Networked multi-robot systems pose challenging problems due to the need of integrating and controlling multiple entities to achieve a common objective. The integration among the robots, often performed via a communication network, allows the cooperation and the coordination of multiple, sometimes heterogeneous, nodes and it may allow to accomplish tasks otherwise impossible for a single robot.

Communication and control architectures for multi-robot systems may span from centralized to distributed architectures. In the first case, one node (e.g., one of the teammates) is able to communicate with all the other nodes and to take decision for the overall team. However, limited communication capabilities, in term of ranges or bandwidth, and limited computational power could make impossible both the communication among all the robots and the adoption of a centralized control strategy. Moreover, the central unit may represent a weakness in terms of reliability since its failure may compromise the overall system.

In distributed systems [see, for example, Bullo et al. (2009) and Mesbahi and Egerstedt (2010)], each robot communicates only with a subset of its teammates and it must take its own decisions only on the base of the received information and of the data from local sensors. Distributed control strategies for multi-robot systems are often based on the so-called consensus task (Olfati-Saber et al., 2007) where, on the basis of only neighbor-to-neighbor information exchange, the robots are required to reach an agreement on a common value regarding a specific variable, exogenous or depending on the state of the single robot (Ren and Beard, 2008). The consensus problem has been mostly investigated for systems characterized by a first-order integrator (Olfati-Saber and Murray, 2004; Cao et al., 2008), or a second-order integrator (Hong et al., 2008; Yu et al., 2010), but some works involving systems with high-order dynamics are also present in the literature (Tuna, 2008; Wang et al., 2008). More in general, the problem of distributed control of multi-robot systems is aimed at achieving a global task (e.g., controlling the geometrical centroid and formation of a team of mobile robots) using only local information and interactions. In Belta and Kumar (2004), a partially decentralized algorithm is proposed to control the network centroid, the variance and the orientation of the system. Many approaches adopt decentralized observers in order to estimate the collective behavior, as in Smith and Hadaegh (2007) where a linear state-feedback control is proposed, or as in Yang et al. (2008) where such an estimate, in conjunction with a local controller, is adopted to minimize a cost function depending on the state of the whole system and aimed at controlling the robots’ formation. An observer-controller scheme for the distributed tracking of time-varying global task variables has been developed in Antonelli et al. (2013, 2014). Time-varying formation control problems for general linear multiagent systems with switching directed topologies have been recently studied in Dong and Hu (2016).

The size of distributed robotic systems may be large and, consequently, the probability of failures of some of the teammates may increase. Despite distributed systems present a potential robustness to faults, the failure of one or more robots, if not properly handled, might jeopardize the task execution. Thus, fault detection and isolation (FDI) strategies become more and more important. Indeed, the detection of faulty robots may allow to exploit the intrinsic redundancy of multi-robot systems in such a way to accomplish the assigned tasks also in the case of failure of one or more units. To the aim, suitable fault accommodation strategies could be considered to correctly handle this situation. Several FDI strategies for single unit systems both in continuous (Zhang et al., 2002) or discrete-time (Caccavale et al., 2013) have been proposed in the literature, but only in the last years the fault diagnosis for multi-robot systems has been object of wider attention from the research community. In Wang et al. (2009), a centralized approach is presented where a central station, aimed at detecting and isolating faults over the whole system, collects information about actuators and sensors coming from each robot. A comparison between a centralized (with a central unit collecting all the information), semi-decentralized (with information exchange only between neighboring robots), and fully decentralized (in which each robot only knows its own state) FDI approaches is presented in Meskin and Khorasani (2009). Observer-based decentralized solutions for detecting and isolating faults in interconnected subsystems are presented in Ferrari et al. (2009) and Zhang and Zhang (2012) where a bank of local adaptive observers, using only measurements and information from neighboring subsystems, are proposed. Observer-based approaches for overlapping linear systems can be found in Stankovic et al. (2010) where local observers are adopted to detect faults on the non-overlapping parts, while a consensus-like strategy is adopted for the overlapping parts.

Usually, distributed FDI schemes allow the detection of faults involving the same robot or its direct neighbors, namely, a direct communication between the faulty teammate and the detecting robot is required. For example, the work in Davoodi et al. (2014) presents a FDI approach for distributed and heterogeneous networked systems where each robot can detect and isolate not only its own faults but also the faults of its direct neighbors. Such limit is overcome in Arrichiello et al. (2015), in which we proposed a distributed FDI algorithm for networked robots where each robot is able to detect faults occurring on any other teammate, even if not in direct communication. In particular, the FDI solution in Arrichiello et al. (2015) builds on a distributed controller-observer schema presented in Antonelli et al. (2013, 2014) where, by means of a local observer, each robot, modeled with a continuous first-order integrator dynamics, estimates the overall state of the team and uses such an estimate to compute its local control input to achieve global tasks. Such approach is extended in Arrichiello et al. (2015) where the same information exchanged by the local observers is used to compute residual vectors whose aim is to allow the detection and the isolation of actuator faults occurring on any robot of the team. A recovery strategy aimed at removing the faulty robot from the team, based on the estimate of the maximum detection time, was developed and presented in Arrichiello et al. (2014a,b), while in Arrichiello et al. (2014c) we extended the strategy also to manage local reactive controls on board each robot to handle possible emerging situations as, for example, the presence of unexpected obstacles. In Marino and Pierri (2015) and Marino et al. (2015), we consider a multi-robot system characterized by a more general and discrete-time linear dynamics.

