and the communication between these vehicles is provided by links I,II and IV. If, at some point, there exists a low coupling between vehicles V1and V3, then these two vehicles do not need to exchange data, i.e., the communication link IV is disabled. An overlapping topology will allow only communication between vehicle V1and V2(link Iis enabled) and between V2and V3 (link II is enabled), resulting the following overlapping topology, {{V1,V 2},{V2,V 3}}, in which V2is the common vehicle. The possible topologies for a platoon formed by four vehicles are presented in table 1. Remark 2. Note that links I,II and III are used to design the coalitions, whereas links IV,Vand VIare only used to ensure direct communication between their corresponding vehicles. For this reason, links IV,Vand VI cannot be used to form a coalition (e.g., the coalition between the vehicles V1and V3, via link IV, is not allowed). In order to illustrate the difference between overlapping typologies and the method used to compute the feedback matrix, an example is used: let us consider the topologies Λ1={V1,V 2},Λ 9={V2,V 3},Λ 6= {V1,V 2,V 3}and Λ2={{V1,V 2},{V2,V 3}}. The methods based on LMI and IK return a feedback gain matrix for each topology, except for Λ2, which is an overlapping topology and only IK can determine a control matrix for it: KΛ1=    KΛ1 11 KΛ1 12 00 KΛ1 21 KΛ1 22 00 00KΛ1 33 0 000KΛ1 44     ,KΛ9=     KΛ9 11 000 0KΛ9 22 KΛ9 23 0 0KΛ9 32 KΛ9 33 0 000KΛ9 44     ,KΛ6=    KΛ6 11 KΛ6 12 KΛ6 13 0 KΛ6 21 KΛ6 22 KΛ6 23 0 KΛ6 31 KΛ6 32 KΛ6 33 0 000KΛ6 44     , KΛ2=    KΛ2 11 KΛ2 12 00 KΛ2 21 KΛ2 22 KΛ2 23 0 0KΛ2 32 KΛ2 33 0 000KΛ2 44     , where KΛz ij ∈R1×n. For each coalition, these two methods compute a new control matrix compared to the method ModK, which computes a single matrix for all topologies and from it all feedback gain matrices are extracted for each topology, including the overlapping topologies: KΛ1=    K11 K12 00 K21 K22 00 00K33 0 000K44   ,KΛ9=   K11 000 0K22 K23 0 0K32 K33 0 000K44   , KΛ6=    K11 K12 K13 0 K21 K22 K23 0 K31 K32 K33 0 000K44   ,KΛ2=   K11 K12 00 K21 K22 K23 0 0K32 K33 0 000K44   , where Kij ∈R1×n. The communication cost also represents a difference between the overlapping coalitions and the classical coalitional method. For example, the coalitions Λ2= {{V1,V 2},{V2,V 3}} and Λ6={V1,V 2,V 3}use the same links {I,II}, but the communication cost is different because in coalition Λ2vehicles V1and V3do not communicate (i.e., link IV is disabled), compared to coalition Λ6, in which link IV is enabled. Table 1 illustrates all communication links and costs required by each coalition. This work proposes two new methods, IK and ModK, to compute the controllers for vehicles, i.e., the feedback gain matrix. The results were compared with the conventional coalitional control based on the LMI approach. The 1464 Ovidiu Pauca et al. / IFAC PapersOnLine 56-2 (2023) 1460–1465 Table 1. Communication topologies of a vehicle platoon formed by four vehicles T Links ξ(Λ) Vehicles T Links ξ(Λ) Vehicles Λ0{∅} 0{V1},{V2},{V3},{V4}Λ7{I,III}2{V1,V 2},{V3,V 4} Λ1{I}1{V1,V 2}Λ8{I,II,III,V I}4{V1,V 2},{V2,V 3,V4} Λ2{I,II}2{V1,V 2},{V2,V 3}Λ9{II}1{V2,V 3} Λ3{I,II,III}3{V1,V 2},{V2,V 3},{V3,V 4}Λ10 {II,III}2{V2,V 3},{V3,V 4} Λ4{III}1{V3,V 4}Λ11 {I,II,III,IV }4{V3,V 4},{V1,V 2,V3} Λ5{I,II,III,IV,V I}5{V1,V 2,V 3},{V2,V 3,V4}Λ12 {II,III,V I}3{V2,V 3,V4} Λ6{I,II,IV }3{V1,V 2,V 3}Λ13 {I,II,III,IV,V,V I}6{V1,V 2,V 3,V 4} 0 50 100 Time[samples] 0 2 4 6 e d 1 [m] ModK IK LMI-based 0 50 100 Time[samples] -4 -3 -2 -1 0 1 e d 2 [m] ModK IK LMI-based 0 50 100 Time[samples] -1 0 1 2 3 e d 3 [m] ModK IK LMI-based 0 50 100 Time[samples] -3 -2 -1 0 1 e d 4 [m] ModK IK LMI-based Fig. 2. Distance error between vehicles 0 20 40 60 80 100 Time[samples] -2 0 2 4 6 e v 1 [m/s] ModK IK LMI-based 0 20 40 60 80 100 Time[samples] -4 -2 0 2 4 e v 2 [m/s] ModK IK LMI-based 0 20 40 60 80 100 Time[samples] -2 0 2 e v 3 [m/s] ModK IK LMI-based 0 20 40 60 80 100 Time[samples] -4 -2 0 2 e v 4 [m/s] ModK IK LMI-based Fig. 3. Velocity error between vehicles methods IK and ModK are allowed to use all possible topologies from table 1, but the one based on LMI uses only the classical topologies (i.e., Λ0,Λ1,Λ 4,Λ 6,Λ 7,Λ 9, Λ12,Λ 13). In Figs. 2-4, the distance errors between vehicles, velocity errors and accelerations of vehicles