In robot systems, mobile robots are expected to operate in environments where it may not be easy to replace batteries or refill the fuel, such as hostile environments.^[1] Therefore, to ensure that the robot team can complete their missions, energy consumption problems must be taken into account at the design phase. Energy consumption caused by both motion and communication belong to the field of energy-optimal problems.^[2–7] In the existing research, the energy-optimal problems are mostly studied by optimal control theory. For example, in Ref. [4], the authors proposed a power control and channel allocation optimization algorithm with low energy consumption for channel allocation game model. Tokekar et al.^[6] studied the problem of finding velocity profiles for car-like robots to minimize the energy consumption while traveling along a given path. In Ref. [7], Egerstedt et al. introduced an open-loop control protocol for multi-robots to achieve rendezvous while minimizing the overall energy consumption. However, this type of open-loop controller cannot achieve adjustment and compensation for deviations of the controlled variables. Therefore, the closed-loop controller is proven to be more favorable to actual robot systems.

Over the past few decades, distributed optimization problems have increasingly attracted attention in science and engineering. In distributed optimization problems, each agent has a local cost function, which is only known by itself. The goal of distributed optimization is to cooperatively optimize a global cost function in the form of the sum of local cost functions of all agents. In the process of optimization, each agent can only use the local information of its neighbors. There are a large number of studies for distributed optimization algorithms in the context of multi-agent systems. Most of these studies focus on the protocols with first-order or second-order linear systems. The adopted methods include sub-gradient method,^[8–10] gradient method,^[11–13] primal-dual method^[14,15], zero-gradient-sum (ZGS) method,^[16–18] and so forth. For example, Nedic et al.^[8] first put forward the distributed discrete-time sub-gradient optimization algorithm with diminishing step size. In Ref. [12], a distributed gradient optimization algorithm was presented for continuous-time multi-agent systems with single-integrator dynamics, which took into account the gradient gain and finite-time convergence. In Ref. [13], the authors proposed a gradient-based distributed optimization algorithm for first-order integrators. A continuous-time zero-gradient-sum algorithm was presented in Ref. [16] to solve the time-varying distributed convex optimization problem. However, the ZGS algorithm needs to restrict the initial states of agents.

In practice, each agent is usually equipped with simple embedded micro-processor which has limited energy and communication bandwidth. Most existing algorithms are updated continuously, which might be inefficient or impractical since they can lead to large communication load and waste excessive resources. To reduce the unnecessary waste of communication and computation resources, event-triggered control laws have been applied to single and double-integrators.^[19–22] Similarly, event-triggered algorithms have also been applied to distributed optimization problems.^[23–29] Lü et al.^[23] focused on the distributed event-triggered sub-gradient algorithm for solving a class of convex optimization problem based on first-order discrete-time multi-agent systems. In Ref. [25], an event triggered zero-gradient-sum algorithm was proposed to solve the optimization problem. To avoid calculating the threshold of the triggering times continuously, Liu et al.^[26] presented a novel distributed event-triggered optimization algorithm based on sampled-data state. In Ref. [29], the distributed event-triggered optimization algorithm was extended from first-order dynamics to second-order dynamics, thereby accomplishing the optimal coordination problems.

Motivated by these considerations, in this paper we focus on the distributed event-triggered optimization problem for multiple nonholonomic wheeled mobile robots. The local cost function of each robot represents its own battery energy consumption, which is related to the state of the hand position. By coordinate transformation, the dynamics of the hand positions of robots are formulated as two groups of first-order integrators. We will solve the energy-optimal problem by using the distributed event-triggered optimization algorithm, and at the same time the states of the hand positions exponentially converge to the optimal point.

