Leaderless maneuver guidance and event-triggered formation control for distributed multi-space-robot systems

In this paper, the distributed displacement-based formation and leaderless maneuver guidance control problems of multi-space-robot systems are investigated by introducing event-triggered control update mechanism. A distributed formation and leaderless maneuver guidance control framework is first constructed, which includes two parallel controllers, namely an improved linear quadratic regulator and a distributed consensus-based formation controller. By applying this control framework, the desired formation configuration of multi-space-robot systems can be achieved and the center of leaderless formation can converge to the target point globally. Second, a pull-based event triggering mechanism is introduced to the distributed formation controller, which makes the control update independent of the events of neighboring robots, avoids unnecessary control updates, and saves the extremely limited energy of space robots. Furthermore, the potential Zeno behaviors have been excluded by proving a positive lower bound for the inter-event times. Finally, numerical simulation verifies the effectiveness of the theoretical results.


Introduction
With the development of space robotics, the cooperative control of multi-space-robot systems (MSRSs) has a wide application prospect in various space missions, such as the construction of large space structures [1,2], space debris removal [3,4], space transportation [5], and the detection of gravitational waves in space [6,7]. As the basis for conducting above complex operations and various tasks, the maneuver guidance and formation control of MSRSs are often needed. Therefore, this topic has attracted significant attention from researchers in various scientific communities.
As stated in Ref. [8], on MSRSs, the purpose of the maneuver guidance and formation control is to drive a group of space robots with autonomous decision-making ability to achieve the required formation geometry from any bounded initial states, while maintaining the configuration to maneuver to the tar-get position on the adjacent orbit together. First, the maneuver guidance problem can be solved by a linear quadratic regulator (LQR) optimal control method [9] or a data-driven control method [10,11]. Second, the rapid development of consensus theory gives a distributed formation control method, which only depends on local information to control MSRSs. A large number of results on distributed formation control exist that can be divided into the following categories: bearingbased formation control [12,13], position-based control [14,15], displacement-based control [16,17], and distance-based control [18,19]. Basic principles and execution differ between these methods. Specially, the displacement-based formation-control method, each robot only needs to measure the displacement between itself and its neighbors based on the specified communication topology and use the displacement information to design the controller to achieve the desired formation configuration.
As is well known, the communication, computing, and energy resources of MSRSs are extremely limited. To reduce the resource occupation and energy consumption of robotic communication and control updates, the sampled-data method is used [20][21][22]. This scheme updates the controller periodically, which reduces the energy consumption to a certain extent. However, it will still cause unnecessary waste. For example, when the state of the robot tends to be stable, the controller will still communicate and update the controller according to a pre-determined period. Therefore, a technique called an event-triggered control scheme [23] has been rapidly developed recently. A triggering condition will be introduced in the eventtriggered control scheme. Communication and controller updates will be carried out only when the triggering condition is met. For example in [23][24][25][26][27][28], when the state error of the system exceeds a certain threshold, the control update takes place. The results show that compared with sample-based control scheme, event-triggered control can effectively reduce the update times of controller and save computing, communication and energy resources. Later, Dimarogonas [29] extended the event-triggered control scheme to multi-agent systems. Because of its strong engineering application value, this scheme has been widely studied in multi-robot control communities [30][31][32]. Although the event-triggered control scheme also has many applications in space missions, the previous works mainly focus on the attitude coop-erative control of MSRSs [33][34][35]. To the best of our knowledge, there is little research on an event-triggered control scheme being applied to the formation configuration control of distributed MSRSs. In addition, a pull-based event-triggered protocol is presented, where each robot merely updates the controller at its own triggering instants [36,37]. Therefore, it can further reduce the communication congestion of the system and save computing resources. It is worth noting that under the event-triggered control scheme, an adverse phenomenon is easily caused, namely, that the system may be triggered countless times in a limited time. We call this triggering behavior Zeno behavior [38,39], and it is not allowed in hardware execution. Therefore, to avoid Zeno behavior, we must ensure that the positive minimum inter-event time (IET) exists.
Based on the above discussion, this paper studies the distributed displacement-based formation and maneuver guidance control problems of MSRS by introducing event-triggered control update mechanisms. Compared with [16] the main contributions can be summarized as follows: First, a distributed formation and maneuver guidance control framework is skillfully constructed, which includes two types of parallel controllers, namely an improved LQR and a distributed consensus-based formation controller. By applying this control framework, the desired formation configuration of MSRS can be achieved and the center of formation can globally converge to the target point in a leaderless manner.
Second, a pull-based event-triggered control update mechanism is introduced to the aforementioned distributed consensus-based formation controller. This type of mechanism makes the control updates independent from the events occurring of neighboring robots. It has the obvious advantages of avoiding unnecessary control updates and samplings, which means that its limited energy can be further saved.
Third, by deriving the analytic expression of the minimum inter-event time, it is proved that there exists a positive lower bound for the inter-event times, and the potential Zeno behaviors of the proposed eventtriggered mechanism are excluded theoretically.
The rest of this paper is organized as follows. Several notations, basic theories of graphs, an orbit coordinate system, and a formal system statement regarding this study are introduced in Sect. 2. In Sect. 3, a distributed event-triggered control scheme for MSRSs is proposed with its feasibility analyzed. A simulation example is given to show the efficacy of our results in Sect. 4. Finally, conclusions are presented in Sect. 5.

