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This paper introduces the problem of determining through distributed consensus the fastest mixing Markov chain with a desired sparsity pattern. In contrast to the centralized optimization-based problem formulation, we develop a novel distributed relaxation by constructing a dynamical system over the cross product of an appropriately patterned set of stochastic matrices. In particular, we define a probability distribution over the set of such patterned stochastic matrices and associate an agent with a random matrix drawn from this distribution. Under the assumption that the network of agents is connected, we employ consensus to achieve agreement of all agents regardless of their initial states. For sufficiently many agents, the law of large numbers implies that the asymptotic consensus limit converges to the mean stochastic matrix, which for the distribution under consideration, corresponds to the chain with the fastest mixing rate, relative to a standard bound on the exact rate. Our approach relies on results that express general element-wise nonnegative stochastic matrices as convex combinations of 0–1 stochastic matrices. Its performance, as a function of the weights in these convex combinations and the number of agents, is illustrated in computer simulations. Because of its differential and distributed nature, this approach can handle large problems and seems likely to be well suited for applications in distributed control and robotics.

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Ability to translate to a goal position under the constrains imposed by complex environmental conditions is a key capability for biological and artificial systems alike. Over billions of years evolutionary processes have developed a wide range of solutions to address mobility needs in air, in water and on land. The efficacy of such biological locomotors is beyond the capabilities of engineering solutions that has been produced to this date. Nature has been and will surely remain to be a source of inspiration for engineers in their quest to bring “real mobility” to their creations. In recent years a new class of dynamic legged terrestrial robotic systems \cite{Autumn-Buehler-Cutkosky.SPIE2005,Raibert.Book1986,Raibert-Blankesport-Nelson.IFAC2008,Saranli-Buehler-Koditschek.IJRR2001} have been developed inspired by, but without mimicking, the examples from the Nature. The experimental work with these platforms over the past decade has led to an improved appreciation of legged locomotion. This paper is an overview of fundamental advantages dynamic legged locomotion offers over the classical wheeled and tracked approaches.

This paper describes the development of a legged robot designed for general locomotion of complex terrain but specialized for dynamical, high-speed climbing of a uniformly convex cylindrical structure, such as an outdoor telephone pole. This robot, the RiSE V3 climbing machine—mass 5.4 kg, length 70 cm, excluding a 28 cm tail appendage—includes several novel mechanical features, including novel linkage designs for its legs and a non-backdrivable, energy-dense power transmission to enable high-speed climbing. We summarize the robot’s design and document a climbing behavior that achieves rapid ascent of a wooden telephone pole at 21 cm/s, a speed previously unachieved—and, we believe, heretofore impossible—with a robot of this scale. The behavioral gait of the robot employs the mechanical design to propel the body forward while passively maintaining yaw, pitch, and roll stability during climbing locomotion. The robot’s general-purpose legged design coupled with its specialized ability to quickly gain elevation and park at a vertical station silently with minimal energy consumption suggest potential applications including search and surveillance operations as well as ad hoc networking.

Legged locomotion on flowing ground (e.g., granular media) is unlike locomotion on hard ground because feet experience both solid- and fluid-like forces during surface penetration. Recent bioinspired legged robots display speed relative to body size on hard ground comparable with high-performing organisms like cockroaches but suffer significant performance loss on flowing materials like sand. In laboratory experiments, we study the performance (speed) of a small (2.3 kg) 6-legged robot, SandBot, as it runs on a bed of granular media (1-mm poppy seeds). For an alternating tripod gait on the granular bed, standard gait control parameters achieve speeds at best 2 orders of magnitude smaller than the 2 body lengths/s (≈60 cm/s) for motion on hard ground. However, empirical adjustment of these control parameters away from the hard ground settings restores good performance, yielding top speeds of 30 cm/s. Robot speed depends sensitively on the packing fraction φ and the limb frequency ω, and a dramatic transition from rotary walking to slow swimming occurs when φ becomes small enough and/or ω large enough. We propose a kinematic model of the rotary walking mode based on generic features of penetration and slip of a curved limb in granular media. The model captures the dependence of robot speed on limb frequency and the transition between walking and swimming modes but highlights the need for a deeper understanding of the physics of granular media.

