Multi-Agent Robot Motion Planning for Rendezvous Applications in a Mixed Environment with a Broadcast Event-Triggered Consensus Controller

Abstract

Finding consensus is one of the most important tasks in multi-agent robot motion coordination research, especially in a communication environment. This justification underlies the use of event-triggered controller in current multi-agent consensus research. However, the communication issue has not been adequately addressed in a broadcast communication environment for rendezvous applications. Therefore, the broadcast event-triggered (BET) controller with a new formulation was designed using the Simultaneous Perturbation Stochastic Algorithm (SPSA). Theorems and relevant proofs were presented. Agent performances with the BET controller were evaluated and compared with the conventional broadcast time-triggered (BTT) controller. The results showed an effective motion generated by a multi-agent robot to reach the rendezvous point based on the Bernoulli distribution and gradient approximation of the agent local controller. The BET controller has proven to work more efficiently than the BTT controller when it reaches convergence in less than 40.42% of time and 21.00% of iterations on average. The utilization of communication channels is slightly reduced for BET, which is 71.09% usage instead of fully utilized by BTT. The threshold value of the event-triggered function (ETF) and SPSA parameters affected agent performances. Future research may consider using an effective and efficient BET controller in a complex communication environment with many variations of graph topology networks.

1. Introduction

1.1 Motivation of the research

The effectiveness and robustness of multi-agent robot systems (MARS) [1, 2, 3, 4] in carrying out tasks as compared to single agent [5, 6, 7] has led to an expansion in the MARS research. The advantages of having MARS that may increase the flexibility and scalability as well as reduce load among agents, makes cooperative MARS research still active until present day. Example of applications with MARS such as medical robots known as the nano [8] and magnetic robot [9] have been used to send medicine directly to the human organ, inspection robots to inspect and clean pipelines in oil and gas industries [10, 11], planetary rovers [12, 13], unmanned aerial vehicles [14, 15, 16, 17, 18], and swarm robots [19, 20]. Due to the importance of MARS in assisting human activities, several researchers have been actively reviewing and discussed a variety of issues related to cooperative MARS [2, 21, 22] such as formation [23, 24, 25, 26, 27], consensus [28, 29, 30], containment [31, 32, 33], tracking [34], and rendezvous [29, 35, 36, 37].

One of the issues that researchers pay a lot of attention to in motion coordination of MARS is consensus subject to the difficulties of agent to find an agreement between agents to a certain degree [22, 38, 39]. Various researchers proposed to develop an effective communication and control system to be used practically in MARS consensus applications. Issues related to limited communication resources [40, 41], time delay [42, 43, 44], and disturbances [45, 46, 47] are among the communication problems which might affect an agent to reach consensus.

Finding strategies to limit the usage of agent resources such as communication [48, 49, 50, 51, 52] and energy [16, 53, 54, 55, 56] has been emphasized in consensus research. This is to ensure that the consensus controller is feasible and practical to be used. Therefore, minimizing communication usage and energy resources is not an option in consensus, especially when the agent is embedded with constrained controller board resources. As a result, an event-based system [35, 47, 57, 58, 59] has been used to reduce communication in terms of the number of communication channels utilised for transmission while also preserving bandwidth coverage. Other examples of research concentrating on decreasing energy resources from trajectory [52] and actuator [53] are energy-awareness or energy-efficiency [16, 60, 61, 62]. The lifespan of a MARS can be increased by conserving resources, which also increases the significance of this research for MARS in practical applications.

In addition, a hybrid controller was developed to provide a robust system that can ensure that the consensus task may be accomplished effectively. Sliding mode controller [46], fuzzy logic techniques [63, 64, 65, 66, 67, 68, 69], model predictive control [70], distributed control [49, 63, 71, 72, 73, 74], and dynamic role assignment [75] are a few examples of hybrid controllers that have been used for multi-agent robot formation, rendezvous, and path planning [76, 77, 78, 79, 80, 81] applications. In order to identify the consensus system efficiently and conserve the agent’s communication and resources, these controllers have been coupled with an event-based system [45, 70, 71, 82, 83].

In relation to wireless network technology, there is a need for a consensus study since the network topologies exist between the agent and its neighbours [29, 73, 84, 85]. For instance, the communication between the UAV ground mobile robot via the network on the air and the ground [86], the communication between the heterogeneous UAV and the satellite and ground station [87], and the communication between the cluster multi-agent and the network topology [88]. Since the agents are connected in a network topology of information flow either directed [89, 90] or undirected [91, 92], one way or two-way communications, one-to-one or broadcast [25, 31, 58, 93, 94, 95, 96, 97, 98], finding consensus can therefore be considered challenging due to the complexity of the network.

The importance of MARS motion coordination consensus research was proven from the above discussion. When communication problems arise in the broadcast and communication contexts, the agent’s performances such as utilization of agent resources and energy levels may be impacted. This served as the impetus for putting up a remedy and plan to create a successful and effective consensus control and communication system. Although some studies on event-triggered have been proposed for consensus research studies such as average consensus [58, 59, 73, 85, 99, 100, 101, 102], leader follower consensus [103], and other consensus or rendezvous control systems [29, 35, 36, 46, 63, 70, 71, 82, 88, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112], research on resolving the communication problem for the application of rendezvous in a broadcast communication environment has yet to be explored.

