Energy-Optimal Goal Assignment of Multi-Agent System with Goal Trajectories in Polynomials
EEnergy-Optimal Goal Assignment of Multi-Agent Systemwith Goal Trajectories in Polynomials
Heeseung Bang,
Student Member, IEEE , Logan E. Beaver,
Student Member, IEEE ,Andreas A. Malikopoulos,
Senior Member, IEEE
Abstract — In this paper, we propose an approach for solvingan energy-optimal goal assignment problem to generate thedesired formation in multi-agent systems. Each agent solves adecentralized optimization problem with only local informationabout its neighboring agents and the goals. The optimizationproblem consists of two sub-problems. The first problem seeksto minimize the energy for each agent to reach certain goals,while the second problem entreats an optimal combinationof goal and agent pairs that minimizes the energy cost. Byassuming the goal trajectories are given in a polynomial form,we prove the solution to the formulated problem exists globally.Finally, the effectiveness of the proposed approach is validatedthrough the simulation.
I. I
NTRODUCTION
Control of swarm systems is an emerging topic in thefields of controls and robotics. Due to their adaptability andflexibility [1], swarm systems have attracted considerableattention in transportation [2], construction [3], andsurveillance [4] applications. As we deploy swarms inexperimental testbeds [5]–[8] and outdoor experiments [9],it is critical to minimize the cost per agent to ensure swarmsare an affordable solution to emerging problems. This isthe driving force behind energy-optimal control algorithms,which reduce the battery storage requirements, and therefore,the cost, of agents while simultaneously expanding theiruseful life.A fundamental problem in swarm systems is theassignment of agents to a particular formation. There is arich literature on the creation of a desired formation, suchgenerating rigid formations from triangular sub-structures[10], [11], crystal growth-inspired algorithms [12], andregion-based formation controllers [13]. It is also possiblefor agents to construct formations using only scalar, bearing,or distance measurements [14], [15], and many formationproblems may be solved using consensus techniques [16].However, only a few of these approaches consider the energycost to individual agents in the swarm.Similar to the efforts reported in [17]–[19], we seek theassignment of a finite number of agents to a set of desiredstates. Our approach leverages optimal control to guaranteeinter-agent collision avoidance while minimizing the energyconsumed by each agent. Unlike [17], our approach is not
This research was supported by the Sociotechnical Systems Center (SSC)at the University of Delaware.The authors are with the Department of Mechanical Engineering,University of Delaware, Newark, DE 19716, USA. (emails: [email protected]; [email protected];[email protected].) pairwise between agents, instead we consider all nearbyagents during goal assignment. Our approach also does notrequire the agents to be assigned to unique goals a priori.Similar to [18], our approach imposes a priority ordering onthe agents to generate assignments and trajectories. However,our approach to prioritization is dynamic and decentralized,as opposed to the global static priority presented in [18].Finally, our approach to assignment only considers the localarea around an agent, unlike the global auction algorithm in[19]. Additionally, we consider the unconstrained energy costrequired to reach a goal during assignment, whereas [17]–[19] only consider the distance to the goal. In other words,our approach considers the energy cost required for the agentto match the goal’s velocity.By leveraging optimal control, we explicitly allow for theprioritization of safety as a hard constraint on the system.Strong guarantees on safety are