Search

This article studies the localization and circumnavigation problem concerned with a cluster of nonholonomic vehicles independently of a uniform global coordinate. Particularly, the clustered vehicles collaboratively locate an unknown target and then circumnavigate it isometrically with desired distance and velocity. However, the vehicles have no access to any position information, including their own absolute positions and others’ relative positions. Instead, they are capable of measuring relative bearings only with respect to their neighbors within their local coordinates. Different local coordinates are not necessarily aligned. Moreover, only some of the vehicles can measure relative bearings with respect to the target. For the solution to the concerned localization and circumnavigation problem, a coordinate-free distributed algorithm composed by two parts is proposed. To be specific, the first distributed localization part guarantees that the vehicles accurately estimate the relative positions with respect to the target and neighbors given a persistently exciting condition. And the second distributed circumnavigation part further guarantees that the vehicles move around the target estimation isometrically with the desired distance and velocity. Also, the necessary persistently exciting condition is shown to be met. Finally, simulation and experiment are both given to confirm the effectiveness of the proposed distributed algorithm.

We address the motion planning problem for a robotic system whose configuration manifold contains a group of rotations. Our approach is applied to a free-floating space robot composed of a three-dimensional base (a spacecraft) and an anthropomorphic onboard manipulator. The robot is actuated by the torques exerted at the joints of the onboard manipulator. A coordinate-free representation of rotations is utilized. The Lagrangian formalism is employed in order to derive a dynamics model of the robot that takes the form of a control system defined on the group of rotations and the joint space of the onboard manipulator. Using the conservation of angular momentum of the robot, a Jacobian motion planning algorithm is designed relying on the Endogenous Configuration Space Approach. The performance of the algorithm is verified by computer simulations.

This paper proposes a coordinate-free controller for a nonholonomic vehicle to circumnavigate a fully-actuated moving target by using range-only measurements. If the range rate is available, our Proportional Derivative (PD)-like controller has a simple structure as the standard PD controller, except the design of an additive constant bias and a saturation function in the error feedback. We show that if the target is stationary, the vehicle asymptotically encloses the target with a predefined radius at an exponential convergence rate, i.e., an exact circumnavigation pattern can be completed. For a moving target, the circumnavigation error converges to a small region whose size is shown proportional to the maneuverability of the target, e.g., the maximum linear speed and acceleration. Moreover, we design a second-order sliding mode (SOSM) filter to estimate the range rate and show that the SOSM filter can recover the range rate in a finite time. Finally, the effectiveness and advantages of our controller are validated via both numerical simulations and real experiments., 13 pages,17 figures

Computing quantum chemical properties of small molecules and polymers can provide insights valuable into physicists, chemists, and biologists when designing new materials, catalysts, biological probes, and drugs. Deep learning can compute quantum chemical properties accurately in a fraction of time required by commonly used methods such as density functional theory. Most current approaches to deep learning in quantum chemistry begin with geometric information from experimentally derived molecular structures or pre-calculated atom coordinates. These approaches have many useful applications, but they can be costly in time and computational resources. In this study, we demonstrate that accurate quantum chemical computations can be performed without geometric information by operating in the coordinate-free domain using deep learning on graph encodings. Coordinate-free methods rely only on molecular graphs, the connectivity of atoms and bonds, without atom coordinates or bond distances. We also find that the choice of graph-encoding architecture substantially affects the performance of these methods. The structures of these graph-encoding architectures provide an opportunity to probe an important, outstanding question in quantum mechanics: what types of quantum chemical properties can be represented by local variable models? We find that Wave, a local variable model, accurately calculates the quantum chemical properties, while graph convolutional architectures require global variables. Furthermore, local variable Wave models outperform global variable graph convolution models on complex molecules with large, correlated systems.

We present an efficient, accurate computational method for a coordinate-free model of flame front propagation of Frankel and Sivashinsky. This model allows for overturned flames fronts, in contrast to weakly nonlinear models such as the Kuramoto–Sivashinsky equation. The numerical procedure adapts the method of Hou, Lowengrub and Shelley, derived for vortex sheets, to this model. The result is a nonstiff, highly accurate solver which can handle fully nonlinear, overturned interfaces, with similar computational expense to methods for weakly nonlinear models. We apply this solver both to simulate overturned flame fronts and to compare the accuracy of Kuramoto–Sivashinsky and coordinate-free models in the appropriate limit. doi:10.1017/S1446181121000079

We present a coordinate-free version of Fefferman’s solution of Whitney’s extension problem in the space Cm−1,1(Rn). While the original argument relies on an elaborate induction on collections of partial derivatives, our proof uses the language of ideals and translation-invariant subspaces in the ring of polynomials. We emphasize the role of compactness in the proof, first in the familiar sense of topological compactness, but also in the sense of finiteness theorems arising in logic and semialgebraic geometry. These techniques may be relevant to the study of Whitney-type extension problems on sub-Riemannian manifolds where global coordinates are generally unavailable.

