Modified Vector Field Research Articles

Functional equivariance and modified vector fields

Functional equivariance and modified vector fields

Robust distributed control within a curve virtual tube for a robotic swarm under self‐localization drift and precise relative navigation

AbstractTo guide the movement of a robotic swarm in a corridor‐like environment, a curve virtual tube with no obstacle inside was designed in our previous work. This article generalizes the controller design to the condition that all robots have self‐localization drifts and precise relative navigation, where the flocking algorithm is introduced to reduce the negative impact of the self‐localization drift. Similar to the “many wrongs principle,” it is shown that the cohesion behavior and the velocity alignment behavior are able to reduce the influence of the position measurement drift and the velocity measurement error, respectively. For convenience in practical use, a modified vector field controller with five control terms is put forward. Finally, the effectiveness of the proposed method is validated by numerical simulations and real experiments.

Modified vector field and nonlinear guidance law for low-cost UAV path following

Purpose The purpose of this paper is to present a high accuracy path following method for low-cost fixed-wing UAVs. Design/methodology/approach The original vector field (VF) algorithm is condensed. A spatial integration mechanism is added to the existing VF and nonlinear guidance law, aiming to decrease steady-state cross-track-error and cope with long-term disturbance. Findings Numerical simulations show the proposed method could diminish steady-state cross-track-error effectively. Test flights show the proposed method is applicable on low-cost fixed-wing UAVs. Practical implications The path following accuracy shown in simulations and test flights indicates the proposed method could be deployed in scenarios including inflight rendezvous, formation, trafficway take-off and landing. Originality/value This paper provides an improved high-accuracy path following method for low-cost fixed-wing UAVs.

Sharp decay estimates for the Vlasov–Poisson system with an external magnetic field

In this paper, we establish sharp decay estimates for the Vlasov–Poisson system with an external magnetic field on Rx3×Rv3. Our arguments are based on the modified vector field method developed in Smulevici (2016) for the classical Vlasov–Poisson system in the 3–D case, and hence we extend some results in Smulevici (2016) to the Vlasov–Poisson system with an external magnetic field.

Modified Vector Field Path-Following Control System for an Underactuated Autonomous Surface Ship Model in the Presence of Static Obstacles

A modified path-following control system using the vector field method for an underactuated autonomous surface ship model is proposed in the presence of static obstacles. With this integrated system, autonomous ships are capable of following the predefined path, while avoiding the obstacles automatically. It is different from the methods in most published papers, which usually study path-following and obstacle collision avoidance, separately. This paper considers the coupled path following and collision avoidance task as a whole. Meanwhile, the paper also shows the heading control design method in the presence of static obstacles. To obtain a strong stability property, a nonlinear autopilot is designed based on the manoeuvring tests of the free-running ship model. The equilibrium point of the controller is globally exponentially stable. For the guidance system, a novel vector field method was proposed, and the proof shows the coupled guidance and control system is uniform semi-global exponentially stable (USGES). To prevent the obstacles near the predefined path, the proposed guidance law is augmented by integrating the repelling field of obstacles so that it can control the ship travel toward the predefined path through the obstacles safely. The repelling field function is given considering the obstacle shape and collision risk using the velocity obstacle (VO) algorithm. The simulations and ship model test were performed to validate the integrated system of autonomous ships.

The stability of the Minkowski space for the Einstein–Vlasov system

We prove the global stability of the Minkowski space viewed as the trivial solution of the Einstein-Vlasov system. To estimate the Vlasov field, we use the vector field and modified vector field techniques developed in [FJS15; FJS17]. In particular, the initial support in the velocity variable does not need to be compact. To control the effect of the large velocities, we identify and exploit several structural properties of the Vlasov equation to prove that the worst non-linear terms in the Vlasov equation either enjoy a form of the null condition or can be controlled using the wave coordinate gauge. The basic propagation estimates for the Vlasov field are then obtained using only weak interior decay for the metric components. Since some of the error terms are not time-integrable, several hierarchies in the commuted equations are exploited to close the top order estimates. For the Einstein equations, we use wave coordinates and the main new difficulty arises from the commutation of the energy-momentum tensor, which needs to be rewritten using the modified vector fields.

