Bearing Rigidity Theory and Its Applications for Control and Estimation of Network Systems: Life Beyond Distance Rigidity

Abstract

Distributed control and estimation of multi-agent systems has received tremendous research attention in recent years due to their potential across many application domains [1], [2]. Here, the term "agent" can represent a sensor, an autonomous vehicle, or any general dynamical system. These multi-agent systems are becoming increasingly attractive because of their robustness against system failure, their ability to adapt to dynamic and uncertain environments, and their economic advantages compared to the implementation of more expensive monolithic systems.Formation control and network localization are two fundamental tasks for multi-agent systems that enable them to perform complex missions. The goal of formation control is to control each agent using local information from neighboring agents so that the entire team forms a desired spatial geometric pattern (see [2] for a recent survey on formation control). While the notion of a formation as a geometric pattern has a natural meaning for robotic systems, it may also correspond to more abstract configurations for the system state of a team of agents. The goal of network localization is to estimate the location of each agent in a network using locally sensed or communicated information from neighboring agents [3]-[6]. Network localization is usually the first step that must be completed before a sensor network provides other services like positioning mobile robots or monitoring areas of interest.For a formation control or network localization task, the type of information available to each agent is an important factor that determines the design of the corresponding control or estimation algorithms. Most of the existing approaches for formation control assume that each agent can obtain the relative positions of their nearest neighbors. In order to obtain relative positions in practice, each agent can measure their absolute positions using, for example, GPS, and then share their positions with their neighbors via wireless communications. This method is, however, not applicable when operating in GPS-denied environments such as indoors, underwater, or in deep space. Furthermore, the absolute accuracy of the GPS may not meet the requirements of high-accuracy formation control tasks. Rather than relying on external positioning systems such as GPS, each agent can use onboard sensors to sense their neighbors.Optical cameras are widely used onboard sensors for ground and aerial vehicles to achieve various sensing tasks due to their characteristics of being low-cost, light-weight, and low-power. It is notable that optical cameras are inherently bearing-only sensors. Specifically, once a target has been recognized in an image, its bearing relative to the camera can be calculated immediately from its pixel coordinate based on the pin-hole camera model [7, Section 3.3]. As a comparison, the range from the target to the camera is more complicated to obtain because it requires additional geometric information of the target and extra estimation algorithms, which may significantly in...

Fixed‐time orientation estimation and network localisation of multi‐agent systems

“…where x ∈ ℝ n and f : ℝ + × ℝ n → ℝ n is possibly discontinuous, which was extensively addressed in [20]. Suppose that the origin is an equilibrium point of the system described in (6).…”

“…Distance-based network localisation problems are well formulated in [3,4], where rigidity of an underlying graph of a given multi-agent system is crucial. Similarly, it has been known that bearing rigidity is essential for bearing-based localisation [5][6][7]. Though rigidity-based localisation methods are beneficial because they require only distance or bearing measurements, their performance heavily depends on highly dense interconnections of agents, which is linked to the rigidity of underlying graphs.…”

Angle Fixability and Angle-Based Sensor Network Localization

“…Lemma 9 implies that at least 3 anchors are required for ASNL in R 2 . Note that for SNL based on bearings measured in the global coordinate frame [12], [13], [15], [16], the minimum number of anchors required for unique localizability in R 2 is 2. This is because the angle between two global bearings can be determined as clockwise or counterclockwise in the global coordinate frame.…”

Consensus of multi‐agent system based on novel avoidance function and unequal avoidance strategy

“…Li et al [8] designed the proportional–integral reference velocity controller for the unmeasurable task‐space velocity, the designed controller could achieve the bearing‐based formation. Zhao et al [9] designed the bearing‐based controller under the bearing rigidity to make the multi‐agent system achieve the target formation. Li et al [10] designed the distributed adaptive controller under the reference acceleration and adaptive control scheme to solve the system uncertainties.…”