Based on an abstract algebraic setting we provide an elementary derivation of the Youla-Kucera parametrization of stabilizing controllers and also an alternative, coordinate free, approach of the problem. For this latter case, in contrast to the Youla-Kucera approach, the parameter set is not universal but its elements can be generated by a universal algorithm. We also emphasise the natural continuity property of this parametrization compared to the Youla-Kucera case. Extending the framework to the LFT loops we show by using elementary tools that every controller which stabilizes the interior loop of the generalized plant also stabilizes the LFT loop.

In this article, we investigate some spectral properties of a singular dissipative fourth-order dissipative operator in lim−4 case at the singular point. For this purpose we construct the character...

This paper investigates new coordinate-free formation control strategies of multi-agent systems to overcome the negative effects of time delays. To this end, we present a single predictor-feedback scheme to compensate the multiple communication delays, assumed to be unknown but bounded and arbitrarily-fast time-varying. Although delays cannot exactly be compensated due to time-varying delay mismatches, the trade-off between robustness and convergence speed can be notably improved if the control gain is suitably designed. Hence, with the objective of enlarging the time-varying delay interval for a given convergence speed, an LMI-based iterative algorithm is presented to solve the control gain synthesis.

This paper conveys attitude and rate estimation without rate sensors by performing a critical comparison, validated by extensive simulations. The two dominant approaches to facilitate attitude estimation are based on stochastic and set-membership reasoning. The first one mostly utilizes the commonly known Gaussian-approximate filters, namely the EKF and UKF. Although more conservative, the latter seems to be more promising as it considers the inherent geometric characteristics of the underline compact state space and accounts -- from first principles -- for large model errors. We address the set-theoretic approach from a control point of view, and we show that it can overcome reported deficiencies of the Bayesian architectures related to this problem, leading to coordinate-free optimal filters. Lastly, as an example, we derive a modified predictive filter on the tangent bundle of the special orthogonal group $\mathbb{TSO}(3)$., Comment: Accept

The isoline tracking of this work is concerned with the control design for a sensing robot to track a given isoline of an unknown 2-D scalar filed. To this end, we propose a coordinate-free controller with a simple PI-like form using only the concentration feedback for a Dubins robot, which is particularly useful in GPS-denied environments. The key idea lies in the novel design of a sliding surface based error term in the standard PI controller. Interestingly, we also prove that the tracking error can be reduced by increasing the proportion gain, and be eliminated for circular fields with a non-zero integral gain. The effectiveness of our controller is validated via simulations by using a fixed-wing UAV on the real dataset of the concentration distribution of PM2.5 in an area of China.

In this work, we derive a coordinate-free formulation of the coupled dynamics of a class of 6DOF aerial manipulators consisting of an underactuated quadrotor equipped with a 2DOF articulated manipulator, and demonstrate that the system is differentially flat with respect to the end effector pose. In particular, we require the center of mass of the entire system to be fixed in the end effector frame, suggesting a reasonable mechanical design criterion. We make use of an inertial decoupling transformation to demonstrate differential flatness, allowing us to plan dynamically feasible trajectories for the system in the space of the 6DOF pose of the end effector, which is ideal for achieving precise manipulator tasks. Simulation results validate the flatness-based planning methodology for our dynamic model, and its usefulness is demonstrated in a simulated aerial videography task.

This paper addresses the optimal orbit rendezvous problem for thrust-vectoring spacecraft. In this scenario, the chaser spacecraft is equipped with only one impulsive thruster fixed in the body frame, while the thrust direction is pointed using attitude control of the spacecraft. An improved particle swarm optimization (PSO) algorithm is employed to generate an optimal multiple-burn rendezvous trajectory in terms of thruster pulse duration and required attitude. A coordinate-free, finite-time, attitude control is developed to ensure that this required thrust vector is met exactly at the prescribed time. The proposed orbit-attitude guidance algorithm is compared with one which utilizes a conventional PSO and is shown to significantly improve performance. In addition, the set-point attitude control is illustrated, in simulation, to deliver the required thrust direction at the prescribed time.