1.2 Related work

The validity of the MARS motion coordination research was proven when several researchers kept improvising the control and communication systems, especially for consensus problems. For better understanding, this section will discuss the two main important findings found in the literature, which have been divided into two categories: the development of MARS consensus control and event-based consensus.

1.2.1 Multi-agent robot system consensus

The consensus of multi-agents is among the issues that have attracted many researchers’ attention in the past few years [22, 38]. An agreement between agents can be achieved via an appropriate communication and control system to reach certain quantities of interest. Therefore, various control and communication issues related to multi-agent consensus have been discussed to show the recent issues in multi-agent consensus. The development of consensus control depends on the issues solved for every single application. Some focus on velocity and positional consensus issues. The consensus on velocity leads to a collective behaviour where the agents can move with a common velocity, while the consensus on position drives all agents to a common position. Formation [27], tracking [113], and rendezvous [114, 115] were among the consensus applications being focused on by researchers.

Since communication will determine the realization of consensus, a recent development of multi-agent consensus research focusing on communication issues such as time delay [42, 112], disturbances [45, 71], and limited resources [64, 70] has been evident. Therefore, an alternative to broadcast [113, 114, 115, 116, 117] and agent-to-agent [102, 118] communication has been introduced, as well as sampling methods such as time-triggered, periodic, aperiodic, or event-triggered. However, the MARS consensus for rendezvous applications was extended in this research. Applied to a group of leaderless agents with a double integrator system, the issue of communication among agents in a broadcast communication environment has been proposed to determine the realization of multi-agent consensus for practical rendezvous applications. The main objective is to find positional consensus among the multi-agents with zero velocities and acceleration when the rendezvous point is achieved.

1.2.2 Event-based consensus

The idea of event-based sampling was proposed by Astrom et al. [119] in early 1999, while many researchers applied the multi-agent control issue [38, 81, 106, 107]. The objective of having ET is to save communication resources and computation, where the sample data will be sent to the agent and the control update will only occur when it meets the event-triggered function (ETF).

The evolution of event-based control shows that it has been applied initially with a single integrator or order [110], followed by a second order [118], and more recently in a fourth order control system [83] for MARS. Based on positive research outcomes, the ET has thus been recently extended with other intelligent control systems such as Sliding Mode Control [70], adaptive fuzzy [64], and Model Predictive Control [69]. These controllers were integrated with an event-based system, which guaranteed the efficiency of the tracking, formation, and positioning systems in terms of resource utilization.

1.3 Contributions

The broadcast controller has been combined with the event-triggered controller using simultaneous perturbation stochastic algorithm (SPSA) for the rendezvous application for MARS which is the main contribution of this paper [113, 114]. The agents should be able to agree on the rendezvous point while maintaining their performance in terms of convergence time and iteration, trajectory pattern, and number of communication channels. The proposed theorems and proofs are well presented and discussed. Besides that, the broadcast time-triggered (BTT) system, known as a traditional sample technique [95, 97, 98], has been used in this case study as a comparison to the broadcast event-triggered (BET) control system. The effectiveness and efficiency of a new BET control system are evaluated and proven based on the results obtained. Lastly, the effects of SPSA parameter and ET controller changes on agent performances were observed and analyzed. This will be yet another contribution that illustrates the relationship between these hybrid BET controller parameters and agent outputs within the context of the broadcast and communication environment.

1.4 Organization of the chapter

The sections of this paper are structured as follows: the first section and second section presents the introduction and formulation of the problem, the third section focuses on the design of a new BET controller with discussion on related theorems and proof, the fourth section explains the effectiveness and efficiency evaluation of multi-agent robot performances with BET and BTT, the fifth section explains the SPSA and ETF effects on agent performances, and the final section is the conclusion with some recommendations for MARS motion coordination research direction.

2. Problem formulation

Assume a dynamic linear group of agents $i$ connected with neighbours $j$in an undirected topology network located within the broadcast communication environment as shown in Figure 1. The agent has to communicate among themselves to reach consensus to rendezvous point $x_d$ while continuously receiving global feedback $B\left(t\right)$ from global controller . This is to satisfy the motion coordination task in a such a way that the value of $P\left(x\left(t\right)\right)=0$which shows the convergence achieved as derived from Eqs. (1) and (2).

Figure 1.

Multi-agent robot working environment.