Cooperative Path Following Control of Fixed-wing Unmanned Aerial Vehicles with Collision Avoidance

In this paper, we propose a novel curved path following scheme for multiple fixed-wing unmanned aerial vehicles (UAVs) that can achieve coordinated curved path following and handle collision avoidance simultaneously. The proposed solution is a hybrid system that combines a path following strategy and a collision avoidance method to ensure collision-free maneuvering of a group of UAVs cooperative flight. A strategy based on a virtual structure and a kinematic model is proposed to derive the cooperative curved path following. Meanwhile, a possible collision will occur when multi-UAVs flight in the form of dense formations, or there are path intersections when the formation is changed. However, all fixed-wing vehicles need to maintain a minimum airspeed and cannot stop before the collision. Therefore, there is a need for a fast and efficient method that can be implemented online, taking into account the physical limitations of the vehicle (minimum speed, turning rate, etc.) to avoid possible collision. We employ a modified vector field histogram (VFH) method to provide real-time collision avoidance, and evasive maneuvers work only if the distance is within the conflict zone. In order to verify the proposed control scheme, dense formation flight tests of small fixed-wing UAVs were implemented.

The stability of the Minkowski space for the Einstein–Vlasov system

This text serves as an introduction to the article [23] written in collaboration with David Fajman and Jérémie Joudioux, and presented at the Laurent Schwarz Seminar in March 2018. In [23], we establish the global stability of the Minkowski space viewed as the trivial solution of the Einstein-Vlasov system. To estimate the Vlasov field, we use the vector field and modified vector field techniques developed in [21, 22]. In particular, the initial support in the velocity variable does not need to be compact. To control the effect of the large velocities, we identify and exploit several structural properties of the Vlasov equation to prove that the worst non-linear terms in the Vlasov equation either enjoy a form of the null condition or can be controlled using the wave coordinate gauge. The basic propagation estimates for the Vlasov field are then obtained using only weak interior decay for the metric components. Since some of the error terms are not time-integrable, several hierarchies in the commuted equations are exploited to close the top order estimates. For the Einstein equations, we use wave coordinates and the main new difficulty arises from the commutation of the energy-momentum tensor, which needs to be rewritten using the modified vector fields.

Autonomous dynamic obstacle avoidance for bacteria-powered microrobots (BPMs) with modified vector field histogram.

In order to broaden the use of microrobots in practical fields, autonomous control algorithms such as obstacle avoidance must be further developed. However, most previous studies of microrobots used manual motion control to navigate past tight spaces and obstacles while very few studies demonstrated the use of autonomous motion. In this paper, we demonstrated a dynamic obstacle avoidance algorithm for bacteria-powered microrobots (BPMs) using electric field in fluidic environments. A BPM consists of an artificial body, which is made of SU-8, and a high dense layer of harnessed bacteria. BPMs can be controlled using externally applied electric fields due to the electrokinetic property of bacteria. For developing dynamic obstacle avoidance for BPMs, a kinematic model of BPMs was utilized to prevent collision and a finite element model was used to characteristic the deformation of an electric field near the obstacle walls. In order to avoid fast moving obstacles, we modified our previously static obstacle avoidance approach using a modified vector field histogram (VFH) method. To validate the advanced algorithm in experiments, magnetically controlled moving obstacles were used to intercept the BPMs as the BPMs move from the initial position to final position. The algorithm was able to successfully guide the BPMs to reach their respective goal positions while avoiding the dynamic obstacles.

Small Data Solutions of the Vlasov-Poisson System and the Vector Field Method

The aim of this article is to demonstrate how the vector field method of Klainerman can be adapted to the study of transport equations. After an illustration of the method for the free transport operator, we apply the vector field method to the Vlasov-Poisson system in dimension 3 or greater. The main results are optimal decay estimates and the propagation of global bounds for commuted fields associated with the conservation laws of the free transport operators, under some smallness assumption. Similar decay estimates had been obtained previously by Hwang, Rendall and Velazquez using the method of characteristics, but the results presented here are the first to contain the global bounds for commuted fields and the optimal spatial decay estimates. In dimension 4 or greater, it suffices to use the standard vector fields commuting with the free transport operator while in dimension 3, the rate of decay is such that these vector fields would generate a logarithmic loss. Instead, we construct modified vector fields where the modification depends on the solution itself. The methods of this paper, being based on commutation vector fields and conservation laws, are applicable in principle to a wide range of systems, including the Einstein-Vlasov and the Vlasov-Nordstrom system.