We investigate a coordinate-free model of flame fronts introduced by Frankel and Sivashinsky; this model has a parameter $\alpha$ which relates to how unstable the front might be. We first prove short-time well-posedness of the coordinate-free model, for any value of $\alpha>0.$ We then argue that near the threshold $\alpha \approx 1,$ the solution stays arbitrarily close to the solution of the weakly nonlinear Kuramoto--Sivashinsky (KS) equation, as long as the initial values are close., Comment: 28 Pages, 0 figures

This paper studies how to control large formations of autonomous agents in the plane, assuming that each agent is able to sense relative positions of its neighboring agents with respect to its own local coordinate system. We tackle the problem by adopting two types of controllers. First, we use the classical gradient-based controllers on three leader agents to meet their distance constraints. Second, we develop other type of controllers for follower agents: utilizing the properties of rooted graphs, one is able to design linear controllers incorporating relative positions between the follower agents and their neighbors, to stabilize the overall large formations. The advantages of the proposed method are fourfold: (i) fewer constraints on neighboring relationship graphs; (ii) simplicity of linear controllers for follower agents; (iii) global convergence of the overall formations; (iv) implementation in local coordinate systems, in no need of a global coordinate system. Numerical simulations show the effectiveness of the proposed method.

This article concerns the definition and identification of qualitative spatial relationships for the full and partial enclosure of spatial regions. The article precisely defines three relationships between regions—“surrounds,” “engulfs,” and “envelops”—highlighting the correspondence to similar definitions in the literature. An efficient algorithm capable of identifying these qualitative spatial relations in a network of dynamic (mobile) geosensor nodes is developed and tested. The algorithms are wholly decentralized, and operate in-network with no centralized control. The algorithms are also “coordinate-free,” able to operate in distributed spatial computing environments where coordinate locations are expensive to capture or otherwise unavailable. Experimental evaluation of the algorithms designed demonstrates the efficiency of the approach. Although the algorithm communication complexity is dominated by an overall worst-case O ( n 2 ) leader election algorithm, the experiments show in practice an average-case complexity approaching linear, O ( n 1.1 ).

The main purpose of this work is to provide application of differential forms in physics. For this purpose, we describe differential forms, exterior algebra in details and then we express Maxwell’s equations by using differential forms. In the theory of pseudo-Riemannian manifolds there will be an important operator, called Hodge Star Operator. Hodge Star Operator arises in the coordinate free formulation of Maxwell’s equation in flat space-time. This operator is an important ingredient in the formulation of Stoke’stheorem.GANIT J. Bangladesh Math. Soc.Vol. 37 (2017) 15-27

Youla parametrization of stabilizing controllers is a fundamental result of control theory: starting from a special, double coprime, factorization of the plant provides a formula for the stabilizing controllers as a function of the elements of the set of stable systems. In this case the set of parameters is universal, i.e., does not depend on the plant but only the dimension of the signal spaces. Based on the geometric techniques introduced in our previous work this paper provides an alternative, geometry based parametrization. In contrast to the Youla case, this parametrization is coordinate free: it is based only on the knowledge of the plant and a single stabilizing controller. While the parameter set itself is not universal, its elements can be generated by a universal algorithm. Moreover, it is shown that on the parameters of the strongly stabilizing controllers a simple group structure can be defined. Besides its theoretical and educative value the presentation also provides a possible tool for the algorithmic development.

Water-mediated ion-pair dissociation is studied by molecular dynamics simulations of NaCl in water. Multidimensional free-energy analysis clarifies the relation between two essential solvent coordi...

In this paper, we got the best linear unbiased predictor of any linear function of the elements of a finite population under coordinate-free models. The optimal predictor of these quantities was obtained in an earlier work considering models with a known diagonal covariance matrix. We extended this result assuming any known covariance matrix. It is shown that in the particular case of the coordinatized models, this general predictor coincides with the optimal predictor of the total population under a regression super population model with correlated observations.

International audience; We produce a coordinate free presentation of some concepts usually involved in incremental mechanics (tangent linear stiffness matrix, stability, loading paths for example) but not always well founded. Thanks to the geometric language of vector bundles, a well defined geometrical object may be associated to each of these tools that allows us to understand some latent difficulties linked with these tools due to the absence of a natural connection and also to extend some of our recent results of linear stability to a non linear framework.