Referring to this environment, the agents have limited knowledge of its environment whereby the system does not provide any global coordinates and the agents will rely more on their relative position during communication. Due to this situation, it may cause high utilization of communication resources such as channel and bandwidth required when the agent must communicate continuously to reach consensus to the desired rendezvous point. This might get worse if the number of agents increased and the location of the target is quite far from the agent. The communication issue must be considered as it might affect the effectiveness and the efficiency of the agent, especially when the agent is supplied with limited power from the microcontroller board. The agent’s performances such as time and iteration are taken as well as trajectory will get effected until convergence is achieved. While various problems had been solved in the scope of agent in broadcast and communication environment such as quantization [97], collision avoidance [115], instability [95], consensus [98, 116], and event-triggered [35, 64, 117, 118], the communication problem within the range of broadcast and communication environment still remains unsolved.

$$P\left(x\right)=\sum _{i=1}^N\left|\left|x_i-x_d\right|\right|$$E1

$${lim}_{t\to \infty }P\left(x\left(t\right)\right)=0$$E2

3. Broadcast event-triggered controller design

Notation: Denote $R$ is the real number, $R_{+}$ is a set of positive real numbers, $n$ is sample size, and $N$ is the population size. The state position of agent in Euclidean coordinates is represented by vector while $x$is expressed by $x_i\left(t\right)={\left|x_1,x_2,x_3\dots .x_n\right|}^T$ $\in R^n$. The function represented by $f\left(x\right)$, $\beta$ SPSA random and deterministic movement, $\alpha$ state function, and $\gamma$ ETF. Assume $e$ is the measurement error, $L_i$ local controller, $G$ global controller, $B$ broadcast signal $\in$ $R,A_i$agent number, $P\left(x\left(t\right)\right)$ is performances index measurement, $t$ $\in N$is discrete time, $(t+1$) $\in N$is next discrete time, and $u_{i,j,R,D1,D2}$ is control input. The agent $i$ is connected with its own neighbor set $j\in N_i$ in undirected graph $G=\left(V,E\right)$where $V$ is vertex represent the agent and $E$is the edge of each vertex represent the connection between $i$ and $j$.

The overall design of this controller is based on the SPSA where the unknown gradient, which is the state position, will be generated using simultaneous perturbation approximation as shown in Figure 2. Measurement of two objective functions is required in this process instead of the overall dimension of agent state position. Firstly, the digital camera will capture the image of the agent’s position and update $x_i\left(t\right)$ in the system. The global controller $G$compute the performances index $P\left(x\left(t\right)\right)$ of the scalar value which indicates the distance between the agent’s and rendezvous point to send $B\left(t\right)$ to the local controller $L_i$. The local controller then computes agent control input $u_i$(t) based on random Bernoulli distribution error and gradient approximation with event-triggered error as shown on the right side of Figure 2. Lastly, each agent $A_i$ will update the next position $x_i\left(t+1\right)$ based on the latest value of control input. This process continuously iterates until all agents reach consensus at the rendezvous point. When the system reaches convergence, agent performances will be evaluated and recorded such as convergence time and iteration, number of channels (NOCs), as well as agent trajectory to determine the effectiveness and efficiency of the system. A detailed description of agent state position, global controller, local controller, ETF, and agent motion are explained in the section below.

Figure 2.

BET controller design.

Agent state position, ${\mathit{x}}_{\mathit{i}}$

The agent state position of $x_i\left(0\right)=\left[x_1\left(0\right),x_2\left(0\right),x_3\left(0\right)\dots \right]\in R^n$ depends on the image from the digital camera. A collective agent position in Cartesian coordinates can be represented as the Euclidean norm of the vector $x_i\left(t\right)=\mid x_1\left(t\right),x_2\left(t\right),x_3\left(t\right)\mid \in R^n$. These positions update will be continuously sent to the global controller until the agent meets the rendezvous point.

Global controller, $\mathit{G}$

The performances of agent were measured based on the degree of agent reaching consensus indicated by the distance between the agent with the rendezvous point as presented by Eqs. (1) and (3). $B\left(t\right)\in R$ represents the output of global controller, $x\left(t\right)\in R^{\mathit{nN}}$is the collective state position of agents, and $P\left(x\right)\in R$ is the objective function of rendezvous.

$$G:B\left(t\right)=P\left(x\left(t\right)\right)$$E3

Local controller, ${\mathit{L}}_{\mathit{i}}$

The local controller or known as distributed controller will determine the agent movement $A_i$at every even and odd $t$ time based on the input from broadcast signal $B\left(t\right)\in R$. The variables of the control system are represented by a column vector as shown in Eq. (4) where ${\delta }_{i1}\left(t\right)$ is the state position, ${\delta }_{i2}\left(t\right)$ is the broadcast signal, ${\delta }_{i3}\left(t\right)$ is the even movement and ${\delta }_{i4}\left(t\right)$ is the odd movement. Three main functions which might affect the output of $u_i\left(t\right)\in R$ (Eq. (5)) are $\alpha \mathit{:}{R}^v\times R\to R^v$ is the state function, $\beta \mathit{:}{R}^{v}XR\to R^v$ is the random and deterministic function of SPSA, and $\gamma \mathit{:}{R}^v\times R\to R^v$ is the standard consensus protocol with ET as stated in Eqs. (6)–(8). As a result, the random error of Bernoulli represented by β even and the deterministic error of SPSA approximation with communication error of ET represented by β odd will determine the agent’s movement as shown in Figures 3 and 4.

