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

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

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

Vector Field Guided Path Following Control:Singularity Elimination and Global Convergence

Vector field guided path following (VF-PF) algorithms are fundamental in robot navigation tasks, but may not deliver the desirable performance when robots encounter singular points where the vector field becomes zero. The existence of singular points prevents the global convergence of the vector field's integral curves to the desired path. Moreover, VF-PF algorithms, as well as most of the existing path following algorithms, fail to enable following a self-intersected desired path. In this paper, we show that such failures are fundamentally related to the mathematical topology of the path, and that by "stretching" the desired path along a virtual dimension, one can remove the topological obstruction. Consequently, this paper proposes a new guiding vector field defined in a higher-dimensional space, in which self-intersected desired paths become free of self-intersections; more importantly, the new guiding vector field does not have any singular points, enabling the integral curves to converge globally to the "stretched" path. We further introduce the extended dynamics to retain this appealing global convergence property for the desired path in the original lower-dimensional space. Both simulations and experiments are conducted to verify the theory.Comment: Accepted by 2020 IEEE 59th Conference on Decision and Control (CDC). This is the full versio

Evolutionary dynamics under periodic switching of update rules on regular networks

Microscopic strategy update rules play an important role in the evolutionary dynamics of cooperation among interacting agents on complex networks. Many previous related works only consider one \emph{fixed} rule, while in the real world, individuals may switch, sometimes periodically, between rules. It is of particular theoretical interest to investigate under what conditions the periodic switching of strategy update rules facilitates the emergence of cooperation. To answer this question, we study the evolutionary prisoner's dilemma game on regular networks where agents can periodically switch their strategy update rules. We accordingly develop a theoretical framework of this periodically switched system, where the replicator equation corresponding to each specific microscopic update rule is used for describing the subsystem, and all the subsystems are activated in sequence. By utilizing switched system theory, we identify the theoretical condition for the emergence of cooperative behavior. Under this condition, we have proved that the periodically switched system with different switching rules can converge to the full cooperation state. Finally, we consider an example where two strategy update rules, that is, the imitation and pairwise-comparison updating, are periodically switched, and find that our numerical calculations validate our theoretical results

Limit Cycles in Replicator-Mutator Dynamics with Game-Environment Feedback

This paper considers the coevolutionary game and environment dynamics under mutations of strategies. Individuals’ game play affects the dynamics of changing environments while the environment in turn affects the decision-making dynamics of individuals through modulating game payoffs. For some such closed-loop systems, we prove that limit cycles will never appear; however, in sharp contrast, after allowing mutations of strategies in these systems, the resulting replicator-mutator dynamics under environmental feedback may well exhibit Hopf bifurcation and limit cycles. We prove conditions for the Hopf bifurcation and thus the existence of stable limit cycles, and also illustrate these results using simulations. For the coevolutionary game and environment system, these stable limit cycles correspond to sustained oscillations of population’s decisions and richness of the environmental resource

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