In this work, we extend the conference papers (Arrichiello et al., 2014a,b,c) by presenting a unified distributed fault-tolerant control (DFTC) approach composed of a distributed observer-controller strategy to achieve global mission, of a fault detection, identification, and recovery (FDIR) strategy to manage unrecoverable faults, of a local fault estimator to accommodate recoverable faults; moreover, the proposed architecture results robust to the activation of local behaviors not included in the global control strategy. Furthermore, with respect to those conference papers, here we provide further analytical details, we consider the case of noisy state measurements, and we improve the numerical validation by providing an integrated case study concerning multi-robot formation control where recoverable/unrecoverable faults and activation of local reactive behavior occur.

Let us introduce the some basic notation adopted throughout the rest of the paper. The generic variable relative to the i-th robot will be denoted as v_i, while the corresponding collective variable, i.e., a stacked vector collecting the variables v_i for all the robots, will be denoted without any subscript/superscript; namely, v=[v1Tv2T…vNT]T where N is the number of robots. The estimate of collective variable v locally made by the i-th robot is denoted as  iv^, while the collection of the estimation vectors of the different robots is denoted as v^⋆=[1v^T 2v^T … Nv^T]T. The estimation error performed by i-th robot is denoted as  iv˜=v−iv^, while the collection of the estimation errors of the different robots is expressed as v˜⋆=[1v˜T 2v˜T … nv˜T]T. Finally, the vector v* represents v⋆=1N⊗v where 1_N is the (N × 1) vector of ones and ⊗ denotes the Kroneker product operator.

Let us consider a multi-robot system composed of N robots, each of them characterized by the following dynamics (i = 1, 2, …, N) (1)x˙i=ui+ϕi(t)+ξi(t,xi), where x_i ∈ ℛⁿ and u_i ∈ ℛⁿ denote, respectively, the i-th robot’s state and input, ϕ_i ∈ ℛⁿ is an additive actuator fault term that is zero in healthy conditions and ξ_i(t, x_i) ∈ ℛⁿ is a term collecting the model uncertainties and disturbances. The following assumptions on the single robot’s model hold:

assumption 1. The model uncertainties ξ_i(t, x_i) are assumed to be norm bounded by a known constant term ξ¯, assumed equal for each robot, i.e., (2)||ξi(t,xi)||≤ξ¯,∀t,∀i=1,2,…,N.

assumption 2. Each robot has direct access only to a noisy measure x_m,i of its own state, i.e., (3)xm,i=xi+ηi<+∞, where η_i ∈ ℛⁿ is the measurement additive noise supposed to be norm bounded by a known positive scalar η¯, assumed equal for each robot, i.e., (4)||ηi(t)||≤η¯<+∞∀t,∀i=1,2,…,N.

Let us define the collective variables of the system as the vectors collecting the variables of all the robots of the network, as (5)x=[x1Tx2T…xNT]T∈RNnu=[u1Tu2T…uNT]T∈RNnϕ=[ϕ1Tϕ2T…ϕNT]T∈RNnξ=[ξ1Tξ2T…ξNT]T∈RNnη=[η1Tη2T…ηNT]T∈RNn.

On the basis of the single robot dynamics [equation (1)], the collective dynamics is expressed as (6)x˙=u+ϕ+ξ, while the collective noisy measurement of the system state is (7)xm=x+η.

The robots are required to achieve a global mission, i.e., a mission depending on the collective state x of the overall team. However, since the collective state measurement is not available to the robots, a distributed observer-controller scheme is adopted in which, for each robot, a local observer provides a local estimate of the collective state,  ix^; then, the controller computes the robot input in a pseudo-centralized manner on the base of the estimated collective state (Antonelli et al., 2013, 2014; Arrichiello et al., 2015).

In addition to the global mission, each robot must achieve also local (reactive) tasks to take into account unexpected events such as the presence of obstacles along its path. The switching between global behavior and reactive behavior is handled by a Local Supervisor according to the current scenario.

In this section, we present the proposed strategy to handle recoverable and unrecoverable actuator faults occurring on any robot of the team. In the presence of a fault, the first step is to decide if the robot objective can be achieved in spite of its occurrence by opportunely reconfiguring the control input. In this case, it is possible to classify the fault as recoverable (Staroswiecki, 2008) (e.g., a bias between the commanded input and the actual one), and active fault-tolerant strategies can be locally activated by single robots, that locally modifies the nominal control law [equation (16)] so as to accommodate the fault by compensating the effects of the fault on its dynamics. Regarding non-recoverable faults (e.g., permanent actuator failure) active strategies are not available, but in multi-robot system the intrinsic redundancy can be exploited in order to define a passive fault tolerance strategy aimed at removing the faulty vehicle from the team and rescheduling the mission needs.

