Firing rate models are dynamical systems widely used in applied and theoretical neuroscience to describe local cortical dynamics in neuronal populations. By providing a macroscopic perspective of neuronal activity, these models are essential for investigating oscillatory phenomena, chaotic behavior, and associative memory processes. Despite their widespread use, the application of firing rate models to associative memory networks has received limited mathematical exploration, and most existing studies are focused on specific models. Conversely, well-established associative memory designs, such as Hopfield networks, lack key biologically relevant features intrinsic to firing rate models, including positivity and interpretable synaptic matrices reflecting the action of long-term potentiation and long-term depression. To address this gap, we propose a general framework that ensures the emergence of rescaled memory patterns as stable equilibria in the firing rate dynamics. Furthermore, we analyze the conditions under which the memories are locally and globally asymptotically stable, providing insights into constructing biologically plausible and robust systems for associative memory retrieval.

To enable the computation of effective randomized patrol routes for single- or multi-robot teams, we present RoSSO, a Python package designed for solving Markov chain optimization problems. We exploit machine-learning techniques such as reverse-mode automatic differentiation and constraint parametrization to achieve superior efficiency compared to general-purpose nonlinear programming solvers. Additionally, we supplement a game-theoretic stochastic surveillance formulation in the literature with a novel greedy algorithm and multi-robot extension. We close with numerical results for a police district in downtown San Francisco that demonstrate RoSSO’s capabilities on our new formulations and the prior work.

Global stability and robustness guarantees in learned dynamical systems are essential to ensure well-behavedness of the systems in the face of uncertainty. We present Extended Linearized Contracting Dynamics (ELCD), the first neural network-based dynamical system with global contractivity guarantees in arbitrary metrics. The key feature of ELCD is a parametrization of the extended linearization of the nonlinear vector field. In its most basic form, ELCD is guaranteed to be (i) globally exponentially stable, (ii) equilibrium contracting, and (iii) globally contracting with respect to some metric. To allow for contraction with respect to more general metrics in the data space, we train diffeomorphisms between the data space and a latent space and enforce contractivity in the latent space, which ensures global contractivity in the data space. We demonstrate the performance of ELCD on the high dimensional LASA, multi-link pendulum, and Rosenbrock datasets.

While monotone operator theory is often studied on Hilbert spaces, many interesting problems in machine learning and optimization arise naturally in finite-dimensional vector spaces endowed with non-Euclidean norms, such as diagonally-weighted $\ell_{1}$ or $\ell_{\infty}$ norms. This paper provides a natural generalization of monotone operator theory to finite-dimensional non-Euclidean spaces. The key tools are weak pairings and logarithmic norms. We show that the resolvent and reflected resolvent operators of non-Euclidean monotone mappings exhibit similar properties to their counterparts in Hilbert spaces. Furthermore, classical iterative methods and splitting methods for finding zeros of monotone operators are shown to converge in the non-Euclidean case. We apply our theory to equilibrium computation and Lipschitz constant estimation of recurrent neural networks, obtaining novel iterations and tighter upper bounds via forward-backward splitting.

We propose and analyze a continuous-time firing-rate neural network, the positive firing-rate competitive network (PFCN), to tackle sparse reconstruction problems with non-negativity constraints. These problems, which involve approximating a given input stimulus from a dictionary using a set of sparse (active) neurons, play a key role in a wide range of domains, including, for example, neuroscience, signal processing, and machine learning. First, by leveraging the theory of proximal operators, we relate the equilibria of a family of continuous-time firing-rate neural networks to the optimal solutions of sparse reconstruction problems. Then we prove that the PFCN is a positive system and give rigorous conditions for the convergence to the equilibrium. Specifically, we show that the convergence depends only on a property of the dictionary and is linear-exponential in the sense that initially, the convergence rate is at worst linear and then, after a transient, becomes exponential. We also prove a number of technical results to assess the contractivity properties of the neural dynamics of interest. Our analysis leverages contraction theory to characterize the behavior of a family of firing-rate competitive networks for sparse reconstruction with and without non-negativity constraints. Finally, we validate the effectiveness of our approach via a numerical example.