are illustrated. From these figures, it results that the states of each vehicle are led to the origin, meaning that the vehicles maintain a desired distance to the vehicle in front. Moreover, using IK and ModK methods, the acceleration of the vehicles has smaller variations compared to the results obtained by the LMI approach. This can be seen also in Figs. 2- 3, where the distance and velocity errors are decreasing smoother to zero compared to the case of LMI. A sudden variation of the acceleration leads to decreased passengers’ 0 50 100 Time[samples] -2 0 2 4 6 8 a 1 [m/s 2 ] ModK IK LMI-based 0 50 100 Time[samples] -8 -6 -4 -2 0 2 a 2 [m/s 2 ] ModK IK LMI-based 0 50 100 Time[samples] -2 0 2 4 6 a 3 [m/s 2 ] ModK IK LMI-based 0 50 100 Time[samples] -4 -2 0 2 4 a 4 [m/s 2 ] ModK IK LMI-based Fig. 4. Acceleration of vehicles comfort and an increase of fuel consumption and emissions. Every N= 10 samples, the platoon, using Algorithm 1 described in Section 3, decides which is the new topology that offers better control performances and reduced fuel cost and communication. The communication topologies and the time when these were used are illustrated in Fig. 5. Using the LMI based method, the first topology chosen is Λ4. This one assumes that vehicles {V3,V 4}have communications links active and exchange information between them. The next two topologies are Λ9and Λ1. This means that only vehicles V2and V3and respectively vehicles V1and V2exchange data between them, while the remaining platoon uses a decentralised controller based only on the information from the local sensors. In the time sample interval [40,50], the links Iand III are active, allowing communication between vehicles {V1,V 2} and {V3,V 4}, i.e., Λ7. The last link active is II, which assumes a communication between vehicles {V2,V 3}, while the remaining platoon uses a decentralised controller. The second method, IK, initially uses the overlapping topology Λ11, in which the links {I,II,III,IV }allow communication between vehicles {V3,V 4}and between {V1,V 2,V 3}. The topology Λ1is used during the sample interval [20,30] and only the communication between vehicles V1and V2is active. In the interval [30,60], the overlapping topology Λ2activates the links {I,II}so that V1communicates with V2, and V2with V3. The rest of the vehicles use a decentralised control solution based on sensors measurements. The last method is the one that obtain the largest reduction of communication costs between vehicles. The method uses only the typologies Λ3 and Λ1. Topology Λ3is an overlapping topology in which the communication links are active only for the groups {V1,V 2},{V2,V 3}and, respectively, the group {V3,V 4}. 0 10 20 30 40 50 60 70 80 90 100 Time[samples] 0 1 2 3 4 5 6 7 8 9 10 11 Topology ModK IK LMI-based Fig. 5. Network mode Topology Λ1assumes that only the vehicles {V1,V 2} are linked. From Figs. 2-5 it can be observed that the decentralised solution, Λ0, is used in all three cases when the states are close to the origin. Also, from these figures, it can be observed that for the methods IK and ModK all the states reach first the origin; from that point on, the communication links are disabled and, the platoon uses only the decentralised control, i.e., topology Λ0. The maximum values of cost functions of the three methods are: JModK = 177.6, JIK = 172.9, JLMI = 203.4, where the cost Jis computed for all simulation time i.e., 100 time samples, and is the same cost as the one used by Algorithms 2 and 3. It can be noticed that methods which use overlapping coalitions, have similar results and outperform the method based on LMI. The Shapley (9) and characteristic function (8) values of the communication network links, (only I,II and III), are illustrated in table 2. The link with the smallest values is I, which connects vehicles {V1,V 2}. Moreover, this result is in accordance with the characteristic values, which have smaller values compared to all topologies with a single link. These values prove that the communication between vehicles V1and V2has the greatest relevance in the performance of the overall platoon. Table 2. Characteristic function and Shapely values Links I II III Method fσ lfσ lfσ l ModK -19.64 -4.49 2.68 0.51 -2.16 -0.14 IK -19.29 -5.41 11.05 2.12 -1.4 -0.98 LMI -21.98 -6.47 14.92 2.27 -4.9 -1.82 6. CONCLUSIONS