Recently, cooperative control of MASs has been widely studied because of its broad applications including swarming,^[1] flocking,^[2,3] rendezvous,^[4] formation control,^[5,6] target pursuit,^[7,8] and automated highway systems. As a fundamental problem in cooperative control of MASs, consensus means that all agents under properly distributed control inputs, called a consensus protocol, converge to a common value. Many consensus protocols have been proposed for single-integrator MASs.^[9–15] However, many agents such as underwater robots are controlled to achieve a desired motion by their accelerations rather than their velocities in reality. Therefore, there is a need to investigate consensus of double-integrator MASs. For many existing consensus protocols for double-integrator MASs, the eventual value to be reached is a function of all agents’ initial states, which is unknown in advance. This is often called the χ-consensus problem.^[16] However, in some practical applications, all agents might be required to track a common moving target. This is often called the consensus tracking (CT) problem. Since the protocols about the χ-consensus cannot be directly used to solve the CT problems, proper consensus tracking protocols (CTPs) for double-integrator MASs should be proposed. Considerable efforts have been made on the CT of double-integrator MASs, where main topics include delays,^[17–19] robustness,^[19–21] finite-time convergence,^[21–23] and containment control.^[24,25]

References [17] and [25] about the CT of double-integrator MASs assumed that continuous-time CTPs are used to guarantee the CT of double-integrator MASs. However, in many cases, when a continuous-time system is equipped with digital sensors and controllers, only sampled-data can be used to synthesise the control laws. Hence, it is important to consider sampled-data-based CT of double-integrator MASs. Recently, references [26]–[32] investigated sampled-data-based CT of double-integrator MASs. Particularly, it was proved in Ref. [32] that when a communication topology is undirected and fixed and there are paths from the target to all agents, choosing an appropriate sampling period can guarantee double-integrator MASs to track an asymptotically unbounded-acceleration and bounded-jerk target (AUABJT) available to only partial agents by using the proposed sampled-data CTP. It is easy to see from the sampled-data CTP proposed in Ref. [32] that each agent needs to know its own and neighbours’ positions and velocities. However, in some practical situations, to save hardware costs, an agent is often not equipped with a velocity sensor. For an agent without a velocity sensor, its velocity information cannot be measured and is thus unavailable to itself and its neighbours. In this case, the sampled-data CTP in Ref. [32] cannot be directly applied to the CT of double-integrator MASs with an AUABJT available to only partial agents based on sampled-data without velocity measurements. Therefore, a question arises that how we can devise a sampled-data CTP without velocity measurements to guarantee that double-integrator MASs track an AUABJT available to only partial agents.

To answer the above question, we propose a sampled-data CTP without velocity measurements by using the numerical derivatives of an agent's and an AUABJT's positions to estimate their velocities. We have theoretically proved and numerically illustrated that the proposed sampled-data CTP without velocity measurements is effective to guarantee that double-integrator MASs with an AUABJT available to only partial agents achieve asymptotic bounded consensus tracking (ABCT).

In this section, we theoretically demonstrate the effectiveness of the protocol (3). Main results are provided below.

In this section, we provide a numerical example to demonstrate the effectiveness of the protocol (3). Consider MASs with five agents, whose communication topology is provided in Fig. 1, where the target is labelled as 0.

Fig. 1.

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	Fig. 1. Communication topology.

Without loss of generality, we assume that all weights of edges are 1. The initial states of five agents are randomly set. It can be verified that when , (4)–(6) hold simultaneously. Figure 2 illustrates the numerical results.

Fig. 2.

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	Fig. 2. (color online) Positions and velocities of systems (1) under protocol (3) with and .

It is clear from Fig. 2 that the positions and velocities of all agents converge to the position and velocity of the target in the asymptotical and bounded sense. This means that the systems (1) under the protocol (3) achieve ABCT. This numerically illustrates the effectiveness of the protocol (3).

In this paper, we considered ABCT of double-integrator MASs with an AUABJT available to partial agents based on sampled-data without velocity measurements. We proposed a sampled-data CTP without velocity measurements and proved that the proposed protocol is effective to guarantee that double-integrator MASs with an AUABJT available to only partial agents achieve ABCT. In this paper, to overcome the difficulties caused by the unavailability of velocity information we used the numerical derivatives of an agent's and an AUABJT's positions to estimate their velocities. Recently, some distributed estimation-based CTPs have been proposed to solve CT of MASs with partial measurements.^[36–39] Motivated by the methods in Refs. [36]–[39], in the future we intend to design appropriately distributed estimation-based CTPs without velocity measurements to deal with the issue considered in this paper. In this paper, we assumed the communication topology to be undirected and fixed. Considering that the communication topology might be directed and switching in some real situations, we will extend this paper's results to the case of a directed switching communication topology.

^[27–31]