Exponentially accurate Hamiltonian embeddings of symplectic A-stable Runge–Kutta methods for Hamiltonian semilinear evolution equations

We prove that a class of A-stable symplectic Runge–Kutta time semi-discretizations (including the Gauss–Legendre methods) applied to a class of semilinear Hamiltonian partial differential equations (PDEs) that are well posed on spaces of analytic functions with analytic initial data can be embedded into a modified Hamiltonian flow up to an exponentially small error. Consequently, such time semi-discretizations conserve the modified Hamiltonian up to an exponentially small error. The modified Hamiltonian is O(hp)-close to the original energy, where p is the order of the method and h is the time-step size. Examples of such systems are the semilinear wave equation, and the nonlinear Schrödinger equation with analytic nonlinearity and periodic boundary conditions. Standard Hamiltonian interpolation results do not apply here because of the occurrence of unbounded operators in the construction of the modified vector field. This loss of regularity in the construction can be taken care of by projecting the PDE to a subspace in which the operators occurring in the evolution equation are bounded, and by coupling the number of excited modes and the number of terms in the expansion of the modified vector field with the step size. This way we obtain exponential estimates of the form O(exp(–c/h1/(1+q))) with c > 0 and q ⩾ 0; for the semilinear wave equation, q = 1, and for the nonlinear Schrödinger equation, q = 2. We give an example which shows that analyticity of the initial data is necessary to obtain exponential estimates.

Vector Field Based Formation Control of Multi-Robot System

Abstract Multi-robot system as formation can perform better than a single robot in many situations like collective movement of objects, search and rescue operations etc. In this paper a vector field framework is used for achieving path planning for a single robot, and a nonlinear controller is designed for tracking this path. The tracking controller is later extended for achieving formation control of multi-robot system. Asymptotic stability of the controller for any reference path is ensured by using Lyapunov theory. The robots may encounter obstacles in their path during tracking or may collide with each other during the switching of formation patterns. Obstacle avoidance and collision avoidance have been achieved by modified vector field. If the terrain is constrained then it needs to switch from one formation to another according to the terrain requirements. Thus the capability of switching of formation or reconfiguration is an important aspect in the control of multiple mobile robots. If one of the member robots is lost, then the other robots continue with the remaining formation structure and thus ensures robustness against loss of team member. From the extensive simulations using Matlab environment it is seen that any formation of mobile robots is achieved in addition to reconfiguration and obstacle avoidance. The formation control algorithm has been experimentally demonstrated by a series of experiments conducted using LEGO NXT Mindstorm robot kit.

Standoff Target Tracking using a Vector Field for Multiple Unmanned Aircrafts

This paper presents strategies for standoff target tracking by a team of unmanned aircrafts using vector field. Many methods to the vector field approach were investigated in other papers, but a modified vector field is introduced to obtain new interesting characteristics in this paper. The modified vector field satisfies more constraints. We introduce two guidance modes to track a target: one is a capturing mode and the other one is a loitering mode. In the capturing mode, aircrafts can arrive at desired positions and time, i.e., unmanned aircrafts can capture a target simultaneously from all sides. After a target is captured, the guidance mode is changed to the loitering mode. Then the relative spacing among aircrafts is controlled by a standoff distance command and a speed command. Hence, they track a target with a desired formation.

Modified Vector Field Histogram with a Neural Network Learning Model for Mobile Robot Path Planning and Obstacle Avoidance

In this work, a Modified Vector Field Histogram (MVFH) has been developed to improve path planning and obstacle avoidance for a wheeled driven mobile robot. It permits the detection of unknown obstacle to avoid collisions by simultaneously a steering the mobile robot toward the target; a regular grid map representation for a work space environment is caurried out. A Neural Network (NN) model is used to learn many critical situations of environment during robot navigation among obstacles using MVFH. Also, digital filter has been utilized for improving the robustness of obstacle avoidance trajectory of mobile robot. The proposed MVFH-NN has been implemented and tested by using MobotSim program simulation and MATLAB . The developed algorithm showed good navigation properties and can be used in complex real world (maze-like environment), it also shows good ability to overcome limitations of the traditional VFH algorithm (like wide candidate valley, narrow hallway and target distance limitation).