Preliminaries
In this section, several mathematical notations and preliminary results that will be used throughout this paper are provided.

Notations
Throughout this paper, R n and R m×n denote the real ndimensional Euclidean vector space and m × n matrix space, respectively. Let I n be an n-dimensional identity matrix. · expresses the Euclidean norm of a vector or the spectral norm of a matrix. M T is the transpose of matrix M. Let λ min (M) and λ max (M) denote the minimum and maximum eigenvalues, respectively. A ⊗ B as the Kronecker product of matrices A and B.

Basic theory of graph
In R n , the communication network topology among N robots can be represented by an undirected graph G = (V, E, A), where V = 1, ..., N and E ⊂ V × V are the vertex set and edges set, respectively. A = [a i j ] ∈ R N ×N is the associated adjacency matrix defined by a i j = 1 if and only if (i, j) ∈ E, and a i j = 0 otherwise. The degree matrix is For more details, the reader is referred to Ref. [40].

Orbit coordinate system
Define a reference coordinate system where the x-axis represents the radial direction, the y-axis represents the in-track direction, and the z-axis represents the crosstrack direction to facilitate the description of the robot's motion, as shown in Fig. 1. Radial refers to the direction of the geocentric radius vector, cross-track refers to the direction normal to the orbit plane (along the orbital angular momentum vector), and in-track refers to the direction perpendicular to both the radial track and cross-track.

System statement
Consider a MSRS with N robots. As illustrated in Fig.  2, our aim is the guidance of all robots to establish a specific formation configuration and maintain it until it reaches the target location of the adjacent orbit: In the orbit coordinate system as shown in Fig. 2, the motion dynamics of the space robot i in MSRS relative to the target point similar to the one considered in [16] can be expressed as: where is the orbital angular velocity of the target point; μ is the geocentric gravitational constant; R 0 is the distance from the target point to the center of the Earth, and R i is the distance from robot i to the center of the Earth; R i = R 0 +r i ; R 0 , and R i are the corresponding moduli.
The dynamics (1) can be linearized and rewritten as a state-space equation as follows [16]: where is the state vector of robot i; U i is control input, which includes two parts, i.e., the maneuver guidance law U Gi and formation control law U Fi ; and A and B are coefficient matrices. If the target point is located on adjacent circular orbit, or if the eccentricity of the orbit is particularly small, A and B, the system matrices, are constants and can be written as where ω is the mean orbital angular velocity of the target point.
For maneuvering guidance, an improved LQR method is adopted, and an optimal control law is designed as follows [16]: where K G = R −1 G B T P G and P G can be obtained by solving the continuous-time algebraic Riccati equation (ARE): The α 1 > 0; Q G ∈ R n×n and R G ∈ R m×m are positive-definite matrices such that has a minimum value.
We can define the formation center of a MSRS as where N is the number of robots in the team, and we define the state vector from the formation center to robot i as is the desired formation configuration. For formation control, a control law is proposed as follows [16]: where is the neighbor tracking error of robot i in formation control; c F > 0 is coupling gain; K F = R −1 F B T P F is a feedback control gain matrix. Similarly, P F can be obtained by solving the continuous-time ARE: where α 2 > 0 and the positive-definite matrices Q F ∈ R n×n and R F ∈ R m×m are similar to those in cost function (5).

Assumption 2
The undirected communication topology G is connected, has a spanning tree, and is balanced.