The paper proposes a bio-inspired coalitional control strategy to regulate distances between the vehicles of a platoon. The novelties of the paper are represented by the use of two new methods based on the minimisation of a cost function for designing the controllers for each topology and the introduction of overlapping topologies, which can better optimise the use of the communication network. The results were compared with a solution based on the LMI approach. The simulations show that the proposed methods with overlapping coalitions outperform the LMI- based classical coalition control approach. REFERENCES Aghababa, M.P. and Saif, M. (2020). Finite time sliding mode control of connected vehicle platoons guaranteeing string stability. In IEEE International Conference on Human-Machine Systems, 1–6. Ayyub, M., Oracevic, A., Hussain, R., Khan, A.A., and Zhang, Z. (2022). A comprehensive survey on clustering in vehicular networks: Current solutions and future challenges. Ad Hoc Networks, 124, 102729. Badue, C., Guidolini, R., Carneiro, R.V., Azevedo, P., Cardoso, V.B., Forechi, A., Jesus, L.F.R., Berriel, R.F., Paixao, T.M., Mutz, F.W., Oliveira-Santos, T., and de Souza, A.F. (2019). Self-driving cars: A survey. CoRR, abs/1901.04407. Bai, H., Arcak, M., and Wen, J. (2011). Introduction, 1–17. Springer New York, New York, NY. Baldivieso-Monasterios, P.R. and Trodden, P.A. (2021). Coalitional predictive control: Consensus-based coalition forming with robust regulation. Automatica, 125, 109380. Barreiro-Gomez, J. and Zhu, Q. (2022). Coalitional stochastic differential games for networks. IEEE Control Systems Letters, 6, 2707–2712. Chanfreut, P., Maestre, J.M., and Camacho, E.F. (2021). A survey on clustering methods for distributed and networked control systems. Annual Reviews in Control, 52, 75–90. Chanfreut, P., Keijzer, T., Ferrari, R.M., and Maestre, J.M. (2020). A topology-switching coalitional control and observation scheme with stability guarantees. IFAC-PapersOnLine, 53(2), 6477–6482. Fang, C., Shangguan, J., Li, H., and Yue, M. (2021). Distributed cooperative control of vehicle platoon based on a composite safety distance. In 6th IEEE International Conference on Advanced Robotics and Mechatronics, 535–540. Leon Calvo, J.A. and Mathar, R. (2018). Connected vehicles coordination: A coalitional game-theory approach. In European Conference on Networks and Communications, 1–6. Lewis, F.L., Zhang, H., Hengster-Movric, K., and Das, A. (2014). Introduction to Synchronization in Nature and Physics and Cooperative Control for Multi-Agent Systems on Graphs, 1–21. Springer London, London. Maestre, J.M., Mu˜noz de la Pe˜na, D., Jim´enez Losada, A., Algaba, E., and Camacho, E. (2014). A coalitional control scheme with applications to cooperative game theory. Optimal Control Applications and Methods, 35(5), 592–608. Maglaras, L.A. and Katsaros, D. (2016). Social clustering of vehicles based on semi-markov processes. IEEE Transactions on Vehicular Technology, 65(1), 318–332. Maxim, A., Pauca, O., M., M.J., and Caruntu, C.F. (2022). Assessment of computation methods for coalitional feedback controllers. In European Control Conference, 1448– 1453. Reynolds, C. (1987). Flocks, herds, and schools: A distributed behavioral model. ACM SIGGRAPH Computer Graphics, 21, 25–34. Youyun, X., Selo, S., Selo, S., and Ronald, A. (2019). Coalitional game theoretical approach for VANET clustering to improve SNR. Journal of Computer Networks and Communications. Ovidiu Pauca et al. / IFAC PapersOnLine 56-2 (2023) 1460–1465 1465 0 10 20 30 40 50 60 70 80 90 100 Time[samples] 0 1 2 3 4 5 6 7 8 9 10 11 Topology ModK IK LMI-based Fig. 5. Network mode Topology Λ1assumes that only the vehicles {V1,V 2} are linked. From Figs. 2-5 it can be observed that the decentralised solution, Λ0, is used in all three cases when the states are close to the origin. Also, from these figures, it can be observed that for the methods IK and ModK all the states reach first the origin; from that point on, the communication links are disabled and, the platoon uses only the decentralised control, i.e., topology Λ0. The maximum values of cost functions of the three methods are: JModK = 177.6, JIK = 172.9, JLMI = 203.4, where the cost Jis computed for all simulation time i.e., 100 time samples, and is the same cost as the one used by