Much recent interest has focused on the design of optimization algorithms from the discretization of an associated optimization flow, i.e., a system of differential equations (ODEs) whose trajectories solve an associated optimization problem. Such a design approach poses an important problem: how to find a principled methodology to design and discretize appropriate ODEs. This paper aims to provide a solution to this problem through the use of contraction theory. We first introduce general mathematical results that explain how contraction theory guarantees the stability of the implicit and explicit Euler integration methods. Then, we propose a novel system of ODEs, namely the Accelerated-Contracting-Nesterov flow, and use contraction theory to establish it is an optimization flow with exponential convergence rate, from which the linear convergence rate of its associated optimization algorithm is immediately established. Remarkably, a simple explicit Euler discretization of this flow corresponds to the Nesterov acceleration method. Finally, we present how our approach leads to performance guarantees in the design of optimization algorithms for time-varying optimization problems.

Flow networks are ubiquitous in natural and engineered systems, and in order to understand and manage these networks, one must quantify the flow of commodities across their edges. This paper considers the estimation problem of predicting unlabeled edge flows from nodal supply and demand. We propose an implicit neural network layer that incorporates two fundamental physical laws: conservation of mass, and the existence of a constitutive relationship between edge flows and nodal states (e.g., Ohm's law). Computing the edge flows from these two laws is a nonlinear inverse problem, which our layer solves efficiently with a specialized contraction mapping. Using implicit differentiation to compute the solution's gradients, our model is able to learn the constitutive relationship within a semi-supervised framework. We demonstrate that our approach can accurately predict edge flows in several experiments on AC power networks and water distribution systems.

Many tasks regarding the monitoring, management, and design of communication networks rely on knowledge of the routing topology. However, the standard approach to topology mapping—namely, active probing with traceroutes—relies on cooperation from increasingly non-cooperative routers, leading to missing information. Network tomography, which uses end-to-end measurements of additive link metrics (like delays or log packet loss rates) across monitor paths, is a possible remedy. Network tomography does not require that routers cooperate with traceroute probes, and it has already been used to infer the structure of multicast trees. This paper goes a step further. We provide a tomographic method to infer the underlying routing topology of an arbitrary set of monitor paths using the joint distribution of end-to-end measurements, without making any assumptions on routing behavior. Our approach, called the Möbius Inference Algorithm (MIA), uses cumulants of this distribution to quantify high-order interactions among monitor paths, and it applies Möbius inversion to “disentangle” these interactions. In addition to MIA, we provide a more practical variant called Sparse Möbius Inference, which uses various sparsity heuristics to reduce the number and order of cumulants required to be estimated. We show the viability of our approach using synthetic case studies based on real-world ISP topologies.

Implicit neural networks, a.k.a., deep equilibrium networks, are a class of implicit-depth learning models where function evaluation is performed by solving a fixed point equation. They generalize classic feedforward models and are equivalent to infinite-depth weight-tied feedforward networks. While implicit models show improved accuracy and significant reduction in memory consumption, they can suffer from ill-posedness and convergence instability.This paper provides a new framework, which we call Non-Euclidean Monotone Operator Network (NEMON), to design well-posed and robust implicit neural networks based upon contraction theory for the non-Euclidean norm $\ell_\infty$. Our framework includes (i) a novel condition for well-posedness based on one-sided Lipschitz constants, (ii) an average iteration for computing fixed-points, and (iii) explicit estimates on input-output Lipschitz constants. Additionally, we design a training problem with the well-posedness condition and the average iteration as constraints and, to achieve robust models, with the input-output Lipschitz constant as a regularizer. Our $\ell_\infty$ well-posedness condition leads to a larger polytopic training search space than existing conditions and our average iteration enjoys accelerated convergence. Finally, we evaluate our framework in image classification through the MNIST and the CIFAR-10 datasets. Our numerical results demonstrate improved accuracy and robustness of the implicit models with smaller input-output Lipschitz bounds. Code is available at https://github.com/davydovalexander/Non-Euclidean_Mon_Op_Net.

This article analyzes the meeting time between a pair of pursuer and evader performing random walks on digraphs. The existing bounds on the meeting time usually work only for certain classes of walks and cannot be used to formulate optimization problems and design robotic strategies. First, by analyzing multiple random walks on a common graph as a single random walk on the Kronecker product graph, we provide the first closed-form expression for the expected meeting time in terms of the transition matrices of the moving agents. This novel expression leads to necessary and sufficient conditions for the meeting time to be finite and to insightful graph-theoretic interpretations. Second, based on the closed-form expression, we set up and study the minimization problem for the expected capture time for a pursuer/evader pair. We report theoretical and numerical results on basic case studies to show the effectiveness of the design.