This paper investigates the control problem of a Dubins-like robot to circumnavigate an unknown target with a prescribed radius in GPS-denied environments. Considering the sensing capabilities of small robots, the current distance to the target is the only measurement. A coordinate free range-only control law is devised to force the robot to achieve the objective of target circumnavigation. The proposed controller can guarantee global convergence irrespective of the initial states and locally exponential convergence near the equilibrium point. Furthermore, the selection of control parameters is independent of the initial states. The effectiveness and advantages of our approach are validated via simulations.

Youla parametrization of stabilizing controllers is a fundamental result of control theory: starting from a special, double coprime, factorization of the plant provides a formula for the stabilizing controllers as a function of the elements of the set of stable systems. In this case the set of parameters is universal, i.e., does not depend on the plant but only the dimension of the signal spaces. Based on the geometric techniques introduced in our previous work this paper provides an alternative, geometry based parametrization. In contrast to the Youla case, this parametrization is coordinate free: it is based only on the knowledge of the plant and a single stabilizing controller. While the parameter set itself is not universal, its elements can be generated by a universal algorithm. Moreover, it is shown that on the parameters of the strongly stabilizing controllers a simple group structure can be defined. Besides its theoretical and educative value the presentation also provides a possible tool for the algorithmic development.

We present a coordinate-free description of Carlsson-Weinshall duality between scene points and camera pinholes and use it to derive a new characterization of primal/dual multi-view geometry. In the case of three views, a particular set of reduced trilinearities provide a novel parameterization of camera geometry that, unlike existing ones, is subject only to very simple internal constraints. These trilinearities lead to new "quasi-linear" algorithms for primal and dual structure from motion. We include some preliminary experiments with real and synthetic data.

The modeling of the different mechanical behaviors of brittle and quasi-brittle materials in tension and compression leads to partitioning of the strain (or stress) tensor into a positive part and a negative part. In this study, applying a recently proposed general method to the two-dimensional (2D) strain and stress tensors, closed-form coordinate-free expressions are obtained for their decompositions which are orthogonal in the sense of an inner product where the forth-order elastic stiffness or compliance acts as a metric. The orthogonal decompositions are given analytically and explicitly for all possible 2D elastic symmetries, i.e., isotropic, orthotropic, square, and totally anisotropic elastic materials. These results can be directly used, for example, in developing phase field methods for modeling and simulating the fracture of isotropic and anisotropic brittle and quasi-brittle materials.

In this work, the divergence and curl operators are obtained using the coordinate free non rigid basis formulation of differential geometry. Although the authors have attempted to keep the presentation self-contained as much as possible, some previous exposure to the language of differential geometry may be helpful. In this sense the work is aimed to late undergraduate or beginners graduate students interested in mathematical physics. To illustrate the development, we graphically present the eleven coordinate systems in which the Laplace operator is separable. We detail the development of the basis and the connection for the cylindrical and paraboloidal coordinate systems. We also present in [1] codes both in Maxima and Maple for the spherical orthonormal basis, which serves as a working model for calculations in other situations of interest. Also in [1] the codes to obtain the coordinate surfaces are given.

This paper concentrates on coordinate-free formation control for directed networks, for which the dynamic motion of each agent is assumed to be governed only by a local control. We develop a graph Laplacian approach to solve the global and exponential formation stablization problem using merely relative position measurements between neighbors. First, to capture the sensing and control architectures that are needed to maintain the shape of a formation, a necessary and sufficient topological condition is proposed. Second, a Laplacian-based control law is developed for the stablization problem of a group of mobile agents to a desired formation shape under both fixed and switching topologies due to temporal node failures. Simulation results are provided to demonstrate that our Laplacian-based formation control strategy is inherently fault-tolerant and robust to node failures.

This paper presents tracking control laws for two different objectives of a nonholonomic system - a spherical robot - using a geometric approach. The first control law addresses orientation tracking using a modified trace potential function. The second law addresses contact position tracking using a right transport map for the angular velocity error. A special case of this is position and reduced orientation stabilization. Both control laws are coordinate free. The performance of the feedback control laws are demonstrated through simulations.