valuable to avoid inter-agentcollisions and to guarantee that agents avoid obstacles in theenvironment. We propose an extension of our previous workon energy-optimal goal assignment and trajectory generation[20], [21]. The main contributions of this paper are: (1) weoptimally determine the arrival time of each agent duringassignment, while we provide a set of sufficient conditionson the goal dynamics to guarantee that the arrival time isfinite; and (2) we propose an event-triggered approach togoal assignment that guarantees all agents will converge to aunique goal. We also provide a numerical demonstration ofour improved assignment and trajectory generation scheme.The remainder of the paper is organized as follows. InSection II, we formulate the optimal goal assignment andtrajectory generation problem. In Section III, we formulatethe goal assignment problem and provide an event-triggeredupdate scheme that guarantees convergence. In Section IV,we explain the trajectory planning scheme, and in SectionV, we quantify the improvement in performance over ourprevious work [20], [21]. Finally, we draw our conclusionsand propose future research directions in Section VI.II. M
ODELING F RAMEWORK
We consider a problem of generating a desired formationby allocating N ∈ N agents into M ∈ N goals, where M ≥ N . The agents and the goals are indexed by the sets A = { , . . . , N } and F = { , . . . , M } , respectively. Forcontinuous time t ∈ R ≥ , each agent i ∈ A obeys double- a r X i v : . [ c s . M A ] J a n ntegrator dynamics, ˙ p i ( t ) = v i ( t ) , (1) ˙ v i ( t ) = u i ( t ) , (2)where p i ( t ) ∈ R and v i ( t ) ∈ R are the time-varyingposition and velocity vectors, and u i ( t ) ∈ R is the controlinput. The control input and velocity of each agent arebounded by || v i ( t ) || ≤ v max , (3) || u i ( t ) || ≤ u max , (4)where v max and u max are the maximum allowable speed andcontrol inputs, and || · || is the Euclidean norm. The state ofeach agent is given by the time-varying vector x i ( t ) = (cid:20) p i ( t ) v i ( t ) (cid:21) . (5)We denote the distance between two agents i, j ∈ A by d ij ( t ) = || p i ( t ) − p j ( t ) || . (6)In order to avoid collisions between agents, we impose thefollowing pairwise constraints for all agents i, j ∈ A , i (cid:54) = j , d ij ( t ) ≥ R, ∀ t ≥ , (7) h (cid:29) R, (8)where R ∈ R > is the radius of a safety disk centered oneach agent, and h ∈ R > is the sensing and communicationhorizon. Next, we define the neighborhood of an agent,which is our basis for local information. Definition 1.
The neighborhood of agent i ∈ A is the time-varying set N i ( t ) = (cid:110) j ∈ A (cid:12)(cid:12)(cid:12) d ij ( t ) ≤ h (cid:111) . Agent i may sense and communicate with every neighboringagent j ∈ N i ( t ) .We also define the notion of desired formation . Definition 2.
The desired formation is the set of time-varying vectors G ( t ) = { p ∗ k ( t ) ∈ R | k ∈ F} .The set G ( t ) can be prescribed offline, i.e., by a designer,or online by a high-level planner. Since we consider thedesired formation with polynomial trajectories, each goal k ∈ F has the form p ∗ k ( t ) = η (cid:88) l =0 c k,l t l , η ≥ , (9)where η is the degree of the polynomial and the coefficients c k,l ∈ R are constant vectors.We impose the following model for the rate of energyconsumption by agent i ∈ A , ˙ E i ( t ) = 12 || u i ( t ) || . (10) Physically, this energy model implies that minimizing L norm of acceleration directly reduces the total energyconsumed by each agent.In our modeling framework, we impose the followingassumptions. Assumption 1.
There are no errors or delays withrespect to communication and sensing within each agent’sneighborhood.
Assumption 2.
The energy cost of communication isnegligible, i.e., the energy consumption is only in the formof (10).
Assumption 3.