Algebraic graph theory is a cornerstone in the study of electrical networks ranging from miniature integrated circuits to continental-scale power systems. Conversely, many fundamental results of algebraic graph theory were laid out by early electrical circuit analysts. In this paper, we survey some fundamental and historic as well as recent results on how algebraic graph theory informs electrical network analysis, dynamics, and design. In particular, we review the algebraic and spectral properties of graph adjacency, Laplacian, incidence, and resistance matrices and how they relate to the analysis, network reduction, and dynamics of certain classes of electrical networks. We study these relations for models of increasing complexity ranging from static resistive direct current (dc) circuits, over dynamic resistor..inductor..capacitor (RLC) circuits, to nonlinear alternating current (ac) power flow. We conclude this paper by presenting a set of fundamental open questions at the intersection of algebraic graph theory and electrical networks.

We study the problem of quickest detection of anomalies in an environment under extreme uncertainties in sensor measurements. The robotic roadmap corresponding to the environment can be represented as a graph with an arbitrary topology. We analyze the Ensemble CUSUM Algorithm for this surveillance problem. We quantify the delay in detection of anomalies using the Ensemble CUSUM Algorithm and also frame an optimization problem to minimize this detection delay. We then provide an upper bound on the optimal detection delay and frame a convex optimization problem to minimize this upper bound. We also propose an efficient policy that achieves this upper bound and can be computed by solving a semidefinite program. We illustrate the efficacy of the Ensemble CUSUM Algorithm using numerical simulations. We observe that the efficient policy outperforms policies based on other well-known Markov chains.

We consider the problem of dynamic vehicle routing under exact-time constraints on servicing demands. Demands are sequentially generated in an environment, and every demand needs to be serviced exactly after a fixed finite interval of time after it is generated. We design routing policies for a service vehicle to maximize the fraction of demands serviced at steady state. The main contributions are as follows. First, we demonstrate that this problem is described by an appropriate directed acyclic graph structure which leads to a computationally efficient routing algorithm based on a longest-path computation. Second, under the assumption of the demands being generated uniformly randomly in the environment and via a Poisson process in time, we provide two analytic lower bounds on the service fraction of the longest path policy. The first bound is relative to an optimal noncausal version of the policy, i.e., a policy based on knowledge of all future demand requests. The second bound is an explicit function of the vehicle dynamics and demand generation rate and, therefore, useful as a design tool. Finally, we present numerical results to support the analytic bounds.

This paper studies prototypical strategies to sequentially aggregate independent decisions. We consider a collection of agents, each performing binary hypothesis testing and each obtaining a decision over time. We assume the agents are identical and receive independent information. Individual decisions are sequentially aggregated via a threshold-based rule. In other words, a collective decision is taken as soon as a specified number of agents report a concordant decision (simultaneous discordant decisions and no-decision outcomes are also handled). We obtain the following results. First, we characterize the probabilities of correct and wrong decisions as a function of time, group size, and decision threshold. The computational requirements of our approach are linear in the group size. Second, we consider the so-called fastest and majority rules, corresponding to specific decision thresholds. For these rules, we provide a comprehensive scalability analysis of both accuracy and decision time. In the limit of large group sizes, we show that the decision time for the fastest rule converges to the earliest possible individual time, and that the decision accuracy for the majority rule shows an exponential improvement over the individual accuracy. Additionally, via a theoretical and numerical analysis, we characterize various speed/accuracy tradeoffs. Finally, we relate our results to some recent observations reported in the cognitive information processing (CIP) literature.

We propose distributed algorithms to automatically deploy a team of mobile robots to partition and provide coverage of a nonconvex environment. To handle arbitrary nonconvex environments, we represent them as graphs. Our partitioning and coverage algorithm requires only short-range, unreliable pairwise “gossip” communication. The algorithm has two components: 1) a motion protocol to ensure that neighboring robots communicate at least sporadically and 2) a pairwise partitioning rule to update territory ownership when two robots communicate. By studying an appropriate dynamical system on the space of partitions of the graph vertices, we prove that territory ownership converges to a pairwise-optimal partition in finite time. This new equilibrium set represents improved performance over common Lloyd-type algorithms. Additionally, our algorithm is an "anytime algorithm'' that also scales well for large teams and can be run by on-board computers with limited resources. Finally, we report on large-scale simulations in complex environments and hardware experiments using the Player/Stage robot control system.

