Normal forms for underactuated mechanical systems with symmetry

We introduce cascade normal forms for underactuated mechanical systems that are convenient for control design. These normal forms include three classes of cascade systems, namely, nonlinear systems in strict feedback form, feedforward form, and nontriangular quadratic form (to be defined). In each case, the transformation to cascade systems is provided in closed-form. We apply our results to the Acrobot, the rotating pendulum, and the cart-pole system

Ultrafast Consensus in Small-World Networks

In this paper, we demonstrate a phase transition phenomenon in algebraic connectivity of small-world networks. Algebraic connectivity of a graph is the second smallest eigenvalue of its Laplacian matrix and a measure of speed of solving consensus problems in networks. We demonstrate that it is possible to dramatically increase the algebraic connectivity of a regular complex network by 1000 times or more without adding new links or nodes to the network. This implies that a consensus problem can be solved incredibly fast on certain small-world networks giving rise to a network design algorithm for ultra fast information networks. Our study relies on a procedure called "random rewiring" due to Watts & Strogatz (Nature, 1998). Extensive numerical results are provided to support our claims and conjectures. We prove that the mean of the bulk Laplacian spectrum of a complex network remains invariant under random rewiring. The same property only asymptotically holds for scale-free networks. A relationship between increasing the algebraic connectivity of complex networks and robustness to link and node failures is also shown. This is an alternative approach to the use of percolation theory for analysis of network robustness. We also show some connections between our conjectures and certain open problems in the theory of random matrices

Exponential ε-tracking and ε-stabilization of second-order nonholonomic SE(2) vehicles using dynamic state feedback

In this paper, we address the problem of ε-tracking and ε-stabilization for a class of SE(2) vehicles with second-order nonholonomic constraints. We introduce a class of transformations called near-identity diffeomorphism that allow dynamic partial feedback linearization of the translational dynamics of this class of SE(2) vehicles. This allows us to achieve global exponential ε-stabilization and ε-tracking (in position) for the aforementioned classes of autonomous vehicles using a coordinate-independent dynamic state feedback. This feedback is only discontinuous w.r.t. the augmented state. We apply our results to ε-stabilization and ε-tracking for an underactuated surface vessel

Global configuration stabilization for the VTOL aircraft with strong input coupling

Trajectory tracking and configuration stabilization for the vertical takeoff and landing (VTOL) aircraft has been so far considered in the literature only in the presence of a slight (or zero) input coupling (i.e., for a small ε). In this paper, our main contribution is to address global configuration stabilization for the VTOL aircraft with a strong input coupling using a smooth static state feedback. In addition, the differentially flat outputs for the VTOL aircraft are automatically obtained as a by-product of applying a decoupling change of coordinates

A Unified Analytical Look at Reynolds Flocking Rules

In this paper, we present a unified theoretical view of the so-called ``Flocking Rules of Reynolds'' introduced in 1987. No equations describing the rules or mathematical models of the mobile agents known as ``boids'' were presented in the original work by Reynolds. We show how to model a group of autonomous mobile agents by dynamic nets and achieve flocking by dissipation of the structural energy of the multi-agent system. As a by-product, we obtain a single protocol called the (alpha,alpha) protocol that encompasses all three flocking rules of Reynolds. We provide geometric interpretations of the advanced forms of some of these flocking rules. Simulation results are provided that demonstrate flocking of 100 agents towards a sink

Flocking with Obstacle Avoidance

In this paper, we provide a dynamic graph theoretical framework for flocking in presence of multiple obstacles. In particular, we give formal definitions of nets and flocks as spatially induced graphs. We provide models of nets and flocks and discuss the realization/embedding issues related to structural nets and flocks. This allows task representation and execution for a network of agents called alpha-agents. We also consider flocking in the presence of multiple obstacles. This task is achieved by introducing two other types of agents called beta-agents and gamma-agents. This framework enables us to address split/rejoin and squeezing maneuvers for nets/flocks of dynamic agents that communicate with each other. The problems arising from switching topology of these networks of mobile agents make the analysis and design of the decision-making protocols for such networks rather challenging. We provide simulation results that demonstrate the effectiveness of our theoretical and computational tools

Agreement Problems in Networks with Directed Graphs and Switching Topology

In this paper, we provide tools for convergence and performance analysis of an agreement protocol for a network of integrator agents with directed information flow. Moreover, we analyze algorithmic robustness of this consensus protocol for the case of a network with mobile nodes and switching topology. We establish a connection between the Fiedler eigenvalue of the graph Laplacian and the performance of this agreement protocol. We demostrate that a class of directed graphs, called balanced graphs, have a crucial role in solving average-consensus problems. Based on the properties of balanced graphs, a group disagreement function (i.e. Lyapunov function) is proposed for convergence analysis of this agreement protocol for networks with directed graphs. This group disagreement function is later used for convergence analysis for the agreement problem in networks with switching topology. We provide simulation results that are consistent with our theoretical results and demonstrate the effectiveness of the proposed analytical tools

Consensus problems in networks of agents with switching topology and time-delays

In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results

Safe Sequential Path Planning Under Disturbances and Imperfect Information

Multi-UAV systems are safety-critical, and guarantees must be made to ensure no unsafe configurations occur. Hamilton-Jacobi (HJ) reachability is ideal for analyzing such safety-critical systems; however, its direct application is limited to small-scale systems of no more than two vehicles due to an exponentially-scaling computational complexity. Previously, the sequential path planning (SPP) method, which assigns strict priorities to vehicles, was proposed; SPP allows multi-vehicle path planning to be done with a linearly-scaling computational complexity. However, the previous formulation assumed that there are no disturbances, and that every vehicle has perfect knowledge of higher-priority vehicles' positions. In this paper, we make SPP more practical by providing three different methods to account for disturbances in dynamics and imperfect knowledge of higher-priority vehicles' states. Each method has different assumptions about information sharing. We demonstrate our proposed methods in simulations.Comment: American Control Conference, 201

Distributed LQR-based Suboptimal Control for Coupled Linear Systems

A well-established distributed LQR method for decoupled systems is extended to the dynamically coupled case where the open-loop systems are dynamically dependent. First, a fully centralized controller is designed which is subsequently substituted by a distributed state-feedback gain with sparse structure. The control scheme is obtained by optimizing an LQR performance index with a tuning parameter utilized to emphasize/de-emphasize relative state difference between interconnected systems. Overall stability is guaranteed via a simple test applied to a convex combination of Hurwitz matrices, the validity of which leads to stable global operation for a class of interconnection schemes. It is also shown that the suboptimality of the method can be assessed by measuring a certain distance between two positive definite matrices which can be obtained by solving two Lyapunov equations