Energy-Preserving Integrators and the Structure of B-series

B-series are a powerful tool in the analysis of Runge–Kutta numerical integrators and some of their generalizations (“B-series methods”). A general goal is to understand what structure-preservation can be achieved with B-series and to design practical numerical methods that preserve such structures. B-series of Hamiltonian vector fields have a rich algebraic structure that arises naturally in the study of symplectic or energy-preserving B-series methods and is developed in detail here. We study the linear subspaces of energy-preserving and Hamiltonian modified vector fields which admit a B-series, their finite-dimensional truncations, and their annihilators. We characterize the manifolds of B-series that are conjugate to Hamiltonian and conjugate to energy-preserving and describe the relationships of all these spaces.

High-order conservative discretizations for some cases of the rigid body motion

Modified vector fields can be used to construct high-order structure–preserving numerical integrators for ordinary differential equations. In the present Letter we consider high-order integrators based on the implicit midpoint rule, which conserve quadratic first integrals. It is shown that these integrators are particularly suitable for the rigid body motion with an additional quadratic first integral. In this case high-order integrators preserve all four first integrals of motion. The approach is illustrated on the Lagrange top (a rotationally symmetric rigid body with a fixed point on the symmetry axis). The equations of motion are considered in the space fixed frame because in this frame Lagrange top admits a neat description. The Lagrange top motion includes the spherical pendulum and the planar pendulum, which swings in a vertical plane, as particular cases.

Conservative discretizations of the Kepler motion

Modified vector fields are used to construct high-order conservative discretizations of the three-dimensional Kepler motion. The numerical integrators preserve the Hamiltonian function, the angular momentum and Runge–Lenz vector. In particular, the exact integrator of the Kepler motion is found. The proposed numerical schemes permit explicit implementation.

The Discrete Moser?Veselov Algorithm for the Free Rigid Body, Revisited

In this paper we revisit the Moser–Veselov description of the free rigid body in body coordinates, which, in the 3 × 3 case, can be implemented as an explicit, second-order, integrable approximation of the continuous solution. By backward error analysis, we study the modified vector field which is integrated exactly by the discrete algorithm. We deduce that the discrete Moser–Veselov (DMV) is well approximated to higher order by time reparametrizations of the continuous equations (modified vector field). We use the modified vector field to scale the initial data of the DMV to improve the order of the approximation and show the equivalence of the DMV and the RATTLE algorithm. Numerical integration with these preprocessed initial data is several orders of magnitude more accurate than the original DMV and RATTLE approach.

A Note on the Construction of Crouch-Grossman Methods

Numerical integration methods based on rigid frames were introduced by Crouch and Grossman. The order theory of these methods were later analyzed by Owren and Marthinsen. The resulting order conditions are difficult to solve due to nonlinear relations on the weights of the methods. In this paper we propose a variant of the Crouch-Grossman method that uses modified vector fields so that the order conditions of this new method coincide with the classical order conditions for Runge-Kutta methods.

Backward Error Analysis for Numerical Integrators

Backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods. This is true in particular for the integration of Hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time. Such results are typically based on two aspects of backward error analysis: (i) It can be shown that the modified vector fields have some qualitative properties which they share with the given problem and (ii) an estimate is given for the difference between the best interpolating vector field and the numerical method. These aspects have been investigated recently, for example, by Benettin and Giorgilli in [ J. Statist. Phys., 74 (1994), pp. 1117--1143], by Hairer in [Ann. Numer. Math., 1 (1994), pp. 107--132], and by Hairer and Lubich in [ Numer. Math., 76 (1997), pp. 441--462]. In this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs. Our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit Taylor series expansion of the numerical method and the corresponding flow maps as in the above-cited works. As an application we discuss the long time integration of chaotic Hamiltonian systems and the approximation of time averages along numerically computed trajectories.