Each agent has a low-level onboardcontroller that can track the generated optimal trajectory.Assumption 1 is employed to characterize the idealizedperformance of our approach. This may be relaxed by usinga stochastic optimal control problem, or robust control, fortrajectory generation. Assumption 2 may be relaxed forthe case with long-distance communication. For that case,the communication cost can be controlled by varying thecommunication horizon h . Assumption 3 may be strongfor certain applications. This assumption may be relaxedby including kinematic constraints in the optimal trajectorygeneration problem, or by employing a robust low-levelcontroller, such as a control barrier function, for tracking.III. O PTIMAL G OAL A SSIGNMENT
The objective of a goal assignment problem is to assigneach agent to a unique goal such that the total energyconsumption of all agents is minimized. We separate thisinto two sub-problems: (1) finding the minimum-energyunconstrained trajectory for each agent to reach every goal,and (2) finding the optimal assignment of agents to goalssuch that total energy consumption is minimized and at mostone agent is assigned to each goal.To solve the first sub-problem, we consider the caseof any agent i ∈ A traveling between two fixed stateswith the energy model in the form of (10). In thiscase, Hamiltonian analysis yields the following optimalunconstrained minimum-energy trajectory [22], u i ( t ) = a i t + b i , (11) v i ( t ) = a i t + b i t + c i , (12) p i ( t ) = a i t + b i t + c i t + d i , (13)where a i , b i , c i , and d i are constant vectors of integration.Thus, we get the minimum required total-energy for agent i to reach the goal k ∈ F , by substituting (11) into (10), thatis, E i,k ( t i,k ) = (cid:90) t i,k || u i ( τ ) || dτ = a i,x + a i,y t i,k + ( a i,x b i,x + a i,y b i,y ) t i,k + ( b i,x + b i,y ) t i,k , (14)here t i,k is the time taken for the agent i to reach thegoal k , and a i = [ a i,x , a i,y ] T , b i = [ b i,x , b i,y ] T are thecoefficients of (11). We solve for the coefficients a i and b i by substituting the boundary conditions into (12) and (13), a i = 12 t i,k ( p i, − p ∗ k ( t i,k )) + 6 t i,k ( v i, + v ∗ k ( t i,k )) , (15) b i = − t i,k ( p i, − p ∗ k ( t i,k )) − t i,k (2 v i, + v ∗ k ( t i,k )) . (16)Here, p i, and v i, are the initial position and velocity ofthe agent i , respectively. Next, we define an optimizationproblem to find the minimum-energy arrival time. Problem 1 (Energy Minimization) . The minimum-energyarrival time for agent i ∈ A traveling to goal k ∈ F isfound by solving the following optimization problem, E ∗ i,k = min t i,k E i,k ( t i,k ) (17)subject to (9) , Proposition 1.
For goal trajectories in the form of (9), therealways exists a globally optimal solution to Problem 1.
Proof.
First we substitute (9) and its time derivative into (15)and (16), which yields equations of the form a i = η (cid:88) l =0 c l,a t l − , (18) b i = η (cid:88) l =0 c l,b t l − , (19)Squaring (18) and (19) and substituting the result into (14)yields an equation of the form E i,k ( t i,k ) = η (cid:88) l =0 α l t l − i,k , (20)where α l are constant numbers, and α η > , α > . Eq.(9) implies that η ≥ , thus (20) always has polynomialand inverted radical terms. Thus, as t → ∞ , the polynomialterms dominate and lim t →∞ E i,k ( t ) = ∞ . (21)As t → + , the inverted radical terms dominate, and lim t → + E i,k ( t ) = ∞ . (22)Finally, u i ( t ) ∈ R implies that E i,k ( t ) ≥ for t ∈ (0 , ∞ ) by (14). From (21), if we select sufficiently small positivenumber ε , there exists γ such that E i,k ( γ ) > E i,k ( ε ) , ∀ γ ∈ (0 , ε ) . Likewise, from (22), for sufficiently large number β ,there exists δ such that E i,k ( β ) < E i,k ( δ ) , ∀ δ ∈ ( β, ∞ ) .This implies that the local minimum in [ ε, β ] is the globalminimum as well. According to the boundness theorem incalculus, a continuous function in the closed interval isbounded on that interval. That is, for the continuous function(20) in [ ε, β ] , there exist real number m and ¯ m such that: m < E i,k ( t ) < ¯ m, ∀ t ∈ [ ε, β ] , (23) and the proof is complete.Proposition 1 enables the agent to consider the energy-optimal arrival time during goal assignment. In contrast, ourprevious work [20], [21] uses a fixed arrival time that isselected offline by a designer.After the energy minimization is complete, each agentassigns itself and its neighbors to unique goals. This isachieved using an assignment matrix A i ( t ) of size |N i ( t ) | × M , which we define next. Definition 3.
The assignment matrix A i ( t ) for each agent i ∈ A maps all agents j ∈ N i ( t ) to a unique goal index g ∈ F . The elements of A i ( t ) are binary valued, and eachagent is assigned to exactly one goal.We determine the assignment matrix by solving adecentralized optimization problem, which we present laterin this section. Next, we define the prescribed goal to showhow the agent uses the assignment matrix. Definition 4.