The main contributions of this paper follow. (i) Most existing results related to distributed optimization problems have concentrated on first-order or second-order linear systems. However, there are abundant nonlinear and even nonholonomic systems in practice, such as mobile robots. Therefore, we propose the distributed event-triggered optimization algorithm to solve the energy-optimal problem for multiple nonholonomic mobile robots. (ii) A distributed event-triggered control scheme is designed to minimize the energy consumption. For each robot, the controller is updated only at its own triggering times. Moreover, the event-triggered algorithm can avoid the Zeno behavior. (iii) It can be proven that the optimal solution is the system equilibrium point and also the consensus point. Furthermore, the exponential convergence rates of position and velocity are presented with the help of Lyapunov stability theory, which is different from the asymptotical convergence in Ref. [13].

The rest of this paper is organized as follows. In Section 2, some basic concepts on algebraic graph theory are introduced and the problem is formulated. Section 3 shows the main results. The distributed event-triggered optimization algorithm for the energy-optimal problem of multiple robots and the detailed stability analysis are also presented. Simulations are given to validate the theoretical results in Section 4. Finally, Section 5 gives a conclusion of this paper.

In this section, a distributed event-triggered optimization algorithm is proposed to achieve the control objective in Eq. (8) and thus solve the optimization problem in Eq. (2).

To avoid the unnecessary resource waste, we construct the control protocol based on the distributed event-triggered strategy. For robot i, the control law is updated only at some discrete times , called the event-triggering sequence, satisfying a certain triggering condition.

To construct the triggering condition, we define the state measurement error for robot i as 9 where is the k-th event-triggering time of robot i.

The event-triggering times for robot i are specified when the following triggering condition 10 is violated; that is, where β_i > 0 will be determined later and . For , is the latest triggering instant of robot j.

Then, the distributed event-triggered optimization control law for robot i is constructed as 11 where α and γ are positive constants. The control law for each robot depends on itself and its neighbours’ latest event-triggering instants.

Let x = col(x₁,…,x_N), e = col(e₁,…,e_N), and ∇F(x) = col(∇f₁(x₁),…,∇f_N(x_N)). The dynamics of the hand positions can be rewritten in the compact form 12

The following lemma illustrates a basic necessary condition for the optimality.

The following lemma indicates the relationship between the equilibrium point of system in Eq. (12) and the optimal solution of problem in Eq. (2).

Conversely, for any equilibrium point of system in Eq. (12), we have e = 0 once the equilibrium of system in Eq. (12) is achieved. Hence, we obtain 13 From Assumption 1, we have . Multiplying the left of Eq. (13) by gives 14 Therefore, which means According to Lemma 2, we obtain that the equilibrium point of system in Eq. (12) is the optimal solution of problem in Eq. (2).

Employ the transformation 15 and system equation (12) is transformed into 16 where h = h₁ +h₂ with h₁ = ∇F(x+e)−∇F(x) and h₂ = ∇F(x)−∇F(x∗). Let 17 where X₁,E₁ ∈ ℝ²,X₂,E₂ ∈ ℝ^2(N−1), and Q is determined by Remark 1. From Eq. (16) and Remark 1, we obtain 18

The derivative of V along the system Eq. (18) yields Recalling that h = h₁ + h₂ = (∇F(x + e) − ∇F(x)) + (∇F(x) − ∇F(x∗)), using Lemma 1 and Assumption 3 along with ||Q⊗I₂|| = 1, we obtain the following estimation: In addition, Note that R^TLR is symmetric and positive definite matrix with eigenvalues λ₂,…,λ_N and thus the eigenvalues of are . Obviously, . Then we have Therefore, we obtain 21 It follows from the triggering condition in Eq. (10) that Hence, where . Recall that ||Q⊗I₂|| = 1, one has 22 By substituting Eq. (22) into Eq. (21), we obtain Since , then Therefore, one has Denote , and we can select appropriate parameters α,γ, and β_i such that μ > 0. One knows from Eq. (20) that 23 where c = V(0). This implies that exponentially converges to 0, that is, x_i exponentially converges to x∗. From Eq. (11) and ∇F(x∗) = 0, u(t) = col(u₁(t),…,u_N(t)) can be written as According to Assumption 3, we obtain exponentially converges to zero and meanwhile e(t) exponentially tends to zero; therefore, u(t) exponentially reaches zero. From Eq. (5), we know that v_i(t) and ω_i(t) exponentially converge to zero. Hence, the orientation q_i3(t) tends to a certain stationary value.

