Guiding Vector Fields for Robot Motion Control

Using a designed vector field to guide robots to follow a given geometric desired path has found a range of practical applications, such as underwater pipeline inspection, warehouse navigation and highway traffic monitoring. It is thus in great need to build a rigorous theory to guide practical implementations with formal guarantees. It is even so when multiple robots are required to follow predefined desired paths or maneuver on surfaces and coordinate their motions to efficiently accomplish repetitive and laborious tasks. In this thesis, we propose and study a specific class of vector field, called guiding vector fields, on the Euclidean space and a general Riemannian manifold, for singlerobot and multi-robot path following and motion coordination. A guiding vector field is generally composed of two terms: a convergence term which enables the integral curves of the vector field to converge to the desired path, and a propagation term which is tangent to the desired path such that propagation along the desired path is ensured. The guiding vector field is completely determined (up to positive coefficients) by a number of twice continuously differentiable real-value functions (called level functions). The intersection of the zero-level sets of these level functions is the desired path to be followed. Since the guiding vector field is not the gradient of any potential function, and also due to the existence of singular points where the vector field vanishes, the theoretical analysis becomes challenging. Therefore, in Part I of the thesis, we derive extensive theoretical results. And then in Part II, we elaborate on how to utilize guiding vector fields with variations in practical applications

Path following control in 3D using a vector field

Using a designed vector field to control a mobile robot to follow a given desired path has found a range of practical applications, and it is in great need to further build a rigorous theory to guide its implementation. In this paper, we study the properties of a general 3D vector field for robotic path following. We stipulate and investigate assumptions that turn out to be crucial for this method, although they are rarely explicitly stated in the existing related works. We derive conditions under which the local path-following error vanishes exponentially in a sufficiently small neighborhood of the desired path, which is key to show the local input-to-state stability (local ISS) property of the path-following error dynamics. The local ISS property then justifies the control algorithm design for a fixed-wing aircraft model. Our approach is effective for any sufficiently smooth desired path in 3D, bounded or unbounded; the results are particularly relevant since unbounded desired paths have not been sufficiently discussed in the literature. Simulations are conducted to verify the theoretical results

Co-Jumps, Co-Jump Tests, and Volatility Forecasting: Monte Carlo and Empirical Evidence

This study classifies jumps into idiosyncratic jumps and co-jumps to quantitatively identify systematic risk and idiosyncratic risk by utilizing high-frequency data. We found that systematic risk occurs more frequently and has larger magnitudes than the idiosyncratic risk in an individual asset, which indicates that volatilities from one sector are largely derived from the contagious effect of other sectors. We further investigated the importance of idiosyncratic jumps and co-jumps to predict the sector-level S&P500 exchange-traded fund (ETF) volatility. It was found that the predictive content of co-jumps is higher than that of idiosyncratic jumps, suggesting that systematic risk is more informative than idiosyncratic risk in volatility forecasting. Additionally, we carried out Monte Carlo experiments designed to examine the relative performances of the four co-jump tests. The findings indicate that the BLT test and the co-exceedance rule of the LM test outperform other tests, while the co-exceedance rule of the LM test has larger power and a smaller empirical size than that of the BLT test

On Wilson’s theorem about domains of attraction and tubular neighborhoods

In this paper, we show that the domain of attraction of a compact asymptotically stable submanifold in a finite-dimensional smooth manifold of an autonomous system is homeomorphic to the submanifold’s tubular neighborhood. The compactness of the submanifold is crucial, without which this result is false; two counterexamples are provided to demonstrate this

Limit cycles analysis and control of evolutionary game dynamics with environmental feedback

Recently, an evolutionary game dynamics model taking into account the environmental feedback has been proposed to describe the co-evolution of strategic actions of a population of individuals and the state of the surrounding environment; correspondingly a range of interesting dynamic behaviors have been reported. In this paper, we provide new theoretical insight into such behaviors and discuss control options. Instead of the standard replicator dynamics, we use a more realistic and comprehensive model of replicator–mutator dynamics, to describe the strategic evolution of the population. After integrating the environment feedback, we study the effect of mutations on the resulting closed-loop system dynamics. We prove the conditions for two types of bifurcations, Hopf bifurcation and Heteroclinic bifurcation, both of which result in stable limit cycles. These limit cycles have not been identified in existing works, and we further prove that such limit cycles are in fact persistent in a large parameter space and are almost globally stable. In the end, an intuitive control policy based on incentives is applied, and the effectiveness of this control policy is examined by analysis and simulations

Integrated Path Following and Collision Avoidance Using a Composite Vector Field

Path following and collision avoidance are two important functionalities for mobile robots, but there are only a few approaches dealing with both. In this paper, we propose an integrated path following and collision avoidance method using a composite vector field. The vector field for path following is integrated with that for collision avoidance via bump functions, which reduce significantly the overlapping effect. Our method is general and flexible since the desired path and the contours of the obstacles, which are described by the zero-level sets of sufficiently smooth functions, are only required to be homeomorphic to a circle or the real line, and the derivation of the vector field does not involve specific geometric constraints. In addition, the collision avoidance behaviour is reactive; thus, real-time performance is possible. We show analytically the collision avoidance and path following capabilities, and use numerical simulations to illustrate the effectiveness of the theory

Guiding Vector Fields for Following Occluded Path

Accurately following a geometric desired path in a two-dimensional (2-D) space is a fundamental task for many engineering systems, in particular mobile robots. When the desired path is occluded by obstacles, it is necessary and crucial to temporarily deviate from the path for obstacle/collision avoidance. In this article, we develop a composite guiding vector field via the use of smooth bump functions and provide theoretical guarantees that the integral curves of the vector field can follow an arbitrary sufficiently smooth desired path and avoid collision with obstacles of arbitrary shapes. These two behaviors are reactive since path (re)planning and global map construction are not involved. To deal with the common deadlock problem, we introduce a switching vector field, and the Zeno behavior is excluded. Simulations are conducted to support the theoretical results

A Novel Vector-Field-Based Motion Planning Algorithm for 3D Nonholonomic Robots

This paper focuses on the motion planning for mobile robots in 3D, which are modelled by 6-DOF rigid body systems with nonholonomic kinematics constraints. We not only specify the target position, but also bring in the requirement of the heading direction at the terminal time, which gives rise to a new and more challenging 3D motion planning problem. The proposed planning algorithm involves a novel velocity vector field (VF) over the workspace, and by following the VF, the robot can be navigated to the destination with the specified heading direction. In order to circumvent potential collisions with obstacles and other robots, a composite VF is designed by composing the navigation VF and an additional VF tangential to the boundary of the dangerous area. Moreover, we propose a priority-based algorithm to deal with the motion coupling issue among multiple robots. Finally, numerical simulations are conducted to verify the theoretical results