Figure 3.

Even movement (random).

Figure 4.

Odd movement (deterministic).

$${\delta }_i\left(t\right)=\left[\begin{array}{c}\begin{array}{c}{\delta }_{i1}\left(t\right)\\ {\delta }_{i2}\left(t\right)\\ {\delta }_{i3}\left(t\right)\end{array}\\ {\delta }_{i4}\left(t\right)\end{array}\right]\in R^n\times R\times R\times R$$E4

$$L_i:\left\{\begin{array}{c}{\delta }_i\left(t+1\right)=\alpha \left({\delta }_i\left(t\right),B\left(t\right)\right)\\ u_i\left(t\right)=\beta \left({\delta }_i\left(t\right),B\left(t\right)\right)+\gamma \left({\delta }_i\left(t\right),B\left(t\right)\right)\end{array}\right.$$E5

$$\alpha \left({\delta }_i\left(t\right),B\left(t\right)\right)=\left[\begin{array}{c}\begin{array}{c}{∆}_i\left(t\right)\\ B\left(t\right)\\ {\delta }_i\left(t+1\right)\end{array}\\ x_j\left(t\right)\end{array}\right]$$E6

$$\beta \left({\delta }_i\left(t\right),B\left(t\right)\right)=\left\{\begin{array}{c}c\left({\delta }_{i3}\left(t\right)\right){∆}_i\mathit{\text{if}}{\delta }_{i3}\left(t\right)\in \left\{0,2,4,\dots .\right\}\\ -c\left({\delta }_{i4}\left(t\right)\right){\delta }_{i1}\left(t\right)-a\left({\delta }_{i4}\left(t\right)\right)\frac{B\left(t\right)-{\delta }_{i2}\left(t\right)}{c\left({\delta }_{i4}\left(t\right)\right)}\ast {\delta }_{i1}^{-1}\left(t\right)\mathit{\text{if}}{\delta }_{i4}\left(t\right)\in \left\{1,3,5,\dots .\right\}\end{array}\right.$$E7

$$\gamma \left({\delta }_i\left(t\right),B\left(t\right)\right)=k\sum _{j\in N_i}{a}_{\mathit{ij}}\mid x_i{\delta }_{i4}\left(t\right)-x_j{\delta }_{i4}\left(t\right)\mid \text{when}{\delta }_{i4}\left(t\right)\in \left\{1,3,5\right\}$$E8

Event-triggered function

The control input of $u_{D2}$ of each agent is determine by the event-triggered process as shown in Figure 5. The purpose of having an event-triggered in the local controller is to reduce communication among agents where the communication among agent $i$ and its neighbour $j\in N_i$ will only happen if the event detected and satisfy the ETF as shown Eq. (9) Eq. (10) indicates the state measurement error between the agent position during the instant event $t_{k+1}^i$and the previous event $t_k^i$ while Eq. (11) is a threshold value or state dependent value which depends on the agent’s state position and neighbour’s state position.

Figure 5.

Agent $i$ event-triggered controller,$\gamma$.

Initially, the state of $x_i\left(t\right)$ is sample by a sampler at every odd time and become $x_i\left(\mathit{nt}\right)$. The event detector will monitor the event based on the given sample. The new update sample state of agent $i,$ $x_i\left(n_{k+1}t\right)$ will be sent to the neighbour $j$ if the event detector detects an event. Right after the neighbours $j$ received the state sent by agent $i$, it will update the agent $i$ state information and store the newly received state values of agent $i,x_i\left(\mathit{nt}\right)$. This state will then be used by the controller and event detector of agent $j$ until the next event is triggered from agent$i$. When there is no event, the sample of agent position $x_i\left(n_{k}t\right)$ will directly be sent to the controller by means of no transmission and a control update will be required at this time. The control signal is held constant until the next event is controlled by having a zero-order holed in the system.

$$f\left(e_i\left(t\right),x_i\left(t\right)\right)\ge {\sigma }_i\left(z_i\left(t\right)\right)$$E9

$$e_i\left(t\right)=x_i\left(t_k^i\right)-x_i\left(t\right)$$E10

$$z_i\left(t\right)=\sum _{j\in N_i}{a}_{\mathit{ij}}\left(x_i\left(t\right)-x_j\left(t\right)\right)$$E11

Agent motion, ${\mathit{A}}_{\mathit{i}}$

The state position of agent $A_i\left(t\right)$ at $\left(t+1\right)$ is based on the latest control input $u_i\left(t\right)$obtained from the local controller $L_i$ as Eq. (12)

$$A_i:x_i\left(t+1\right)=x_i\left(t\right)+u_i\left(t\right)$$E12

3.1 Proposed theorems

Two relevant theorems were proposed which are convergence of the average consensus of distributed controller as Theorem 1 and convergence of BET consensus controller as Theorem 2. The theorems provided with proof derived from theoretical studies as well as proven in the real experiment.