This paper focuses on the problem of patrolling an environment with a team of autonomous agents. Given a set of strategically important locations (viewpoints) with different priorities, our patrolling strategy consists of 1) constructing a tour through the viewpoints, and 2) driving the robots along the tour in a coordinated way. As performance criteria, we consider the weighted refresh time, i.e., the longest time interval between any two visits of a viewpoint, weighted by the viewpoint's priority. We consider the design of both optimal trajectories and distributed control laws for the robots to converge to optimal trajectories. First, we propose a patrolling strategy and we characterize its performance as a function of the environment and the viewpoints priorities. Second, we restrict our attention to the problem of patrolling a nonintersecting tour, and we describe a team trajectory with minimum weighted refresh time. Third, for the tour patrolling problem and for two distinct communication scenarios, namely the Passing and the Neighbor-Broadcast communication models, we develop distributed algorithms to steer the robots toward a minimum weighted refresh time team trajectory. Finally, we show the effectiveness and robustness of our control algorithms via simulations and experiments.

The subject of this paper is the patrolling of an environment with the aid of a team of autonomous agents. We consider both the design of open-loop trajectories with optimal properties and of distributed control laws converging to optimal trajectories. As performance criteria, the refresh time and the latency are considered, i.e., respectively, time gap between any two visits of the same region and the time necessary to inform every agent about an event occurred in the environment. We associate a graph with the environment, and we study separately the case of a chain, tree, and cyclic graph. For the case of chain graph, we first describe a minimum refresh time and latency team trajectory and propose a polynomial time algorithm for its computation. Then, we describe a distributed procedure that steers the robots toward an optimal trajectory. For the case of tree graph, a polynomial time algorithm is developed for the minimum refresh time problem, under the technical assumption of a constant number of robots involved in the patrolling task. Finally, we show that the design of a minimum refresh time trajectory for a cyclic graph is NP-hard, and we develop a constant factor approximation algorithm.

This paper focuses on Euler angles and on the decomposition of rotations. We consider arbitrary rotation axes that are not necessarily mutually orthogonal; we characterize the set of rotation matrices that admit Euler angles about arbitrary rotation axes; and we provide a single set of Euler angle formulas that applies to any selection of rotation axes. The results are presented and derived in a coordinate-free setting, where no reference frames are required, and no components of any array or matrix are manipulated.

Recent years have witnessed great advancements in the science and technology of autonomy, robotics, and networking. This paper surveys recent concepts and algorithms for dynamic vehicle routing (DVR), that is, for the automatic planning of optimal multivehicle routes to perform tasks that are generated over time by an exogenous process. We consider a rich variety of scenarios relevant for robotic applications. We begin by reviewing the basic DVR problem: demands for service arrive at random locations at random times and a vehicle travels to provide on-site service while minimizing the expected wait time of the demands. Next, we treat different multivehicle scenarios based on different models for demands (e.g., demands with different priority levels and impatient demands), vehicles (e.g., motion constraints, communication, and sensing capabilities), and tasks. The performance criterion used in these scenarios is either the expected wait time of the demands or the fraction of demands serviced successfully. In each specific DVR scenario, we adopt a rigorous technical approach that relies upon methods from queueing theory, combinatorial optimization, and stochastic geometry. First, we establish fundamental limits on the achievable performance, including limits on stability and quality of service. Second, we design algorithms, and provide provable guarantees on their performance with respect to the fundamental limits.

This paper addresses a novel visibility-based pursuit-evasion problem in which a team of searchers with limited range sensors must coordinate to clear any evaders from an unknown planar environment. We present a distributed algorithm built around guaranteeing complete coverage of the frontier between cleared and contaminated areas while expanding the cleared area. Our frontier-based algorithm can guarantee detection of evaders in unknown, multiply-connected planar environments which may be non-polygonal. We also detail a method for storing and updating the global frontier between cleared and contaminated areas without building a global map or requiring global localization, which enables our algorithm to be truly distributed. We demonstrate the functionality of the algorithm through Player/Stage simulations.

The most widely applied resource allocation strategy is to balance, or equalize, the total workload assigned to each resource. In mobile multi-agent systems, this principle directly leads to equitable partitioning policies in which (i) the workspace is divided into subregions of equal measure, (ii) there is a bijective correspondence between agents and subregions, and (iii) each agent is responsible for service requests originating within its own subregion. In this paper, we provide the first distributed algorithm that provably allows m agents to converge to an equitable partition of the workspace, from any initial configuration, i.e., globally. Our approach is related to the classic Lloyd algorithm, and provides novel insights into the properties of power diagrams. Simulation results are presented and discussed.