For agent i ∈ A , the prescribed goal is p ai ( t ) ∈ (cid:8) p ∗ k ∈ G | a ik = 1 , a ik ∈ A i ( t ) , k ∈ F (cid:9) . (24)Since the prescribed goal is determined using only localinformation, it is possible that two agents with differentneighborhoods will prescribe themselves the same goal. Tosolve this problem, each agent must know which agent itis competing with and which one has priority for the goal.This motivates our definitions of competing agents and thepriority indicator function. Definition 5.
The set of competing agents for agent i ∈ A is given by C i ( t ) = (cid:8) j ∈ N i ( t ) | p aj ( t ) = p ai ( t ) , i (cid:54) = j (cid:9) . (25)The information about competing agent is updatedwhenever a new agent enters the neighborhood of agent i . Ifthere is at least one competing agent, that is |C i ( t ) | ≥ , thenall agents j ∈ C i ( t ) must compare their priority indicatorfunction, which we define next. Definition 6.
For each agent i ∈ A , we define the priorityindicator function I i : A \ { i } → { , } . We say that thatagent i ∈ A has priority over agent j ∈ A \ { i } , if and onlyif I i ( j ) = 1 . Additionally, I i ( j ) = 1 if and only if I j ( i ) = 0 .The functional form of the priority indicator functionis determined offline by a designer and is the same forall agents. By Assumption 1 the information required toevaluate priority is instantaneously and noiselessly measuredand communicated between agents. Following this policy,the agent with no priority is permanently banned from itsprescribed goal. Definition 7.
We denote the set of banned goals for agent i ∈ A as B i ( t ) ⊂ F . (26)lements are never removed from B i ( t ) , and a goal g ∈ F is added to B i ( t ) , if p ai ( t ) = p ∗ g ( t ) ∈ G and I i ( j ) = 0 forany j ∈ C i ( t ) \ { i } .Agent i ∈ A assigns itself a prescribed goal by solvingthe following optimization problem, where we include thebanned goals as constraints. Problem 2 (Goal Assignment) . Each agent i ∈ A selectsits prescribed goal (Definition 4) by solving the followingbinary program:min a jk ∈ A i (cid:40) (cid:88) j ∈N i ( t ) (cid:88) k ∈F a jk E ∗ j,k (cid:41) (27)subject to: (cid:88) k ∈F a jk = 1 , j ∈ N i ( t ) , (28) (cid:88) j ∈N i ( t ) a jk ≤ , k ∈ F , (29) a jk = 0 , ∀ j ∈ N i ( t ) , k ∈ B j ( t ) , (30) a jk ∈ { , } . Next, we present Algorithm 1, which describes our event-driven protocol for assigning agents to goals using thecompeting agent set, priority indicator function, and bannedgoal set.
Algorithm 1:
Event-driven algorithm to determinethe prescribed goal for each agent i ∈ A .Solve Problem 2;Determine prescribed goal;Generate optimal trajectory to assigned goal; if |C i ( t ) | ≥ then Compare I i ( j ) for all j ∈ C i ( t ) ; if any I i ( j ) = 0 then Add current goal to B i ( t ) ;Solve Problem 2;Determine prescribed goal;Generate optimal trajectory to assigned goal; endendProposition 2 (Solution Existence) . A solution to Problem2 always exists.
Proof.
Let B ( t ) = (cid:83) i ∈A B i ( t ) be the set of all goals whichany agent is banned from. Let n b ( t ) = |B ( t ) | , then based onAlgorithm 1, there must be exactly n b ( t ) agents assigned tothe n b ( t ) banned goals. Thus, any agent i ∈ A must assignat most N − n b ( t ) agents to M − n b ( t ) goals when solvingProblem 2. As M ≥ N , M − n b ( t ) ≥ N − n b ( t ) , and thefeasible space of Problem 2 is always non-empty.Each agent i ∈ A initially solves Problem 2 to assign itselfto a goal, and re-solves Problem 2 whenever its neighborhood N i ( t ) switches and the set of competing agents becomesnon-empty. It is possible that several agents may assignthemselves to the same goal. If it is the case, all conflictingagents repeat the banning and assignment process until allagents are assigned to a unique goal. Next, using Proposition1 and Proposition 2, we propose Theorem 1 which guaranteesconvergence of all agents to a unique goal in a finite time. Theorem 1.