Theorem 1: Control input from communication between the agent and neighbours will affect the agent linear discrete system dynamics. When the system violates the ETF function, the measurement error will be sent by the agent to the neighbours to update the control input whereas if the system satisfies the ETF function, the information will not be sent. The average consensus among the agents is marginally stabilize at $t\to \infty$and reach a steady state error as proven in Gershgorin circle when the Perron matrix eigenvalue is equal to $\lambda$and Lyapunov stability that indicates the definite negative when $V<0.$

Proof 1: Eigenvalues of Perron matrix and Gershgorin circle unit

The dynamic of the system is represented in Eq. (13) by substituting the state measurement error,

$$\begin{array}{ll}{x}_i\left(t+1\right)& =x_i\left(t\right)+u_i\left(t\right)\\ & =\left(\widehat{x_i}\left(t\right)-e_i\left(t\right)\right)+u_i\left(t\right)\\ & =\left(\widehat{x_i}\left(t\right)-e_i\left(t\right)\right)+(\mathrm{\sigma }{a}_{\mathit{ij}}\left(\widehat{x_j}\left(t\right)-\widehat{x_i}\left(t\right)\right)\\ & =-e_i\left(t\right)+\mathrm{\sigma }{a}_{\mathit{ij}}\widehat{x_j}\left(t\right)+\left(1-\mathrm{\sigma }{a}_{\mathit{ij}}\right)\widehat{x_i}\left(t\right))\end{array}$$E13

The Perron matrix can be used to represent the agent’s next position in discrete time by using a normalized Laplacian matrix as shown in Eq. (14),

$$\begin{array}{ll}{x}_i\left(t+1\right)& =x_i\left(t\right)+u_i\left(t\right)\\ & =x_i\left(t\right)+\frac{1}{1+d_i}{a}_{\mathit{ij}}\sum _{j\in N_i}\left(x_j\left(t\right)-x_i\left(t\right)\right)\end{array}$$

since $a_{\mathit{ij}}\sum _{j\in N_i}\left(x_j\left(t\right)-x_i\left(t\right)\right)$is equivalent to $–\mathit{Lx}\left(t\right)$, thus

$$\begin{array}{l}=x_i\left(t\right)+\frac{1}{1+d_i}\left(-Lx\left(t\right)\right)\\ =x_i\left(t\right)+\left(-{\left(I+D\right)}^{-1}\right)Lx\left(t\right))\\ =x_i\left(t\right)-{\left(I+D\right)}^{-1}\left)Lx\left(t\right)\right)\\ =I-{\left(I+D\right)}^{-1}\left(I+A\right)x\left(t\right)\\ ={\left(I+D\right)}^{-1}\left(I+A\right)x\left(t\right)\\ =P_{e}x\left(t\right)\end{array}$$E14

By substituting agent measurement error into Eq. (14), the latest agent state ${\mathrm{x}}_{\mathrm{i}}$(${\mathrm{t}}_{\mathrm{k}}^{\mathrm{i}}$) =$\widehat{x_i\left(t\right)}$ is equivalent to

$$x\left(t+1\right)=P_e\left(e_i\left(\mathrm{t}\right)+x_i\left(t\right)\right)$$E15

The new state position of Eq. (15) will be sent to the connected neighbors for the control update when the ETF is violated, event occurs and the error $e_i\left(t\right)$ will simultaneously counted and updated. When ETF is satisfied, there is no event and communication happening between the agent and neighbors which might cause the value of error equal to 0. At this point, the agent state value will remain the same when the sampled data is not updated.

Even though the agent’s next state value $x\left(t+1\right)$ as shown in Eq. (15) above affected from the state measurement error, the average consensus among connected agents was guaranteed to be achieved when the eigenvalues of square matrix $P$ is strictly contained in the Gershgorin circle criterion. The values of eigen in Gershgorin circle unit was expected to be ${\lambda }_{10}=1$ and the eigenvalues of Perron matrix for a topology of 10 agents strongly connected with an undirected graph are${\lambda }_1=0,{\lambda }_2=0.5\pm 0.5i$and${\lambda }_{10}=1.$

Proof 2: Lyapunov stability

Eq. (16) of Lyapunov when $\mathit{Lx}\triangleq y={\left[y_1,y_2,\dots .y_N\right]}^T$, represents as Eq. (17)

$$\mathit{Vx}\left(t\right)=\frac{1}{2}x{\left(t\right)}^T\mathit{Lx}\left(t\right)$$E16

$$\begin{array}{ll}\dot{V}=\frac{\partial V}{\partial x}.\frac{\mathit{dx}}{\mathit{dt}}& =x^{T}L\dot{x}-x^{T}L\left(\mathit{Lx}+\mathit{Le}\right)\\ & =-\mathit{LLx}{x}^T-\mathit{LLe}{x}^T\\ & =-y^{T}y-y^T\mathit{Le}\end{array}$$E17

