arXiv subjects

Daniel E. Koditschek

Publications and source records attributed to Daniel E. Koditschek.

32 records · Page 2Linked to original sources

arXiv2018

Technical Report: Reactive Navigation in Partially Known Non-Convex Environments

This paper presents a provably correct method for robot navigation in 2D environments cluttered with familiar but unexpected non-convex, star-shaped obstacles as well as completely unknown, convex obstacles. We presuppose a limited range onboard sensor, capable of recognizing, localizing and (leveraging ideas from constructive solid geometry) generating online from its catalogue of the familiar, non-convex shapes an implicit representation of each one. These representations underlie an online change of coordinates to a completely convex model planning space wherein a previously developed online construction yields a provably correct reactive controller that is pulled back to the physically sensed representation to generate the actual robot commands. We extend the construction to differential drive robots, and suggest the empirical utility of the proposed control architecture using both formal proofs and numerical simulations.

cs.RO↗

arXiv2018

A dynamical system for prioritizing and coordinating motivations

We develop a dynamical systems approach to prioritizing and selecting multiple recurring tasks with the aim of conferring a degree of deliberative goal selection to a mobile robot confronted with competing objectives. We take navigation as our prototypical task, and use reactive (i.e., vector field) planners derived from navigation functions to encode control policies that achieve each individual task. We associate a scalar "value" with each task representing its current urgency and let that quantity evolve in time as the robot evaluates the importance of its assigned task relative to competing tasks. The robot's motion control input is generated as a convex combination of the individual task vector fields. Their weights, in turn, evolve dynamically according to a decision model adapted from the literature on bioinspired swarm decision making, driven by the values. In this paper we study a simple case with two recurring, competing navigation tasks and derive conditions under which it can be guaranteed that the robot will repeatedly serve each in turn. Specifically, we provide conditions sufficient for the emergence of a stable limit cycle along which the robot repeatedly and alternately navigates to the two goal locations. Numerical study suggests that the basin of attraction is quite large so that significant perturbations are recovered with a reliable return to the desired task coordination pattern.

math.DS↗

arXiv2018

Technical Report: Sensor-Based Reactive Symbolic Planning in Partially Known Environments

This paper considers the problem of completing assemblies of passive objects in nonconvex environments, cluttered with convex obstacles of unknown position, shape and size that satisfy a specific separation assumption. A differential drive robot equipped with a gripper and a LIDAR sensor, capable of perceiving its environment only locally, is used to position the passive objects in a desired configuration. The method combines the virtues of a deliberative planner generating high-level, symbolic commands, with the formal guarantees of convergence and obstacle avoidance of a reactive planner that requires little onboard computation and is used online. The validity of the proposed method is verified both with formal proofs and numerical simulations.

cs.RO↗

arXiv2016

Navigation Functions for Convex Potentials in a Space with Convex Obstacles

Given a convex potential in a space with convex obstacles, an artificial potential is used to navigate to the minimum of the natural potential while avoiding collisions. The artificial potential combines the natural potential with potentials that repel the agent from the border of the obstacles. This is a popular approach to navigation problems because it can be implemented with spatially local information that is acquired during operation time. Artificial potentials can, however, have local minima that prevent navigation to the minimum of the natural potential. This paper derives conditions that guarantee artificial potentials have a single minimum that is arbitrarily close to the minimum of the natural potential. The qualitative implication is that artificial potentials succeed when either the condition number-- the ratio of the maximum over the minimum eigenvalue-- of the Hessian of the natural potential is not large and the obstacles are not too flat or when the destination is not close to the border of an obstacle. Numerical analyses explore the practical value of these theoretical conclusions.