Let any agent i ∈ A be assigned to a goal k ∈F under our proposed banning and reassignment approach(Definitions 5 - 7) and polynomial goal trajectories (9). If thesolution to Problem 1 is never increasing, i.e., E ∗ i,k ( t ) ≥ E ∗ i,k ( t ) for sequential assignments of agent i to goal k attimes t , t ∈ R ≥ , where t > t , then all agents arrive attheir unique assigned goal in finite time. Proof.
First, for each agent i ∈ A assigned to a goal k ∈ F , Proposition 1 implies that a finite arrival time, t i,k always exists. Second, Propsition 2 implies that a solutionto the assignment problem (Problem 2) always exists. Thisis sufficient to satisfy the premise of the AssignmentConvergence Theorem presented in [21], which guaranteesall agents arrive at a unique goal in finite time.IV. O PTIMAL P ATH P LANNING
After being assigned to a goal with the optimal arrivaltime, each agent must find the energy-optimal trajectory toreach their assigned goal. For trajectory generation, eachagent plans over the horizon [0 , t i,k ] ⊂ R ≥ , where t = 0 isthe current time and t = t i,k is the optimal arrival time. Theinitial and final states of each agent i ∈ A is p i (0) = p i , v i (0) = v i , (31) p i ( t i,k ) = p ai ( t i,k ) , v i ( t i,k ) = ˙ p ai ( t i,k ) , (32)where t i,k is the argument that minimizes Problem 1. Toavoid collisions we impose a safety constraint to all agentswith lower priority, d ij ( t ) ≥ R, ∀ j ∈ { ξ ∈ A | I i ( ξ ) = 0 } , (33) ∀ t ∈ [ t i , t i,k ] . Next, we formulate the decentralized optimal pathplanning problem.
Problem 3 (Path Planning) . For each agent i ∈ A assignedto goal k ∈ F , the optimal path can be found by solving thefollowing optimal control problem, min u i ( t ) (cid:90) t i,k || u i ( τ ) || dτ (34)subject to: (1) , (2) , (3) , (4) , given: (31) , (32) . We derive the analytical solution to this problem byfollowing the standard methodology used in optimal controlproblems with state and control constraints [22]–[25]. First,we consider the unconstrained solution, given by (11) - (13).If the solution violates any of the constraints, then it isconnected with the new arc corresponding to the violatedig. 1: Simulation result for the proposed method with h = ∞ constraint. This yields a set of the algebraic equation thatare solved simultaneously using the boundary conditions ofProblem 3 and interior conditions between the arcs. Thisprocess is repeated until no constraints are violated, whichyields the feasible solution for Problem 3.The solution is a piecewise-continuous state trajectorycomposed of the following optimal motion primitives [21]:1) no constraints are active,2) one safety constraint is active,3) multiple safety constraints are active,4) one state/control constraint is active, and5) multiple state/control constraint are active.For the full derivation of the solution for each case, see [21].V. S IMULATION R ESULTS
In this section, we present a series of simulation resultsto evaluate the effectiveness of the proposed method. Allthe simulations were conducted with N = M = 10 agentsand goals. The velocity of all the goals are given by thepolynomials v ∗ ( t ) = (cid:20) v ∗ x ( t ) v ∗ y ( t ) (cid:21) = (cid:20) . t − . t + 0 . t . t + 0 . (cid:21) . (35)We randomly selected the initial positions of the agents in R , which we then fixed for each simulation.To demonstrate the effect of the energy-optimal arrivaltime (Problem 1), we compared the simulation results ofthe proposed method with that of the previous method [21],as shown in Fig. 1 and Fig. 2. We selected T = 5 for thetime parameter of the previous method. To remove the effectof decentralization on the performance, we set the sensingdistance h = ∞ for both cases. Proposed method Previous methodEnergy consumption 0.69 kJ/kg 7.86 kJ/kgTotal arrival time 4.57 s 5 s