Eq. (17) will be Eq. (18) with Laplacian matrix,

$$\begin{array}{ll}\dot{V}& =\sum _i{y}_i^2-\sum _i\sum _{j\in N_i}{y}_i\left(e_i-e_j\right)\\ & =\sum _i{y}_i^2-\sum _i\left|N_i\right|+\sum _i\sum _{j\in N_i}{y}_i{e}_j\end{array}$$E18

For$a>0,\dot{V}$ can bound by using inequality$\left|\mathit{xy}\right|\le \frac{a}{2}{x}^2+\frac{1}{2a}{y}^2$, thus

$$\dot{V}\le -\sum _i{y}_i^2+\sum _{i}a\left|N_i\right|y_i^2+\sum _i\frac{1}{2a}\left|N_i\right|y_i^2+\sum _i\sum _{j\in N_i}\frac{1}{2a}{y}_j^2$$E19

Since the communication graph is undirected,

$$\sum _i\sum _{j\in N_i}\frac{1}{2a}{e}_j^2=\sum _i\sum _{j\in N_i}\frac{1}{2a}{e}_i^2=\sum _i\frac{1}{2a}\mid N_i\mid e_i^2$$E20

Therefore, the final Lyapunov derivation will be negative definite for 0$<{\sigma }_i<1$ which shows that the system was stable.

$$\dot{V}\le -\sum _i\left(1-a\left|N_i\right|\right)y_i^2+\sum _i\frac{1}{a}\mid N_i\mid e_i^2$$

$$\dot{V}\le \sum _i\left({\sigma }_i-1\right)\left(1-a|N_i|\right)y_i^2$$E21

Theorem 2: Given the objective function of BET consensus controller as stated in Eq. (1), the vector of $x\left(\mathrm{t}\right)\in R^{\mathit{nN}}$ which indicates the movement of agent will keep changing until satisfying $\nabla P\left(x\left(t\right)\right)=0$. Let $L_i$, $G$and ETF be given by Eqs. (1)–(11). If BET fulfil Theorem 1 and conditions below,

(B1) ${∆}_{i1}\left(t\right),{∆}_{i2}\left(t\right),\dots \dots .{∆}_{\mathit{\text{iN}}}\left(t\right)$are Bernoulli random probability distribution.

(B2) $a\left(t\right)=a\left(t+1\right)$and $c\left(t\right)=c\left(t+1\right)$ for every $t\in \left\{0,2,4,\dots \right),{\sum }_{t=0}^{\infty }a\left(t\right)=\infty ,$and ${\sum }_{t=0}^{\infty }c\left(t\right)=\infty$

(B3) $x\left(t\right)$is stable under gradient system approximation of $\dot{x\left(t\right)=-\nabla P\left(x\left(t\right)\right)}$where $x\left(t\right)\in R^{\mathit{nN}}$and the stability is in the Lyapunov sense.

(B4) $E[P\left(x\left(t\right)+c\left(t\right)∆\left(t\right)\right]$is bounded for all $t\in N$

(B5) ${sup}_{t\in N}\left|\left|x\left(t\right)\right|\right|<\infty w.p.1$

(B6)$P\left(x\right)$ , the performance index value change at $t$ time

(B7) 0 <$k<1/N$, then $\underset{t\to \infty }{lim}{x}_i\left(t\right)-x_j\left(t\right)=0$

(B8) $f\left(e_i\left(t\right),x_i\left(t\right)\right)\le {\sigma }_{\mathrm{i}}\left(z_i\left(t\right)\right),$then $e_i\left(t\right)=0,$ else $e_i\left(t\right)=\widehat{x_{i\left(t\right)}}$-$x_{i\left(t\right)}$

then,

$\underset{t\to \infty }{lim}x\left(t\right)=x_d$ with probability 1 is achieved.

Proof 3: Consensus among agent with BET

1. The agent’s movement is subject to the global (Eq. (1)), local (Eq. (5), and event-triggered (Eq. (9)) functions from ($t=0$) until $\left(t\to \infty \right)$ and achieved convergence with probability 1 under the above B1–B8 conditions.

2. The next agent state position at even time $t\in \left\{0,2,4,\dots ..\right\}$ and odd time $\left\{1,3,5,\dots ..\right\}$ will depend on Eqs. (21) and (22).

   $$B\left(t\right)=P\left(x\left(t\right)\right)$$E22
   $$B\left(t+1\right)=P\left(x\left(t\right)\right)+\left(c\left(t\right)+{∆}_i\left(t\right)\right)$$E23

3. For $t\in \left\{0,2,4,\dots ..\right\}$, the agent movement with BET is represented by Eq. (24).

   $$x\left(t+2\right)=x\left(t\right)-\left[a\left(t\right)\frac{B\left(t+1\right)-B\left(t\right)}{c\left(t\right)}\ast {∆}_i^{-1}\left(t\right)\right]-k\sum _{j\in N_i}{a}_{\mathit{ij}}\left|x_i-x_j\right|$$E24