[Departement_IRSTEA]Eaux [ADD1_IRSTEA]Hydrosystèmes et risques naturels; International audience; In this paper we extend the Kinematic Structural Stability (KISS) issues obtained in the punctual linear case to a full non linear framework. Kinematic Structural Stability (KISS) refers to the stability of conservative or non-conservative elastic systems under kinematic constraints. We revisite and sometimes introduce some new mechanical and geometrically-related concepts, using the systematic language of vector bundles: transversality-stability, tangent stiffness tensor, loading paths and KISS. We bring an intrinsic geometrical meaning to these concepts and we extend the main results to a general nonlinear framework. The lack of connection on the configuration manifold in order to make derivatives of sections is bypassed thanks to the key rule of the nil section of the cotangent bundle. Two mechanical examples illustrate the interest of the new concepts, namely a discrete nonconservative elastic system under non-conservative loading (Ziegler's column) and, in more details, a four-sphere granular system with nonconservative interactions called diamond pattern.

In learning by demonstration, the generalization of motion trajectories far away from the set of demonstrations is often limited by the dependency of the learned models on arbitrary coordinate references. Trajectory shape descriptors have the potential to remove these dependencies by representing demonstrated trajectories in a coordinate-free way. This paper proposes a constraint-based optimization framework to generalize demonstrated rigid-body motion trajectories to new situations starting from the shape descriptor of the demonstration. Experimental results indicate excellent generalization capabilities showing how, starting from only a single demonstration, new trajectories are easily generalized to novel situations anywhere in task space, such as new initial or target positions and orientations, while preserving similarity with the demonstration. The results encourage the use of trajectory shape descriptors in learning by demonstration to reduce the number of required demonstrations.

Tracking a desired Pointing Direction and simultaneously obtaining a reference Angular Velocity (PDAV) around the pointing direction constitutes a very involved and compli- cated motion encountered in a variety of robotic, industrial and military applications. In this paper through the utilization of global analysis and simulation techniques, the smooth closed- loop vector fields induced by the geometric PDAV controller from [1], are visualized to gain a deeper understanding of its global stabilization properties. First through the calculation of a coordinate-free form of the closed-loop linearized dynamics, the local stability of each equilibrium of the system is analyzed. The results acquired by means of eigenstructure analysis, are used in predicting the frequency of complex precession/nutation oscillations that arise during PDAV trajectory tracking; an important tool in actuator selection. Finally, by utilizing vari- ational integration schemes, the flow converging to the desired equilibrium and the flow “close” to the stable manifold of the saddle equilibrium of the closed-loop system is visualized and analyzed. Results offer intimate knowledge of the closed-loop vector fields bestowing to the control engineer the ability to anticipate and have an estimate of the evolution of the solutions.

This paper presents an implementation of a Meshless Local Strong Form Method that allows users to write elegant code expressed in terms of abstract mathematical objects, such as operators and fields, consequently avoiding working directly with matrix and array indices, which is tedious and error prone. This is achieved by using object oriented programming techniques for definition of abstract concepts and leveraging C++'s powerful templating mechanism. It is demonstrated that code written this way has little-to-no performance overhead compared to classical numerical code while being more expressive and readable, which gravely shortens model development and testing phases. The overall functionality of presented implementation is illustrated on numerical examples from classical thermodynamics, linear elasticity and fluid dynamics in one, two and three dimensions.

Identifying the boundary nodes of a wireless sensor network (WSN) is one of the prerequisites for healing coverage holes, which belongs to the main problems of QoS in a network. Most of the existing solutions for this problem rely on the usage of coordinates or a relatively even distribution of sensors on the underlying network. However, as equipping localization devices on sensors usually means considerably higher cost, coordinates are often unavailable in low-budget networks. And it is often hard to guarantee the distribution of nodes. Therefore, identifying the boundary nodes without coordinates still faces difficulties. In this paper, we propose a distributed algorithm to solve this problem. In our method, a checking node first finds out all the potential triangles that with vertices from the 1-hop neighbors of the node. Then, the sensor keeps triangles containing it by calculating angles. At last, triangles with gap edges are identified by angle comparison to avoid wrong identification of a boundary node which has a U shaped ring locates beside. Illustrated simulation results show the performance effectiveness of our method, especially in networks with random and uneven sensor deployment.