Algorithms 2 and 3. It can be noticed that methods which use overlapping coalitions, have similar results and outperform the method based on LMI. The Shapley (9) and characteristic function (8) values of the communication network links, (only I,II and III), are illustrated in table 2. The link with the smallest values is I, which connects vehicles {V1,V 2}. Moreover, this result is in accordance with the characteristic values, which have smaller values compared to all topologies with a single link. These values prove that the communication between vehicles V1and V2has the greatest relevance in the performance of the overall platoon. Table 2. Characteristic function and Shapely values Links I II III Method fσ lfσ lfσ l ModK -19.64 -4.49 2.68 0.51 -2.16 -0.14 IK -19.29 -5.41 11.05 2.12 -1.4 -0.98 LMI -21.98 -6.47 14.92 2.27 -4.9 -1.82 6. CONCLUSIONS The paper proposes a bio-inspired coalitional control strategy to regulate distances between the vehicles of a platoon. The novelties of the paper are represented by the use of two new methods based on the minimisation of a cost function for designing the controllers for each topology and the introduction of overlapping topologies, which can better optimise the use of the communication network. The results were compared with a solution based on the LMI approach. The simulations show that the proposed methods with overlapping coalitions outperform the LMI- based classical coalition control approach. REFERENCES Aghababa, M.P. and Saif, M. (2020). Finite time sliding mode control of connected vehicle platoons guaranteeing string stability. In IEEE International Conference on Human-Machine Systems, 1–6. Ayyub, M., Oracevic, A., Hussain, R., Khan, A.A., and Zhang, Z. (2022). A comprehensive survey on clustering in vehicular networks: Current solutions and future challenges. Ad Hoc Networks, 124, 102729. Badue, C., Guidolini, R., Carneiro, R.V., Azevedo, P., Cardoso, V.B., Forechi, A., Jesus, L.F.R., Berriel, R.F., Paixao, T.M., Mutz, F.W., Oliveira-Santos, T., and de Souza, A.F. (2019). Self-driving cars: A survey. CoRR, abs/1901.04407. Bai, H., Arcak, M., and Wen, J. (2011). Introduction, 1–17. Springer New York, New York, NY. Baldivieso-Monasterios, P.R. and Trodden, P.A. (2021). Coalitional predictive control: Consensus-based coalition forming with robust regulation. Automatica, 125, 109380. Barreiro-Gomez, J. and Zhu, Q. (2022). Coalitional stochastic differential games for networks. IEEE Control Systems Letters, 6, 2707–2712. Chanfreut, P., Maestre, J.M., and Camacho, E.F. (2021). A survey on clustering methods for distributed and networked control systems. Annual Reviews in Control, 52, 75–90. Chanfreut, P., Keijzer, T., Ferrari, R.M., and Maestre, J.M. (2020). A topology-switching coalitional control and observation scheme with stability guarantees. IFAC-PapersOnLine, 53(2), 6477–6482. Fang, C., Shangguan, J., Li, H., and Yue, M. (2021). Distributed cooperative control of vehicle platoon based on a composite safety distance. In 6th IEEE International Conference on Advanced Robotics and Mechatronics, 535–540. Leon Calvo, J.A. and Mathar, R. (2018). Connected vehicles coordination: A coalitional game-theory approach. In European Conference on Networks and Communications, 1–6. Lewis, F.L., Zhang, H., Hengster-Movric, K., and Das, A. (2014). Introduction to Synchronization in Nature and Physics and Cooperative Control for Multi-Agent Systems on Graphs, 1–21. Springer London, London. Maestre, J.M., Mu˜noz de la Pe˜na, D., Jim´enez Losada, A., Algaba, E., and Camacho, E. (2014). A coalitional control scheme with applications to cooperative game theory. Optimal Control Applications and Methods, 35(5), 592–608. Maglaras, L.A. and Katsaros, D. (2016). Social clustering of vehicles based on semi-markov processes. IEEE Transactions on Vehicular Technology, 65(1), 318–332. Maxim, A., Pauca, O., M., M.J., and Caruntu, C.F. (2022). Assessment of computation methods for coalitional feedback controllers. In European Control Conference, 1448– 1453. Reynolds, C. (1987). Flocks, herds, and schools: A distributed behavioral model. ACM SIGGRAPH Computer Graphics, 21, 25–34. Youyun, X., Selo, S., Selo, S., and Ronald, A. (2019). Coalitional game theoretical approach for VANET clustering to improve SNR. Journal of Computer Networks and Communications.