   The Eq. (25) had been simplified to Eq. (26) after substituting Eqs. (22) and (23) into Eq. (25)

   $$\mathit{x}\left(t+2\right)=x\left(t\right)-a\left(t\right)\frac{(P\left(x\left(t\right)+c\left(t\right){∆}_i\left(t\right)\right)-P\left(x\left(t\right)\right)}{c\left(t\right)}{∆}_i^{-1}\left(t\right)-k{a}_{\mathit{ij}}\sum _{j\in N_i}\left(x_i-x_j\right)$$E25
   $$x\left(t+2\right)=x\left(t\right)-a\left(t\right)d\left(x\left(t\right),{∆}_i(t\right),c\left(t\right))-k{a}_{\mathit{ij}}\sum _{j\in N_i}\left(x_i-x_j\right)$$E26

4. The collective of agent dynamic movement will reach convergence when all agents meet rendezvous point of $x_d$after executing Theorem 1 and Theorem 2.

5. If the condition of broadcast with SPSA convergence satisfy (B1–B6) and conditions of ET satisfy (B7 and B8), it shows that all agents reached the rendezvous desired target$x_d$. The broadcast settings of B1 and B2 as well as B7 and B8 are imposed tuning for $L_i$and $G$ as defined in Eqs. (1)–(11).

4. Agent performances

4.1 Broadcast event-triggered controller

The 10 homogeneous agents were located at 10 initial positions within the workspace area with 1 target point in the 2D Cartesian coordinates as shown in Figure 6a. The BET controller was designed completely with the SPSA algorithm with local and global controllers. Optimal settings of the SPSA based on investigation conducted [120, 121] was used as a reference and the final value applied for this case study which are $a=0.2,A_1=30,{\alpha }_1=0.6,c=1,{\gamma }_1=0.06,a_k=a/{\left(t+1+A\right)}^{{\alpha }_1}$and $c_k=c/{\left(t+1\right)}^{{\gamma }_1}$. With the local controller, the agent will determine the next position based on the stochastic and deterministic rule during even and odd times. Since the agent will communicate among them, the error in communication will also affect the agent’s next position. The global controller at the same time will broadcast the signal continuously to the agent to update on the agent’s location with the rendezvous point.

Figure 6.

Agent motion and location when t = 50, 100, 400, and 900. a. t = 50. b. t = 100. c. t = 400. d. t = 900.

From this, it was proven that the controller works effectively by creating a motion of agents to reach consensus among them until they reach the rendezvous point in average of 84.24 s and 676.6 iteration in 10 times run. The movement was recorded in a one-time run as shown in Figure 6. From Table 1, the NOC was proven to be reduced by 21.79% with a total of 5520 channels used based on the below calculation,

Time (sec)	20.4	40.8	61.2	81.6	102	122.4	142.8	163.2	183.6	204	225.2
Iteration	100	200	300	400	500	600	700	800	900	1000	1104
NOC	248	181	177	255	436	500	500	500	500	500	520

Table 1.

NOCs per 100 iterations.

$$\text{Total of iteration}=1104/2=552\text{iterations}.$$

$$\text{Time}\mathrm{per}\text{iteration}=225.42/1104=0.204\mathrm{s}.$$

Each iteration is equivalent to 10 channels which carried a total of 552*10 = 5520 channels.

In order to prove the first theorem, which consensus achieved among the agents, the results shown in Figure 7 were proven. The agents were proven to reach the average local consensus successfully when it met at the average point at $t$=6 sec (Theorem 1). The eigenvalues of square matrix of Perron illustrated in Figure 8 showed that the value was in the range of the Gershgorin circle (Proof 1) which indicated that the system was marginally stable, and it reached steady state values (Proof 2). Thus, it can be clearly understood that the average consensus among agents was achieved via communication among agents as shown in Eqs. (27) and (28).

Figure 7.

Agent state trajectories in reaching average consensus.

Figure 8.

Eigenvalues of Perron matrix for undirected graph.

$$x_i\left(t\right)=\sum _{j\in N_i}{a}_{\mathit{ij}}\left(x_j\left(t\right)-x_i\left(t\right)\right)$$E27

$${\alpha }_c=\frac{1}{n}\sum _i{x}_i\left(0\right)$$E28

4.2 Broadcast event-triggered and broadcast time-triggered performances comparison

The BET and BTT controllers were proven effective to find consensus to rendezvous target point. The performances of convergence time and iteration, trajectory, the NOC utilization were among the evaluated to compare the effectiveness of both the controllers. Compared with the two different sampling methods that has been integrated into the broadcast communication controller, BET showed faster convergence and less number of iterations than BTT. BET was proven to be able to simplify the convergence process to meet rendezvous, leading 57.15 s with 177 number of iterations against BTT based on average value found in 10 times test run. These readings depended on controller performances with a different sampling system, TT and ET produced inconsistent results in ten times run as illustated in Figures 9 and 10. The stochastic error and deterministic error will determine the next agent’s random movement and next agent’s deterministic movement until rendezvous was achieved.

Figure 9.

Convergence time.

Figure 10.

Number of iteration.