In this section, we present a numerical simulation case study where a team of 4 robots (N = 4), using the proposed distributed fault-tolerant control approach, moves in a plane (n = 2) keeping an assigned formation. During the execution of the mission some of the robots are subject to recoverable faults, unrecoverable faults, as well as reactive behavior activations due to obstacle avoidance.

Gains k_c, k_o, K, and G in equations (15), (19), (29), and (30) were, respectively, set to k_c = 4, k_o = 6, K = 3I₂, and G = 0.01I₂. The measurement noise η_i in equation (3) is assumed to be bounded by 0.001 m (η¯=0.001), while the process noise in equation (1) is bounded by 0.005 m/s (ξ¯=0.005 in equation (2)). The directed network topology shown in Figure 2 is assumed for the information exchange.

The desired path of the centroid σ_1,d(t) is reported in Figure 3 (dashed blue line), while the desired formation σ_2,d(t) is constant and it corresponds to a regular polygon formation of side 0.5 m around the desired centroid. During the simulation, two faults of different nature are considered. In detail, a recoverable fault occurs on robot 2 at t_f,2 = 19 s and it has the following expression: ϕ2=[0,0]T,fort<tf,2[0.3,−0.35]T(1−e−t−tf,20.45),fort≥tf,2.

A non-recoverable fault on robot 1, consisting in a complete stop of the robot, is assumed occurring at t_f,1 = 41.8 s. The vehicle paths are shown in Figure 3 (green dashed lines for healthy robots 3 and 4, and red lines for robots 1 and 2 subject to faults). Figure 4 shows the components of ϕ2=[ϕ2,x,ϕ2,y]T and its estimate ϕ^2=[ϕ^2,x,ϕ^2,y]T performed by the local fault estimator on board robot 2; the figure also shows that, depending on the local observer gains, the measurement and process noise affects the fault estimates. In this sense, based on equation (1) the process noise is seen as a (bounded) fault affecting the robot dynamics.

Figure 5 shows the residuals calculated by each robot of the team and relative to the faulty robot 1. In detail, the residual norms ||ir1|| (i = 1, 2, 3, 4) (continuous blue lines) and the corresponding thresholds (dashed green lines) are plotted. The plots also show that, being the fault unrecoverable, all the residuals are kept below the corresponding threshold as long as the fault is not affecting the robot, while the residuals computed by each robot overcome the thresholds after the fault occurrence even if the faulty robot has not a direct communication link with all the teammates.

Figures 6–8 show that the residuals relative to other robots, namely, ||irj|| (i = 1, 2, 3, 4 and j ≠ 1), do not overcome the corresponding thresholds; this means that no fault is detected on robots 2, 3, 4. In particular, with regard to robot 2, that is affected by a recoverable fault after t_f,2 = 19 s, the residuals do not overcome the threshold since the fault is locally compensated by the controller according to the strategy in Section 5.1, and it does not need to be globally detected and recovered.

It is worth noticing from Figure 3 that the robots are also able to avoid obstacles along their paths via the activation of local reactive behaviors, and that, as shown in Figures 5–8, the activation of the local behaviors do not effect the residual components. Concerning the FDIR strategy described in Section 5.2.2, Figure 3 shows that, after the exclusion of robot 1, the formation is rearranged according to a regular triangular formation. The maximum detection time Δt_d,max in equation (42) is set to 5 s.

In Figure 9, the norms of the state estimation error ∥x˜⋆∥ (top) and the task errors ∥σ˜1∥ and ∥σ˜2∥ (bottom) are shown. The figure shows that these errors grow during the obstacle avoidance and the fault detection phases and decreases toward a neighborhood of the origin in normal conditions. Finally, Figure 10 reports the control input u_i (∀i) in equation (1). The top plot is relative to robot 1 that is affected by a non-recoverable fault; thus, the input becomes zero after the fault occurrence. Concerning the other plots, the discontinuities on the inputs is due to the faulty robot exclusion and formation rearrangement.

In this paper, a distributed fault-tolerant control strategy for networked robots was presented. The proposed approach builds on a distributed controller-observer architecture, a fault detection, isolation and recovery strategy for unrecoverable faults, and a local fault estimation and compensation for recoverable faults. The proposed approach also results robust to the activation of reactive behaviors acting at local levels and not included in the global control strategy.

The proposed architecture allows to manage in an integrated manner the different issues related to distributed formation control and fault management, while keeping limited the communication burden; i.e., the information exchanged among neighbors for state estimation purposes are also used for fault management without further requirements. The approach was successfully tested via numerical simulations, and the results of a single integrated case study involving the occurrence of the different kinds of faults as well as the activation of reactive behaviors was presented to validate the approach.

All authors listed have made substantial, direct, and intellectual contribution to the work and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.