math.DS↗

arXiv2016

The Penn Jerboa: A Platform for Exploring Parallel Composition of Templates

We have built a 12DOF, passive-compliant legged, tailed biped actuated by four brushless DC motors. We anticipate that this machine will achieve varied modes of quasistatic and dynamic balance, enabling a broad range of locomotion tasks including sitting, standing, walking, hopping, running, turning, leaping, and more. Achieving this diversity of behavior with a single under-actuated body, requires a correspondingly diverse array of controllers, motivating our interest in compositional techniques that promote mixing and reuse of a relatively few base constituents to achieve a combinatorially growing array of available choices. Here we report on the development of one important example of such a behavioral programming method, the construction of a novel monopedal sagittal plane hopping gait through parallel composition of four decoupled 1DOF base controllers. For this example behavior, the legs are locked in phase and the body is fastened to a boom to restrict motion to the sagittal plane. The platform's locomotion is powered by the hip motor that adjusts leg touchdown angle in flight and balance in stance, along with a tail motor that adjusts body shape in flight and drives energy into the passive leg shank spring during stance. The motor control signals arise from the application in parallel of four simple, completely decoupled 1DOF feedback laws that provably stabilize in isolation four corresponding 1DOF abstract reference plants. Each of these abstract 1DOF closed loop dynamics represents some simple but crucial specific component of the locomotion task at hand. We present a partial proof of correctness for this parallel composition of template reference systems along with data from the physical platform suggesting these templates are anchored as evidenced by the correspondence of their characteristic motions with a suitably transformed image of traces from the physical platform.

cs.RO↗

arXiv2016

A Hybrid Dynamical Extension of Averaging

We extend a smooth dynamical systems averaging technique to a class of hybrid systems with a limit cycle that is particularly relevant to the synthesis of stable legged gaits. After introducing a definition of hybrid averageability sufficient to recover the classical result, we provide a simple illustration of its applicability to legged locomotion and conclude with some rather more speculative remarks concerning the prospects for further generalization of these ideas.

cs.RO↗

arXiv2015

Discriminative Measures for Comparison of Phylogenetic Trees

In this paper we introduce and study three new measures for efficient discriminative comparison of phylogenetic trees. The NNI navigation dissimilarity $d_{nav}$ counts the steps along a "combing" of the Nearest Neighbor Interchange (NNI) graph of binary hierarchies, providing an efficient approximation to the (NP-hard) NNI distance in terms of "edit length". At the same time, a closed form formula for $d_{nav}$ presents it as a weighted count of pairwise incompatibilities between clusters, lending it the character of an edge dissimilarity measure as well. A relaxation of this formula to a simple count yields another measure on all trees --- the crossing dissimilarity $d_{CM}$. Both dissimilarities are symmetric and positive definite (vanish only between identical trees) on binary hierarchies but they fail to satisfy the triangle inequality. Nevertheless, both are bounded below by the widely used Robinson-Foulds metric and bounded above by a closely related true metric, the cluster-cardinality metric $d_{CC}$. We show that each of the three proposed new dissimilarities is computable in time $O(n^2)$ in the number of leaves $n$, and conclude the paper with a brief numerical exploration of the distribution over tree space of these dissimilarities in comparison with the Robinson-Foulds metric and the more recently introduced matching-split distance.

q-bio.PE↗

arXiv2015

On the Optimality of Napoleon Triangles

An elementary geometric construction known as Napoleon's theorem produces an equilateral triangle built on the sides of any initial triangle: the centroids of each equilateral triangle meeting the original sides, all outward or all inward, comprise the vertices of the new equilateral triangle. In this note we observe that two Napoleon iterations yield triangles with useful optimality properties. Two inner transformations result in a (degenerate) triangle whose vertices coincide at the original centroid. Two outer transformations yield an equilateral triangle whose vertices are closest to the original in the sense of minimizing the sum of the three squared distances.

math.OC↗

arXiv2015

Voronoi-Based Coverage Control of Heterogeneous Disk-Shaped Robots

In distributed mobile sensing applications, networks of agents that are heterogeneous respecting both actuation as well as body and sensory footprint are often modelled by recourse to power diagrams --- generalized Voronoi diagrams with additive weights. In this paper we adapt the body power diagram to introduce its "free subdiagram," generating a vector field planner that solves the combined sensory coverage and collision avoidance problem via continuous evaluation of an associated constrained optimization problem. We propose practical extensions (a heuristic congestion manager that speeds convergence and a lift of the point particle controller to the more practical differential drive kinematics) that maintain the convergence and collision guarantees.

