Bearing rigidity and formation stabilization for multiple rigid bodies in SE(3)

In this work, we first distinguish different notions related to bearing rigidity in graph theory and then further investigate the formation stabilization problem for multiple rigid bodies. Different from many previous works on formation control using bearing rigidity, we do not require the use of a shared global coordinate system, which is enabled by extending bearing rigidity theory to multi-agent frameworks embedded in the three dimensional special Euclidean group SE(3) and expressing the needed bearing information in each agent's local coordinate system. Here, each agent is modeled by a rigid body with 3 DOFs in translation and 3 DOFs in rotation. One key step in our approach is to define the bearing rigidity matrix in SE(3) and construct the necessary and sufficient conditions for infinitesimal bearing rigidity. In the end, a gradient-based bearing formation control algorithm is proposed to stabilize formations of multiple rigid bodies in SE(3)

Enclosing a moving target with an optimally rotated and scaled multiagent pattern

We propose a novel control method to enclose a moving target in a two-dimensional setting with a team of agents forming a prescribed geometric pattern. The approach optimises a measure of the overall agent motion costs, via the minimisation of a suitably defined cost function encapsulating the pattern rotation and scaling. We propose two control laws which use global information and make the agents exponentially converge to the prescribed formation with an optimal scale that remains constant, while the team's centroid tracks the target. One control law results in a multiagent pattern that keeps a constant orientation in the workspace; for the other, the pattern rotates with constant speed. These behaviours, whose optimality and steadiness are very relevant for the task addressed, occur independently from the target's velocity. Moreover, the methodology does not require distance measurements, common coordinate references, or communications. We also present formal guarantees of collision avoidance for the proposed approach. Illustrative simulation examples are provided

Cooperative Visual-Inertial Sensor Fusion: Fundamental Equations

International audienceThis paper provides a new theoretical and basic result in the framework of cooperative visual-inertial sensor fusion. Specifically, the case of two aerial vehicles is investigated. Each vehicle is equipped with inertial sensors (accelerometer and gyroscope) and with a monocular camera. By using the monocular camera, each vehicle can observe the other vehicle. No additional camera observations (e.g., of external point features in the environment) are considered. First, the entire observable state is analytically derived. This state includes the relative position between the two aerial vehicles (which includes the absolute scale), the relative velocity and the three Euler angles that express the rotation between the two vehicle frames. Then, the basic equations that describe this system are analytically obtained. In other words, both the dynamics of the observable state and all the camera observations are expressed only in terms of the components of the observable state and in terms of the inertial measurements. These are the fundamental equations that fully characterize the problem of fusing visual and inertial data in the cooperative case. The last part of the paper describes the use of these equations to achieve the state estimation through an EKF. In particular, a simple manner to limit communication among the vehicles is discussed. Results obtained through simulations show the performance of the proposed solution, and in particular how it is affected by limiting the communication between the two vehicles

Bearing-Only Consensus and Formation Control under Directed Topologies

We address the problems of bearing-only consensus and formation control, where each agent can only measure the relative bearings of its neighbors and relative distances are not available. We provide stability results for the Filippov solutions of two gradient-descent laws from non-smooth Lyapunov functions in the context of differential inclusion. For the consensus and formation control problems with undirected sensing topologies, we prove finite-time and asymptotic convergence of the proposed non-smooth gradient flows. For the directed consensus problem, we prove asymptotic convergence using a different non-smooth Lyapunov function given that the sensing graph has a globally reachable node. Finally, For the directed formation control problem we prove asymptotic convergence for directed cycles and directed acyclic graphs and also introduce a new notion of bearing persistence which guarantees convergence to the desired bearings

Cooperative Visual-Inertial Sensor Fusion: Fundamental Equations and State Determination in Closed-Form

International audienceThis paper investigates the visual and inertial sensor fusion problem in the cooperative case and provides new theoretical and basic results.Specifically, the case of two agents is investigated. Each agent is equipped with inertial sensors (accelerometer and gyroscope) and with a monocular camera. By using the monocular camera, each agent can observe the other agent. No additional camera observations (e.g., of external point features in the environment) are considered.First, the entire observable state is analytically derived. This state contains the relative position between the two agents (which includes the absolute scale), the relative velocity, the three Euler angles that express the rotation between the two local frames and all the accelerometer and gyroscope biases.Then, the basic equations that describe this system are analytically obtained. %In other words, both the dynamics of the observable state and all the camera observations are expressed only in terms of the components of the observable state and in terms of the inertial measurements. These are the fundamental equations that fully characterize the problem of fusing visual and inertial data in the cooperative case. The last part of the paper describes the use of these equations to obtain a closed-form solution that provides the observable state in terms of the visual and inertial measurements provided in a short time interval. The impact of the presence of the bias on the performance of this closed-form solution is also investigated and a simple and effective method to obtain the gyroscope bias is proposed.Extensive simulations clearly show that the proposed method is successful. It is worth noting that it is possible to automatically retrieve the absolute scale and simultaneously calibrate the gyroscopes not only without any prior knowledge, but also without external point features in the environment

Bearing-Only Formation Control Using an SE(2) Rigidity Theory

International audienceThis paper considers a formation control problem for a team of agents that are only able to sense the relative bearings from their local body frame to neighboring agents. It is further assumed that the sensing graph is inherently directed and a common reference frame is not known to all of the agents. Each agent is tasked with maintaining predetermined bearings with their neighbors. Using the recently developed rigidity theory for SE(2) frameworks [1], we propose a gradient-type controller to stabilize the formation. The central construct in the SE(2) rigidity theory for this work is the directed bearing rigidity matrix. We show that a necessary condition for the local stabilization of desired formation is for the corresponding SE(2) framework to be minimally infinitesimally rigid

Bearing-Only Formation Control Using an SE(2) Rigidity Theory

International audienceThis paper considers a formation control problem for a team of agents that are only able to sense the relative bearings from their local body frame to neighboring agents. It is further assumed that the sensing graph is inherently directed and a common reference frame is not known to all of the agents. Each agent is tasked with maintaining predetermined bearings with their neighbors. Using the recently developed rigidity theory for SE(2) frameworks [1], we propose a gradient-type controller to stabilize the formation. The central construct in the SE(2) rigidity theory for this work is the directed bearing rigidity matrix. We show that a necessary condition for the local stabilization of desired formation is for the corresponding SE(2) framework to be minimally infinitesimally rigid

Bearing-Only Formation Control Using an SE(2) Rigidity Theory

International audienceThis paper considers a formation control problem for a team of agents that are only able to sense the relative bearings from their local body frame to neighboring agents. It is further assumed that the sensing graph is inherently directed and a common reference frame is not known to all of the agents. Each agent is tasked with maintaining predetermined bearings with their neighbors. Using the recently developed rigidity theory for SE(2) frameworks [1], we propose a gradient-type controller to stabilize the formation. The central construct in the SE(2) rigidity theory for this work is the directed bearing rigidity matrix. We show that a necessary condition for the local stabilization of desired formation is for the corresponding SE(2) framework to be minimally infinitesimally rigid