[EN] Aim: The increasing availability of Big Biomedical Data is leading to large research data samples collected over long periods of time. We propose the analysis of the kinematics of data probability distributions over time towards the characterization of data temporal variability. Methods: First, we propose a kinematic model based on the estimation of a continuous data temporal trajectory, using Functional Data Analysis over the embedding of a non-parametric statistical manifold which points represent data temporal batches, the Information Geometric Temporal (IGT) plot. This model allows measuring the velocity and acceleration of data changes. Next, we propose a coordinate-free method to characterize the oriented seasonality of data based on the parallelism of lagged velocity vectors of the data trajectory throughout the IGT space, the Auto-Parallelism of Velocity Vectors (APVV) and APVVmap. Finally, we automatically explain the maximum variance components of the IGT space coordinates by means of correlating data points with known temporal factors from the domain application. Materials: Methods are evaluated on the US National Hospital Discharge Survey open dataset, consisting of 3,25M hospital discharges between 2000 and 2010. Results: Seasonal and abrupt behaviours were present on the estimated multivariate and univariate data trajectories. The kinematic analysis revealed seasonal effects and punctual increments in data celerity, the latter mainly related to abrupt changes in coding. The APVV and APVVmap revealed oriented seasonal changes on data trajectories. For most variables, their distributions tended to change to the same direction at a 12-month period, with a peak of change of directionality at mid and end of the year. Diagnosis and Procedure codes also included a 9-month periodic component. Kinematics and APVV methods were able to detect seasonal effects on extreme temporal subgrouped data, such as in Procedure code, where Fourier and autocorrelation methods were not able to. The automated explanation of IGT space coordinates was consistent with the results provided by the kinematic and seasonal analysis. Coordinates received different meanings according to the trajectory trend, seasonality and abrupt changes. Discussion: Treating data as a particle moving over time through a multidimensional probabilistic space and studying the kinematics of its trajectory has turned out to a new temporal variability methodology. Its results on the NHDS were aligned with the dataset and population descriptions found in the literature, contributing with a novel temporal variability characterization. We have demonstrated that the APVV and APVVmat are an appropriate tool for the coordinate-free and oriented analysis of trajectories or complex multivariate signals. Conclusion: The proposed methods comprise an exploratory methodology for the characterization of data temporal variability, what may be useful for a reliable reuse of Big Biomedical Data repositories acquired over long periods of time., This work was supported by UPV grant No. PAID-00-17, and projects DPI2016-80054-R and H2020-SC1-2016-CNECT No. 727560.

The state of stresses in non-homogeneous bars due to eccentric axial loads is examined. The governing formulae are written in coordinate free invariant forms by means of vector and tensor algebra. The concepts of centric and eccentric axial loads for non-homogeneous bars are introduced. A general solution is developed to obtain the kern of cross section of arbitrary shape. Examples show how the results derived can be applied to determine the kern of a given inhomogeneous cross section.

This paper presents a coordinate free pre-metric formulation of charge free Maxwell-Minkowski electrodynamics, and of the developed by the authors non-linear Extended Electrodynamics. First we introduce some formal relations from multilinear algebra and differential geometry to be used further. Then we recall and appropriately modify the existing pre-metric formulation of linear charge free electrodynamics in pre-relativistic and relativistic forms as preparation to turn to corresponding pre-metric nonlinearization. After some preliminary examples and notes on nonlinearization, we motivate our view for existence and explicit formulation of time stable subsystems of the physical field objects considered. Section 5 presents the formal results of our approach on the pre-metric nonlinear formulations in static case, in time-dependent case, and in space-time formulation. In the Conclusion we give our general view on "why and how to nonlinearize". The Appendix gives a possible formal extension of our aproach to many subsystem field objects., 17 pages, no figures. arXiv admin note: text overlap with arXiv:1508.04824

It is undoubtedly interesting, from a practical perspective, to solve the problem of multirobot formation stabilization in a decentralized fashion, while allowing the agents to rely only on their independent onboard sensors (e.g., cameras), and avoiding the use of leader robots or global reference frames . However, a key observation that serves as motivation for the work presented in this chapter is that the available controllers satisfying these conditions generally fail to provide global stability guarantees. In this chapter, we provide novel theoretical tools to address this issue; in particular, we propose coordinate-free formation stabilization algorithms that are globally convergent. The common elements of the control methods we describe are that they rely on relative position information expressed in each robot’s independent frame, and that the absence of a shared orientation reference is dealt with by introducing locally computed rotation matrices in the control laws. Specifically, three different nonlinear formation controllers for mobile robots are presented in the chapter. First, we propose an approach relying on global information of the team, implemented in a distributed networked fashion. Then, we present a purely distributed method based on each robot using only partial information from a set of formation neighbors. We finally explore formation stabilization applied to a target enclosing task in a 3D workspace. The developments in this chapter pave the way for novel vision-based implementations of control tasks involving teams of mobile robots, which is the leitmotif of the monograph. The controllers are formally studied and their performance is illustrated with simulations.