This paper presents a coordination algorithm for mobile autonomous robots. Relying on distributed sensing, the robots achieve rendezvous, i.e., they move to a common location. Each robot is a point mass moving in a simply connected, nonconvex, unknown environment according to an omnidirectional kinematic model. It is equipped with line-of-sight limited-range sensors, i.e., it can measure the relative position of any object (robots or environment boundary) if and only if the object is within a given distance and there are no obstacles in between. The perimeter minimizing algorithm is designed using the notions of robust visibility, connectivity-preserving constraint sets, and proximity graphs. The algorithm provably achieves rendezvous if the interagent sensing graph is connected at any time during the evolution of the group. Simulations illustrate the theoretical results and the performance of the proposed algorithm in asynchronous setups and with measurement errors, control errors, and nonzero robot size. Simulations to illustrate the importance of visibility constraints and comparisons with the optimal centralized algorithm are also included.

A control strategy inspired by the hunting tactics of ladybugs is presented to simultaneously achieve sensor coverage and exploration of an area with a group of networked robots. The controller is distributed in that it requires only information local to each robot, and adaptive in that it modifies its behavior based on information in the environment. The ladybug controller is developed as a modification to a basic coverage control law, first for the non-adaptive case, then for the adaptive case. Stability is proven for both cases with a Lyapunov-type proof. Results of numerical simulations are presented.

This paper presents a new method for reactive collision avoidance for mobile robots in complex and cluttered environments. Our technique is to adapt the ldquodivide and conquerrdquo approach of the nearness-diagram+ navigation (ND+) method to generate a single motion law which applies for all navigational situations. The resulting local path planner considers all the visible obstacles surrounding the robot, not just the closest two. With these changes our new navigation method generates smoother motion while avoiding obstacles. Results from comparisons with ND+ are presented as are experiments using Erratic mobile robots.

In this paper, we address discrete-time pursuit-evasion games in the plane where every player has identical sensing and motion ranges restricted to closed disks of given sensing and stepping radii. A single evader is initially located inside a bounded subset of the environment and does not move until detected. We propose asweep-pursuit-capturepursuer strategy to capture the evader and apply it to two variants of the game. The first involves a single pursuer and an evader in a bounded convex environment, and the second involves multiple pursuers and an evader in a boundaryless environment. In the first game, we give a sufficient condition on the ratio of sensing to stepping radius of the players that guarantees capture. In the second, we determine the minimum probability of capture, which is a function of a novel pursuer formation and independent of the initial evader location. The sweep and pursuit phases reduce both games to previously studied problems with unlimited range sensing, and capture is achieved using available strategies. We obtain novel upper bounds on the capture time and present simulation studies that address the performance of the strategies under sensing errors, different ratios of sensing to stepping radius, greater evader speed, and a different number of pursuers.

This work studies optimal sensor placement and motion coordination strategies for mobile sensor networks. For a target tracking application with range sensors, we investigate the determinant of the Cramer Rao Lower Bound and compute it in the 2D and 3D cases, characterizing the global minima in the 2D case. We propose motion coordination algorithms that steer the mobile sensor network to an optimal deployment and that are amenable to a decentralized implementation. Finally, our numerical simulations illustrate how the proposed algorithms lead to improved performance of an extended Kalman filter in a target tracking scenario.

This paper investigates the behavior of a group of autonomous robots evolving in a polygonal environment according to a "move away from the closest neighbor" heuristic. We demonstrate that this distributed coordination algorithm optimizes an aggregate cost function that measures how uniformly distributed are the robots in their environment. Our technical approach based on non-smooth analysis and computational geometry unveils a sphere-packing problem. The algorithm is implemented in a testbed of indoor mobile robots equipped with sonar. We develop novel approaches for improving single point sonar scan performance. These algorithms are then shown to have improved reliability, resolution and speed in distributed environments as compared to other scanning methods.

This paper presents control and coordination algorithms for groups of vehicles. The focus is on autonomous vehicle networks performing distributed sensing tasks, where each vehicle plays the role of a mobile tunable sensor. The paper proposes gradient descent algorithms for a class of utility functions which encode optimal coverage and sensing policies. The resulting closed-loop behavior is adaptive, distributed, asynchronous, and verifiably correct.