cs.RO↗

arXiv2015

Coordinated Robot Navigation via Hierarchical Clustering

We introduce the use of hierarchical clustering for relaxed, deterministic coordination and control of multiple robots. Traditionally an unsupervised learning method, hierarchical clustering offers a formalism for identifying and representing spatially cohesive and segregated robot groups at different resolutions by relating the continuous space of configurations to the combinatorial space of trees. We formalize and exploit this relation, developing computationally effective reactive algorithms for navigating through the combinatorial space in concert with geometric realizations for a particular choice of hierarchical clustering method. These constructions yield computationally effective vector field planners for both hierarchically invariant as well as transitional navigation in the configuration space. We apply these methods to the centralized coordination and control of $n$ perfectly sensed and actuated Euclidean spheres in a $d$-dimensional ambient space (for arbitrary $n$ and $d$). Given a desired configuration supporting a desired hierarchy, we construct a hybrid controller which is quadratic in $n$ and algebraic in $d$ and prove that its execution brings all but a measure zero set of initial configurations to the desired goal with the guarantee of no collisions along the way.

cs.RO↗

arXiv2015

Event-Selected Vector Field Discontinuities Yield Piecewise-Differentiable Flows

We study a class of discontinuous vector fields brought to our attention by multi-legged animal locomotion. Such vector fields arise not only in biomechanics, but also in robotics, neuroscience, and electrical engineering, to name a few domains of application. Under the conditions that (i) the vector field's discontinuities are locally confined to a finite number of smooth submanifolds and (ii) the vector field is transverse to these surfaces in an appropriate sense, we show that the vector field yields a well-defined flow that is Lipschitz continuous and piecewise-differentiable. This implies that although the flow is not classically differentiable, nevertheless it admits a first-order approximation (known as a Bouligand derivative) that is piecewise-linear and continuous at every point. We exploit this first-order approximation to infer existence of piecewise-differentiable impact maps (including Poincaré maps for periodic orbits), show the flow is locally conjugate (via a piecewise-differentiable homeomorphism) to a flowbox, and assess the effect of perturbations (both infinitesimal and non-infinitesimal) on the flow. We use these results to give a sufficient condition for the exponential stability of a periodic orbit passing through a point of multiply intersecting events, and apply the theory in illustrative examples to demonstrate synchronization in abstract first- and second-order phase oscillator models.

math.DS↗

arXiv2015

Universal Memory Architectures for Autonomous Machines

We propose a self-organizing memory architecture for perceptual experience, capable of supporting autonomous learning and goal-directed problem solving in the absence of any prior information about the agent's environment. The architecture is simple enough to ensure (1) a quadratic bound (in the number of available sensors) on space requirements, and (2) a quadratic bound on the time-complexity of the update-execute cycle. At the same time, it is sufficiently complex to provide the agent with an internal representation which is (3) minimal among all representations of its class which account for every sensory equivalence class subject to the agent's belief state; (4) capable, in principle, of recovering the homotopy type of the system's state space; (5) learnable with arbitrary precision through a random application of the available actions. The provable properties of an effectively trained memory structure exploit a duality between weak poc sets -- a symbolic (discrete) representation of subset nesting relations -- and non-positively curved cubical complexes, whose rich convexity theory underlies the planning cycle of the proposed architecture.

cs.AI↗

arXiv2014

Anytime Hierarchical Clustering

We propose a new anytime hierarchical clustering method that iteratively transforms an arbitrary initial hierarchy on the configuration of measurements along a sequence of trees we prove for a fixed data set must terminate in a chain of nested partitions that satisfies a natural homogeneity requirement. Each recursive step re-edits the tree so as to improve a local measure of cluster homogeneity that is compatible with a number of commonly used (e.g., single, average, complete) linkage functions. As an alternative to the standard batch algorithms, we present numerical evidence to suggest that appropriate adaptations of this method can yield decentralized, scalable algorithms suitable for distributed/parallel computation of clustering hierarchies and online tracking of clustering trees applicable to large, dynamically changing databases and anomaly detection.

stat.ML↗

arXiv2009

Sensitive dependence of the motion of a legged robot on granular media

Legged locomotion on flowing ground ({\em e.g.} granular media) is unlike locomotion on hard ground because feet experience both solid- and fluid-like forces during surface penetration. Recent bio-inspired legged robots display speed relative to body size on hard ground comparable to 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) six-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 two orders of magnitude smaller than the 2 body lengths/s ($\approx 60$ 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.

physics.bio-ph↗