Lemma 1 (Young's Inequality) For any real numbers a and b and any
Lemma 2 [41] Under Assumption 1, given any Q > 0, there exists a unique P > 0 satisfying Lemma 3 [42] Under Assumption 1, for any κ > 0, there exists a positive-definite solution P > 0 to the following Riccati inequality: Lemma 4 [23] The Lipschitz continuity on compact sets of f (x, u) and k(x) implies that f (x, k(x + e)) is also Lipschitz continuous, and we can thus obtain f (x, k (x, e)) ≤ L x + L e .
Remark 1 Under Assumptions 1 and 2, although the maneuver guidance and formation control of MSRSs can be realized, the controller needs continuous update, which will result in an unnecessary waste of resources.
To solve this problem, we introduce the event-triggered control scheme, which updates the controller only when a specific event is triggered, effectively reducing, in turn, the update frequency of the controller.

Main results
In this section, an event-triggered control scheme for the formation control problem is presented to reduce the frequency of controller update. The event-triggered control scheme is designed as follows: where the increment sequence t i k i∈N ,k=0,1,2... are the times when the controller performs calculations and updates, and The state measurement error of robot i is defined as and we define e(t) = e T 1 (t), e T 2 (t), ..., e T N (t) T .
The event-triggered time sequence t i 0 , t i 1 , t i 2 , · · · of robot i will be determined by the following triggering condition: where t ∈ [t i k , t i k+1 ), K = B T P, ρ ∈ (0, 1) is a designed parameter that can be arbitrarily chosen, and λ N = λ max (H ). Obviously, the triggering condition is also distributed, which only must judge whether to update the controller with the information from neighbors.
Theorem 1 Under assumptions 1 and 2, the maneuver guidance and formation control of MSRS (2) can be achieved using the maneuver guidance law (3) and distributed event-triggered formation control scheme (12) with the event-triggered condition (15).
T , and then we havė A MSRS is said to achieve the maneuver guidance and the desired formation η if, for any given bounded initial states, Consider the following Lyapunov function candidate: where P F is a positive definite matrix and satisfies Eq.
DenotingÃ =Ā T P F + P FĀ ,B = c F P F B K F , then By using Young's inequality, we can obtain and By using Lemma 2, we can obtain Furthermore, if we can guarantee that with 0 < ρ < 1, thenV Let λ 1 , λ 2 , ..., λ N be the eigenvalues of matrix H , satisfying 0 < λ 1 ≤ λ 2 , ... ≤ λ N . Since the matrix H is symmetric, then there exists an orthogonal matrix M such that It is clear to see that M T M = I N and H = M J M T . Definingξ(t) = (M T ⊗ I n )ξ(t), the following inequality then holds: Considering inequality (11) Obviously, we must only satisfy that inequality (24) is valid. A sufficient condition for (24) Therefore, for each robot in a MSRS, a sufficient condition is Thus,V (t) < 0 if inequality (30) is satisfied. Furthermore, ξ i will asymptotically converge to zero, which means that, under the maneuver guidance law (3) and the distributed event-triggered control scheme (12) with the triggering condition (15), the maneuver guidance and formation control of MSRS (2) can be achieved. The proof is completed.
The Zeno behavior is excluded according to the following theorem.
Theorem 2 Consider the MSRS (2) with a distributed event-triggered control scheme (12) and distributed event-triggered condition (15). No robot will exhibit Zeno behavior.
Proof If inequality (30) is satisfied, the following inequality is also satisfied: with 0 < ρ < 1. We can now determine the inter-event times by observing the dynamics of (H ⊗ K )e(t) Then, the inter-execution times are bounded by the time it takes for (t) to evolve from 0 to √ ρ, which implies Therefore, for all robots, Zeno behavior can be excluded. The proof is completed. (2). To ensure a distributed formation control mechanism, the dynamics of the center of the formation is expressed aṡ

Theorem 3 Considering a group of space robots without leadership, the dynamics is shown in
and the communication topology is required to be balanced.
Proof Calculating the time derivative of the formationcenter states It can be seen from (34) that the calculation of X 0 must obtain global information. To ensure that the formation is a distributed structure, and that the constraint The formation is distributed if and only if (35) equals 0, and the dynamics of the formation center satisfieṡ X 0 = (A − B K G )X 0 =ĀX 0 . This means a i j = a ji ; that is, the communication topology must be balanced. The proof is completed.
Remark 2 This paper introduces the event-triggered mechanism into the formation control of MSRS. The original MSRS has been transformed from continuoustime control to discrete-time control. Therefore, although the proposal and proof of Theorem 3 are similar to that of Theorem 4 in [16], they are still necessary to be handled with care.

