Safety analysis of high confidence systems requires guaranteed bounds on the probabilities of events of interest. Establishing the correctness of algorithms that aim to compute such bounds is challenging. We address this problem in three steps. First, we use monadic transition systems (MTS) in the category of sets as a framework for modeling discrete time systems. MTS can capture different types o

The Kuramoto model of coupled phase oscillators is often used to describe synchronization phenomena in nature. Some applications, e.g., quantum synchronization and rigid-body attitude synchronization, involve high-dimensional Kuramoto models where each oscillator lives on the n-sphere or SO(n). These manifolds are special cases of the compact, real Stiefel manifold St(p,n). Using tools from optimi

This paper introduces a new lifting method for analyzing convergence of continuous-time distributed synchronization/consensus systems on the unit sphere. Points on the d-dimensional unit sphere are lifted to the (d+1)-dimensional Euclidean space. The consensus protocol on the unit sphere is the classical one, where agents move toward weighted averages of their neighbors in their respective tangent

In the multi-agent systems setting, this paper addresses continuous-time distributed synchronization of columns of rotation matrices. More precisely, k specific columns shall be synchronized and only the corresponding k columns of the relative rotations between the agents are assumed to be available for the control design. When one specific column is considered, the problem is equivalent to synchr

The finite-time attitude synchronization problem is considered in this paper, where the rotation of each rigid body is expressed using the axis-angle representation. Two discontinuous and distributed controllers using the vectorized signum function are proposed, which guarantee almost global and local convergence, respectively. Filippov solutions and nonsmooth analysis techniques are adopted to ha

This paper establishes novel results regarding the global convergence properties of a large class of consensus protocols for multiagent systems that evolve in continuous time on the n-dimensional unit sphere or n-sphere. For any connected, undirected graph and all \in \mathbb{N} \backslash \lbrace 1\rbrace$, each protocol in said class is shown to yield almost global consensus. The feedback laws a

This paper addresses synchronization of Euclidean transformations over graphs. Synchronization in this context, unlike rendezvous or consensus, means that composite transformations over loops in the graph are equal to the identity. Given a set of non-synchronized transformations, the problem at hand is to find a set of synchronized transformations approximating well the non-synchronized transforma

This paper addresses the problem of synchronizing orthogonal matrices over directed graphs. For synchronized transformations (or matrices), composite transformations over loops equal the identity. We formulate the synchronization problem as a least-squares optimization problem with nonlinear constraints. The synchronization problem appears as one of the key components in applications ranging from

In this paper, for any homogeneous system of agents with linear continuous time dynamics, we formulate an optimal control problem. In this problem a convex cost functional of the control signals of the agents shall be minimized, while the outputs of the agents shall coincide at some given finite time. This is an instance of the rendezvous or finite time consensus problem. We solve this problem wit

This paper addresses the consensus problem and the formation problem on SE(3) in multi-agent systems with directed and switching interconnection topologies. Several control laws are introduced for the consensus problem. By a simple transformation, it is shown that the proposed control laws can be used for the formation problem. The design is first conducted on the kinematic level, where the veloci

The alignment of a set of objects by means of transformations plays an important role in computer vision. Whilst the case for only two objects can be solved globally, when multiple objects are considered usually iterative methods are used. In practice the iterative methods perform well if the relative transformations between any pair of objects are free of noise. However, if only noisy relative tr

This paper addresses attitude synchronization problems for systems of multiple rigid-body agents with directed interconnection topologies. Two scenarios which differ in available information are considered. In the first scenario the agents can obtain their rotations and angular velocities relative to an inertial reference frame and transmit these information to their neighbors, while in the second

We consider the problem of attitude synchronization for systems of rigid body agents with directed topologies. Two different scenarios for the rotation matrices of the agents are considered. In the first scenario, the rotations are contained in a convex subset of SO(3), which is a ball of radius less than π/2, whereas in the second scenario the agents are contained in a subset of SO(3), which is a

This paper addresses the attitude synchronization problem in multi-agent systems with directed and switching interconnection topologies. Two cases for the synchronization problem are discussed under different assumptions about the measurable information. In the first case the agents can measure their rotations relative to a global reference coordinate frame, whilst in the second case they can only

In this paper, we give a distributed solution to the problem of making a team of nonholonomic robots reach consensus about their orientations using monocular cameras. We consider a scheme where the motions of the robots are decided using nearest-neighbor rules. Each robot is equipped with a camera and can only exchange visual information with a subset of the other robots. The main contribution of

Here we address the problem of moving a camera from an initial pose to a final pose. The trajectory between the two poses is subject to constraints on the camera motion and the visibility, where we have bounds on the allowed velocities and accelerations of the camera and require that a set of point features are visible for the camera. We assume that the pose is possible to retrieve from the observ

This paper addresses the attitude synchronization problem of multiple rigid body agents in SO(3) with directed and switching interconnection topologies. Using the axis-angle representation of the orientation, a distributed controller based on differences between the orientations of agents in a global frame is proposed. In the case of the balanced interconnection graph, the attitude synchronization

In the present paper we consider the problem of attitude synchronization for a system of rigid body agents. We provide distributed kinematic control laws for two different synchronization problems. In the two problems the objective is the same, i.e., to synchronize the orientations of the agents, but what is assumed to be measurable by the agents differs. In problem 1 the agents measure their own

In this paper we study the output consensus problem for systems of agents with linear continuous, time invariant dynamics. We derive control laws with minimal energy that solves the problem, while using only relative information. Instead of considering a fixed communication topology for the agents, and derive the optimal control for that topology, we derive the optimal control law for any communic

In this paper we consider the problem of constructing feedback control laws for a system of n agents that shall synchronize their attitudes in SO(3). We propose distributed controllers for two synchronization problems, in which the objective is the same, to synchronize the orientations, but what the agents can perceive or communicate differs. In the first problem the agents can measure their orien