We are interested in the development of a variety of legged robot platforms intended for operation in unstructured outdoor terrain. In such settings, the traditions of rational engineering design, driven by analytically informed and computationally assisted studies of robot-environment models, remain ineffective due to the complexity of both the robot designs and the terrain in which they must operate. Instead, empirical trial and error often drives the necessary incremental and iterative design process, hence the development of such robots remains expensive both in time and cost, and is often closely dependent upon the substrate properties of the locomotion terrain. This paper describes a series of concurrent but increasingly coordinated software development efforts that aim to diminish the gap between easily interfaced and physically sound computational models of a real robot’s operation in a complex natural environment. We describe a robot simulation environment in which simple robot modifications can be easily prototyped along and “played” into phenomenological models of contact mechanics. We particularly focus on the daunting but practically compelling example of robot feet interacting granular media, such as gravel or sand, offering a brief report of our progress in deriving and importing physically accurate but computationally tractable phenomenological substrate models into the robot execution simulation environment. With a goal of integration for future robot prototyping simulations, we review the prospects for diminishing the gap between the integrated computational models and the needs of physical platform development.

Arthropods with their extraordinary locomotive capabilities have inspired roboticists, giving rise to major accomplishments in robotics research over the past decade. Most notably bio-inspired hexapod robots using only task level open-loop controllers [22, 9] exhibit stable dynamic locomotion over highly broken and unstable terrain. We present experimental data on the dynamics of Sprawl- Hex — a hexapod robot with adjustable body sprawl — consisting of time trajectory of full body configuration and single leg ground reaction forces. The dynamics of SprawlHex is compared and contrasted to that of insects. SprawlHex dynamics has qualitative similarities to that of insects in both sagittal and horizontal plane. SprawlHex presents a step towards construction of an effective physical model to study arthropod locomotion.

This paper presents an integrated, systems-level view of several novel design and control features associated with the biologically inspired, hexapedal, RiSE (Robots in Scansorial Environments) robot. RiSE is the first legged machine capable of locomotion on both the ground and a variety of vertical building surfaces including brick, stucco, and crushed stone at speeds up to 4 cm/s, quietly and without the use of suction, magnets, or adhesives. It achieves these capabilities through a combination of bioinspired and traditional design methods. This paper describes the design process and specifically addresses body morphology, hierarchical compliance in the legs and feet, and sensing and control systems that enable robust and reliable climbing on difficult surfaces. Experimental results illustrate the effects of various behaviors on climbing performance and demonstrate the robot’s ability to climb reliably for long distances.

Our objective is to provide experimentalists with neuromechanical control hypotheses that can be tested with kinematic data sets. To illustrate the approach, we select legged animals responding to perturbations during running. In the following sections, we briefly outline our dynamical systems approach, state our over-arching hypotheses, define four neuromechanical control architectures (NCAs) and conclude by proposing a series of perturbation experiments that can begin to identify the simplest architecture that best represents an animal’s controller.

This paper addresses problems of robot navigation with nonholonomic motion constraints and perceptual cues arising from onboard visual servoing in partially engineered environments. We propose a general hybrid procedure that adapts to the constrained motion setting the standard feedback controller arising from a navigation function in the fully actuated case. This is accomplished by switching back and forth between moving “down” and “across” the associated gradient field toward the stable manifold it induces in the constrained dynamics. Guaranteed to avoid obstacles in all cases, we provide conditions under which the new procedure brings initial configurations to within an arbitrarily small neighborhood of the goal.

We summarize simulation results on a sample of visual servoing problems with a few different perceptual models. We document the empirical effectiveness of the proposed algorithm by reporting the results of its application to outdoor autonomous visual registration experiments with the robot RHex guided by engineered beacons.

Legged robots are by nature strongly non-linear, high-dimensional systems whose full complexity permits neither tractable mathematical analysis nor comprehensive numerical study. In consequence, a growing body of literature interrogates simplified “template” [1], [2] models—to date almost exclusively confined to sagittal- or horizontal-plane motion—with the aim of gaining insight into the design or control of the far messier reality. In this paper we introduce a simple bounding-in-place (“BIP”) model as a candidate frontal plane template for straight-ahead level ground running and explore its use in formulating hypotheses about whether and why rolling motion is important in legged locomotion. Numerical study of left-right compliance asymmetry in the BIP model suggests that compliance ratios yielding lowest steady state roll suffer far longer disturbance recovery transients than those promoting greater steady state roll. We offer preliminary experimental data obtained from video motion capture data of the frontal plane disturbance recovery patterns of a RHex-like hexapod suggesting a correspondence to the conclusions of the numerical study. Fig. 1. EduBot [19], a RHex-like [20] hexapedal robot.

Jonathan Clark was supported by the IC Postdoctoral Fellow Program under grant number HM158204-1-2030. Samuel Burden was supported by the SUNFEST REU program at the University of Pennsylvania. This work was also partially supported by the NSF FIBR grant #0425878. For more information: Kod*Lab