Next, we will show that any inter-event intervals are lower bounded by a positive number, which guarantees that there is no Zeno behavior.

By applying Newton–Leibnitz formula and noting that , we deduce that According to the above inequality, we have 24 Note that on the basis of the triggering condition in Eq. (10), we hence obtain 25 Multiplying Eq. (25) by and denote , one knows where , , and .

Since exp(q)≥ 1+q for any q ∈ ℝ, we have . For any robot i, In the conclusion, the Zeno behavior can be excluded.

In this section, some numerical results are given to verify the theoretical results.

Consider the unicycle system with five wheeled mobile robots satisfying Eq. (1), whose interaction topology is depicted in Fig. 2 with weights of all edges as 1. Thus, the largest eigenvalue of the Laplacian matrix is λ₅ = 5. The initial states are chosen as (q₁₁,q₁₂,q₁₃)(0) = (1, 0, 0), (q₂₁,q₂₂,q₂₃)(0)=(1.5, 0.5, 0), (q₃₁,q₃₂,q₃₃)(0)=(1, 1,π), , and (q₅₁,q₅₂,q₅₃)(0)=(0.1, 1, 0). Define the hand position by ρ = 0.5. The local battery consumption functions are defined as f_i(x_i) = 50||x_i(t)−x_i(0)||². It can be easily calculated that the optimal value x∗ = [1, 0.71]^T. It is observed that the above local cost functions are strongly convex and their gradients are Lipschitz. After the calculation of the above local cost function , the minimum strongly-convex constant is χ = 100 and the maximum Lipschitz constant is σ = 200. Choose α = 0.6, γ = 0.01, β₁ = 0.04, β₂ = 0.04, β₃ = 0.03, β₄ = 0.02, and β₅ = 0.045 satisfying Eq. (19). Then, by Theorem 1, the optimization problem in Eq. (2) can be solved by the distributed optimization control law in Eq. (11) with the event-triggering condition in Eq. (10). The simulation results are shown in Figs. 3–6, indicating the effectiveness of the proposed energy-optimal strategy. Figure 3 shows the states of five robots’ hand positions converge to the optimal point x∗ = [1, 0.71]^T. Figure 4 shows the event-triggering times of five robots. The measurement error and threshold evolution of the robots are shown in Fig. 5. From Fig. 6, we can see that the orientations converge to fixed values, and the linear velocities reach zero; that is, they will keep still when the robots reach the optimal point.

Fig. 2.

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	Fig. 2. Communication topology for robots.

Fig. 3.

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	Fig. 3. State trajectories of xi.

Fig. 4.

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	Fig. 4. Event-triggering times of five robots.

Fig. 5.

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	Fig. 5. Measurement error and threshold evolution of robots.

Fig. 6.

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	Fig. 6. The evolution of orientation states and linear velocities of robot i.

In this paper, we have presented an energy-optimal strategy such that the states of the robots’ hand positions achieve consensus while minimizing the overall energy consumption of robots. We have formulated this problem as the distributed optimization problem. By coordinate transformation, the dynamics of the hand positions are formulated as two groups of first-order integrators. A distributed event-triggered optimization algorithm is hence presented and guaranteed that the equilibrium point is the optimal solution. Based on the Lyapunov theory, sufficient conditions are obtained to make the states of the robots’ hand positions exponentially converge to the optimal point of the global cost function. At the same time, the velocities and orientations converge to zero and certain fixed values, respectively. Moreover, the Zeno behavior is avoided.

Our further research considers that the robots rendezvous at some point while the overall battery energy consumption reaches the minimum in a finite time. The constraint condition that each robot moves only in its own area will also be considered.