This paper presents motion planning algorithms for underactuated systems evolving on rigid rotation and displacement groups. Motion planning is transcribed into (low-dimensional) combinatorial selection and inverse-kinematics problems. We present a catalog of solutions for all underactuated systems on SE(2), SO(3) and SE(2) /spl times/ /spl Ropf/ classified according to their controllability properties.

The snakeboard is shown to possess two decoupling vector fields, and to be kinematically controllable. Accordingly, the problem of steering the snakeboard from a given configuration at rest to a desired configuration at rest is posed as a constrained static nonlinear inversion problem. An explicit algorithmic solution to the problem is provided, and its limitations are discussed. An ad hoc solution to the nonlinear inversion problem is also exhibited.

This paper presents a computationally efficient method for deriving coordinate representations for the equations of motion and the affine connection describing a class of Lagrangian systems. We consider mechanical systems endowed with symmetries and subject to nonholonomic constraints and external forces. This method is demonstrated on two robotic locomotion mechanisms known as the snake board and the roller racer. The resulting coordinate representations are compact and lead to straightforward proofs of various controllability results.

This paper describes decentralized control laws for the coordination of multiple vehicles performing spatially distributed tasks. The control laws are based on a gradient descent scheme applied to a class of decentralized utility functions that encode optimal coverage and sensing policies. These utility functions are studied in geographical optimization problems and they arise naturally in vector quantization and in sensor allocation tasks. The approach exploits the computational geometry of spatial structures such as Voronoi diagrams.

This paper studies a class of hybrid mechanical systems that locomote by switching between constraints defining different dynamic regimes. We develop a geometric framework for modeling smooth phenomena such as inertial forces, holonomic and nonholonomic constraints, as well as discrete features such as transitions between smooth dynamic regimes through plastic and elastic impacts. We focus on devices that are able to switch between constraints at an arbitrary point in the configuration space. This class of hybrid mechanical control systems can be described in terms of affine connections and jump transition maps that are linear in the velocity. We investigate two notions of local controllability, the equilibrium and kinematic controllability, and provide sufficient conditions for each of them. The tests rely on the assumption of zero velocity switches. We illustrate the modeling framework and the controllability tests on a planar sliding, clamped, and rolling device. In particular, we show how the analysis can be used for motion planning.

We introduce the notion of kinematic controllability for second-order underactuated mechanical systems. For systems satisfying this property, the problem of planning fast collision-free trajectories between zero velocity states can be decoupled into the computationally simpler problems of path planning for a kinematic system followed by time-optimal time scaling. While this approach is well known for fully actuated systems, until now there has been no way to apply it to underactuated dynamic systems. The results in this paper form the basis for efficient collision-free trajectory planning for a broad class of underactuated mechanical systems including manipulators and vehicles in space and underwater environments.

We introduce the notion of kinematic controllability for second-order underactuated mechanical systems. For systems satisfying this property, the problem of planning fast collision-free trajectories between zero velocity states can be decoupled into the computationally simpler problems of path planning for a kinematic system followed by time-optimal time scaling. While this approach is well known for fully actuated systems, until now there has been no way to apply it to underactuated dynamic systems. The results in this paper form the basis for efficient collision-free trajectory planning for a class of underactuated mechanical systems including manipulators and vehicles in space and underwater environments.

We analyze a class of mechanisms that locomote by switching between constraints. Because of the hybrid nature of such systems, most of the existing analysis tools, developed primarily for smooth systems, can not be directly applied. Our aim is to exploit the special structure provided by Lagrangian mechanics to study the controllability of this class of mechanisms. We base the analysis on a series representation of the evolution of the system. Our main result is a description of trajectories involving switches between constraints at nonzero velocity (impacts) in the presence of large inertial forces (drift). The analysis provides a basis for local motion planning. The results are applied to an example of a two-link planar mechanism that can locomote by clamping one of the links.

We consider the convergence analysis of the sign algorithm for adaptive filtering when the input processes are uncorrelated and Gaussian and a fixed step size /spl mu/>0 is used. Exact recursive equations for the covariance matrix of the deviation error are established for any step size /spl mu/>0. Asymptotic time-averaged convergence for the mean-absolute deviation error, mean-square deviation error, and for the signal mean-square estimation error are established. These results are shown to hold for arbitrary step size /spl mu/>0.>