Numerical simulation
To intuitively verify the effectiveness of the theoretical results reported in this paper, we simulate and verify the proposed maneuver guidance and distributed event-triggered formation control algorithms through numerical experiment in MATLAB.
Considering a MSRS composed of four space robots represented by S R 1 , S R 2 , S R 3 , and S R 4 and a virtual formation center represented by S R 0 , respectively, the communication topology used is given as shown in Fig.  3 with the corresponding H matrix (36): It is supposed that the initial states of the four space robots in orbital frame are and α 1 = 1 × 10 −3 , α 2 = 3 × 10 −3 . The values of the parameters in Eqs. (12) and (15) are c F = 1 and ρ = 0.1. For the convenience of comparison, the time-triggered period is set to 0.001s. The simulation results are shown in Fig. 4-9.
In the simulation results, Fig. 4 shows the trajectory of MSRS and the formation process of formation configuration. It can be clearly seen that each space robot in MSRS starts from any bounded initial position, forms the expected rectangular formation within a certain time and then maintains the rectangular formation configuration to maneuver to the target point. Figure 5a, b shows the initial formation configuration and the final formation configuration of the MSRS in Fig. 4, respectively. As can be seen from Fig. 5(b), the final formation configuration is a rectangle with a side length of 100 m. Figure 6 shows the state change trajectories of four space robots in MSRS. Figure 7 shows the change of   inter-event times of four robots with time t. It can be seen from Fig. 6 that about 3000 s, four space robots reach the target position and maintain the expected rectangular formation configuration. Due to the large changes in the displacement and speed of the robot in the early stage, the event conditions will be triggered frequently. However, it can be seen from Fig. 7 that the event interval is still far greater than 0.001 s. The detailed comparison between the time-trigger periods and the average inter-event times of the four space robots can be seen in Table 1. In addition, it can be seen from Fig. 6 that after 3000 s, with the stability of the system, the inter-event time gradually increases with time t. Obviously, during the whole maneuver process, Distance from virtual center(s) Fig. 9 Robots' deviation from formation center the update times of the controllers of MSRS are significantly reduced, so as to save the limited resources of the system. Figure 8 shows the evolution trajectory of formation control inputŨ Fi (i = 1, 2, 3, 4) of each robot in MSRS from 0 to 100 s. It can be clearly seen from the subgraph thatŨ Fi remains unchanged between two adjacent event-triggered instants. That is, compared with the control U Fi in [16], the formation controller of the robot generates fewer updates in this time period.
Finally, Fig. 9 shows the variation of the deviation between the four space robots and the center of the formation with time in the process of MSRS maneuvering. The results show that all the deviation converge to the desired η i (i = 1, 2, 3, 4) of the formation.

Remark 3
The design of the event-triggered conditions in inequality 15 enables us to obtain larger inter-event times (as shown in Fig. 7); the distributed formation control method makes each robot in the multi-spacerobot systems relies only on local information to execution autonomous control; the final numerical experiment results in this paper show that the introduction of the event-triggered mechanism can guarantee the control performance (as shown in Fig. 4) and achieve the desired control goal while effectively reduce the frequency of sampling and control updating.

Conclusions
This paper studied the maneuver guidance and distributed pull-based formation control of leaderless MSRS under a time-invariant communication topology. Through the introduction of the pull-based eventtriggered control update mechanism, the update time of the controller is determined by a distributed event-triggered condition, so as to effectively reduce the update frequency of the controller and save the limited resources of MSRS. At the same time, it ensures that MSRS will not have Zeno behavior under the new formation control algorithm. Finally, the effectiveness of the results is verified by numerical experiment in MATLAB.
The method proposed in this paper effectively reduces the update frequency of the controller of each robot in MSRS, but from the given event-triggered condition, it can be judged that the robot still needs to maintain continuous communication with its neighbors, which still occupies a lot of unnecessary communication resources. Therefore, in the next step, we will consider the event-triggered communication mechanism to decrease the number of communications at the same time. In addition, the design of event-triggered conditions in this paper is conservative. How to design better event-triggered conditions and further enlarge the inter-event times is also the research direction of our future work.