The agent trajectory was observed at every 50 interval of iterations to see the pattern of movement and the measurement error. For BET, it was clearly seen that agent movement was scattered at the first few iterations and gathered towards 60% of the overall process while the BTT move closer and systematically at the beginning until the end of the process as shown in Figures 11 and 12. This showed that the BTT communication gave effect to the agent consensus since the agent continuously communicate with the neighbour at every time whereas the BET communication only gave significant effects when the ETF condition is violated which results to the pattern of measurement error being lesser with BTT as compared to BET as shown in Figure 13. Thus, in terms of movement, the BTT was more accurate compared to the BET.

Figure 11.

Agent state trajectories (BET).

Figure 12.

Agent state trajectories (BTT).

Figure 13.

Agent measurement error (BTT and BET).

The NOCs witin 100 iterations was recorded. For BET, the NOCs can be reduced and save up to 29% utilization as compared to BTT as shown in Table 2. This was because the BET controller will only allow the information exchange between the agent and neighbours to occur when it violated the ETF but for BTT, the information exchange happened at every time-triggered.

Iteration	100	200	300	400	500	600	700	800
NOC (BET)	248	181	177	255	436	500	500	500
NOC (BTT)	500	500	500	500	500	500	500	500

Table 2.

NOCs of BET and BTT.

In overall comparison, the efficiency of BET was more than BTT sine the time taken and number of iterations were lesser. In terms of trajectory, there were better accuracy for TT since it was closer to the target trajectory as compared to ET which was less systematic from the effects of the communication error. Therefore, the implementation of ET into the agent broadcast has an effect towards the overall agent performances in terms of agent time and iteration, utilization of channel as well as agent trajectory as summarized in Table 3. The significant affect was thus proven. It was shown that the second theorem was proven not only in theory but also through the practical experiements.

Type of sampling	Time (sec)	Number of iteration	No channel	Trajectory	Efficiency
ET	84.24	677	70%	Scattered	High
TT	141.4	854	100%	Systematic	Low

Table 3.

Agent performances with BET and BTT in 10 times run.

5. Broadcast event-triggered parameter effects

5.1 Event-triggered function threshold value,${\sigma }_i$

The threshold value (Eq. (9)) has been varied to observe the effects on agent performance in terms of NOC utilization. The optimal threshold value for this case study was within the range of 10 and 30. The results show that the convergence and NOC gets affected when the value of the threshold is less or more than the optimal range, as shown in Table 4.

Threshold, ${\sigma }_i$ of ET	Rendezvous status	Percentage NOC usage
${\sigma }_i$<10	Meet rendezvous with full communication	96.07%
10≤ ${\sigma }_i$≤30	Meet rendezvous with less communication	93.73%
${\sigma }_i$ $>3$0	Does not meet rendezvous	86.93%

Table 4.

Communication and rendezvous status with three different ${\sigma }_i$ values.

5.2 Simultaneous perturbation stochastic algorithm parameters

Based on the BET’s ideal parameters, which were covered in Section 4.1, the gain $a_k$ ranged from 0.025 to 0.003 and the $c_k$ ranged from 0.662 to 1 for iterations 1–194. The gain, which also has an impact on agent performance in terms of convergence time, iterations, and trajectory movements of the agent to reach rendezvous point, was shown to be impacted by changes in each broadcast parameter coefficient. Table 5 shows the example of SPSA parameter settings applied for a different cases, which are basic investigation of SPSA by James Spall [104], as well as broadcast and broadcast event-triggering for MARS motion coordination applied in this research project.

SPSA variables	By James Spall	Broadcast	Broadcast + ET
$a$	0.027	0.21	0.2
$A_1$	20	30	30
${\alpha }_1$	0.602	0.6	0.6
$c$	1	1	1
${\gamma }_1$	0.101	0.06	0.06
$\sigma$	—	—	30

Table 5.

Optimal settings of SPSA for three different case studies.

6. Conclusions

The BET consensus controller was proven to work for multi-agent robots to reach rendezvous points in a broadcast communication environment. The agents will only communicate among themselves when they satisfy the event-triggered condition to produce an effective and efficient way of motion coordination to reach the desired rendezvous point. The proposed controller system reached convergence with minimum time and iteration as well as reduced channel utilization, which can claim better performances as compared with a conventional BTT system. However, the trajectory pattern was not as systematic as the BTT system. A thorough analysis was conducted to highlight the effect of parameters on agent performances. In the future, the artificial intelligent controller and advanced controller can be embedded with event-triggered systems to produce a more robust and optimal controller for a practical multi-agent motion coordination application. Besides that, it is recommended to apply it to large numbers of agents, such as swarm robots, and to different types of graph topologies for various MARS applications. This controller can be further explored and expanded for many more multi-agent motion coordination systems, such as static or dynamic communication environments.

Acknowledgments

Thanks to UTM for providing their available software and robotic platform. This work was supported by Rabdan Academy, Abu Dhabi, United Arab Emirates.

Conflict of interest

The authors declare no conflict of interest.