TABLE I: Numerical result for comparison between theproposed method and the previous one. Fig. 2: Simulation result for the previous method with h = ∞ Fig. 3: Total energy consumption of each agentsNumerical results are shown in Table I. The proposedmethod reduced the total energy consumption by 91.2%compared to the previous method. This result shows that, insome cases, not only energy consumption but also the totaltime required to achieve the desired formation is improved.We attribute this improvement to our algorithm selecting theoptimal arrival time through Problem 1, rather than using afixed arrival time. The energy use of each agent for bothcases are given in Fig. 3, and all the agents consumed aminimum of 83.8% to a maximum of 97.2% less energythan the previous method.Next, we simulated the agents with various sensingdistances to understand its effect on performance. Weimplemented a priority indicator function based on the h [m] min. separation E t f Total bans[cm] [kJ/kg] [s] ∞ a) h = 0 . m (b) h = 0 . m(c) h = 1 m (d) h = 1 . m Fig. 4: Trajectory of each agents with different sensing distances.neighborhood size, energy cost, and index of each agent asdescribed in [21]. The results are shown in Table II, andFig. 4 illustrates the trajectories generated by the agents withvarious values of h . As with our previous work, [20], theresults in Table II show no correlation between the sensingdistance and energy consumption. With respect to the agents’initial position and the desired formation, some informationforces the agent to select the goal that is further than the onethe agent would choose without that information, resultingin extra energy consumption. This process is shown in Fig.4. Compared to (a), the trajectory of one agent (shown withthe orange line) gets longer and longer in (b), (c), and (d).The agent with a longer sensing distance may select a bettergoal at the beginning due to its extra information about otheragents. However, as shown in Table II, this may increase thenumber of banned goals, resulting in a higher number ofassignments and reducing performance.VI. C ONCLUSION
In this paper, we proposed an extension of our previouswork on energy-optimal goal assignment and trajectorygeneration. The goal assignment task was separated intotwo sub-problems that include (1) finding energy-optimalarrival time and (2) assigning each agent to a unique goal. With the goal dynamics in the form of polynomials, weproved that our proposed approach guarantees that all agentsarrive at a unique goal in finite time. We validated theeffectiveness of our approach through simulation. Comparedto previous work, we have shown a significant reduction inenergy consumption.Future work should consider how the initial position of theagents and desired formation affects energy consumption.Quantifying the relationship between sensing distance andperformance is another interesting area of research, as wellas adapting agent memory and other information structuresto the problem. Finally, using recent results constraint-drivenoptimal control [26] to generate agent trajectories in real timeis another compelling research direction.R
EFERENCES[1] H. Oh, A. R. Shirazi, C. Sun, and Y. Jin, “Bio-inspired self-organisingmulti-robot pattern formation: A review,”
Robotics and AutonomousSystems , vol. 91, pp. 83–100, 2017.[2] B. Chalaki, L. E. Beaver, and A. A. Malikopoulos, “Experimentalvalidation of a real-time optimal controller for coordination of cavs ina multi-lane roundabout,” in , 2020, pp. 504–509.[3] Q. Lindsey, D. Mellinger, and V. Kumar, “Construction with quadrotorteams,”
Autonomous Robots , 2012.[4] J. Cortes, “Global formation-shape stabilization of relative sensingnetworks,” in
Proceedings of the American Control Conference , 2009.5] D. Pickem, P. Glotfelter, L. Wang, M. Mote, A. Ames, E. Feron, andM. Egerstedt, “The Robotarium: A remotely accessible swarm roboticsresearch testbed,” in
IEEE International Conference on Robotics andAutomation , 7 2017, pp. 1699–1706.[6] A. Stager, L. Bhan, A. A. Malikopoulos, and L. Zhao, “A scaled smartcity for experimental validation of connected and automated vehicles,”in , 2018,pp. 130–135.[7] M. Rubenstein, C. Ahler, and R. Nagpal, “Kilobot: A low cost scalablerobot system for collective behaviors,” in
Proceedings of the 2012IEEE International Conference on Robotics and Automation , 2012.[8] L. E. Beaver, B. Chalaki, A. M. Mahbub, L. Zhao, R. Zayas, and A. A.Malikopoulos, “Demonstration of a Time-Efficient Mobility SystemUsing a Scaled Smart City,”
Vehicle System Dynamics , vol. 58, no. 5,pp. 787–804, 2020.[9] G. V´as´arhelyi, C. Vir´agh, G. Somorjai, T. Nepusz, A. E. Eiben, andT. Vicsek, “Optimized flocking of autonomous drones in confinedenvironments,”
Science Robotics , vol. 3, no. 20, 2018.[10] J. Guo, Z. Lin, M. Cao, and G. Yan, “Adaptive control schemesfor mobile robot formations with triangularised structures,”
IETControl Theory & Applications , vol. 4, no. 9, pp. 1817–1827, 2010.[Online]. Available: http://digital-library.theiet.org/content/journals/10.1049/iet-cta.2009.0513[11] Y. Hanada, G. Lee, and N. Y. Chong, “Adaptive Flocking of aSwarm of Robots Based on Local Interactions,” in
IEEE SwarmIntelligence Symposium , 2007, pp. 340–347. [Online]. Available: http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4223194[12] Y. Song and J. M. O’Kane, “Forming repeating patterns ofmobile robots: A provably correct decentralized algorithm,” in
IEEEInternational Conference on Intelligent Robots and Systems , vol. 2016-Novem, 2016, pp. 5737–5744.[13] C. C. Cheah, S. P. Hou, and J. J. E. Slotine, “Region-based shapecontrol for a swarm of robots,”
Automatica , 2009.[14] J. O. Swartling, I. Shames, K. H. Johansson, and D. V. Dimarogonas,“Collective Circumnavigation,”
Unmanned Systems , vol. 02, no. 03,pp. 219–229, 2014.[15] Z. Lin, M. Broucke, and B. Francis, “Local control strategies for groups of mobile autonomous agents,”
IEEE Transactions onAutomatic Control , vol. 49, no. 4, pp. 622–629, 2004.[16] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus andcooperation in networked multi-agent systems,”
Proceedings of theIEEE , vol. 95, no. 1, pp. 215–233, 2007.[17] M. Turpin, N. Michael, and V. Kumar, “CAPT: Concurrent assignmentand planning of trajectories for multiple robots,”
International Journalof Robotics Research , vol. 33, no. 1, pp. 98–112, 2014.[18] M. Turpin, K. Mohta, N. Michael, and V. Kumar, “Goal Assignmentand Trajectory Planning for Large Teams of Aerial Robots,”
Proceedings of Robotics: Science and Systems , vol. 37, pp. 401–415,2013.[19] D. Morgan, G. P. Subramanian, S.-J. Chung, and F. Y. Hadaegh,“Swarm assignment and trajectory optimization using variable-swarm,distributed auction assignment and sequential convex programming,”
International Journal of Robotics Research , vol. 35, no. 10, pp. 1261–1285, 2016.[20] L. E. Beaver and A. A. Malikopoulos, “A Decentralized ControlFramework for Energy-Optimal Goal Assignment and TrajectoryGeneration,” in
IEEE 58th Conference on Decision and Control , 2019,pp. 879–884.[21] ——, “An Energy-Optimal Framework for Assignment and TrajectoryGeneration in Teams of Autonomous Agents,”
Systems & ControlLetters , vol. 138, April 2020.[22] A. A. Malikopoulos, C. G. Cassandras, and Y. J. Zhang,“A decentralized energy-optimal control framework for connectedautomated vehicles at signal-free intersections,”
Automatica , vol. 93,pp. 244–256, 2018.[23] A. E. J. Bryson and Y.-C. Ho,
Applied Optimal Control: Optimization,Estimation, and Control . John Wiley and Sons, 1975.[24] A. A. Malikopoulos, L. E. Beaver, and I. V. Chremos, “Optimal timetrajectory and coordination for connected and automated vehicles,”
Automatica , vol. 125, 2021.[25] I. M. Ross,
A Primer on Pontryagin’s Principle in Optimal Control ,2nd ed., E. Solon, Ed. San Francisco: Collegiate Publishers, 2015.[26] L. E. Beaver, M. Dorothy, C. Kroninger, and A. A. Malikopoulos,“Energy-Optimal Motion Planning for Agents: Barycentric Motion andCollision Avoidance Constraints,” in arxiv:2009.00588arxiv:2009.00588