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High-speed Flight in an Ergodic Forest

Sertac Karaman, Emilio Frazzoli

arXiv:1202.0253v1cs.ROcs.SYeess.SY

TL;DR

The paper asks how fast a bird can navigate a randomly generated forest when the obstacle statistics, rather than their precise locations and sizes, are known. It analyzes the problem using ergodic and percolation theory, establishing a critical-speed transition for ergodic forests and studying speed bounds and a percolation model for a Poisson-forest special case.

  • Problem

    The paper asks for the maximum speed at which flight through a randomly generated obstacle field can be maintained indefinitely with probabilistic collision-avoidance guarantees.

    Source: §I · p. 1

  • Method

    The paper analyzes the existence of infinite collision-free trajectories in ergodic forests and derives critical-speed bounds and an equivalent percolation model for a Poisson-forest special case.

    Source: §I · p. 1, §VIII · p. 17

  • Results

    In ergodic forests, speeds below a critical speed permit an infinite collision-free trajectory almost surely, whereas speeds above it do not.

    Source: §I · p. 1

  • Takeaways & Limitations

    The Poisson-forest analysis approximately computes a phase diagram through Monte Carlo simulations using the equivalent percolation model.

    Source: §VIII · p. 17

    The paper assumes ergodicity for its phase-transition result and defers limited-range bird sensing to future work.

    Source: §VIII · p. 17

Abstract

from arXiv · show

Inspired by birds flying through cluttered environments such as dense forests, this paper studies the theoretical foundations of a novel motion planning problem: high-speed navigation through a randomly-generated obstacle field when only the statistics of the obstacle generating process are known a priori. Resembling a planar forest environment, the obstacle generating process is assumed to determine the locations and sizes of disk-shaped obstacles. When this process is ergodic, and under mild technical conditions on the dynamics of the bird, it is shown that the existence of an infinite collision-free trajectory through the forest exhibits a phase transition. On one hand, if the bird flies faster than a certain critical speed, then, with probability one, there is no infinite collision-free trajectory, i.e., the bird will eventually collide with some tree, almost surely, regardless of the planning algorithm governing the bird's motion. On the other hand, if the bird flies slower than this critical speed, then there exists at least one infinite collision-free trajectory, almost surely. Lower and upper bounds on the critical speed are derived for the special case of a homogeneous Poisson forest considering a simple model for the bird's dynamics. For the same case, an equivalent percolation model is provided. Using this model, the phase diagram is approximated in Monte-Carlo simulations. This paper also establishes novel connections between robot motion planning and statistical physics through ergodic theory and percolation theory, which may be of independent interest.

I. INTRODUCTION

The paper frames high-speed flight through a randomly generated forest as a motion-planning problem with probabilistic guarantees. For ergodic forests, it establishes a critical-speed phase transition and studies bounds and percolation models in a Poisson-forest special case.

I. INTRODUCTION

  • The problem asks for the maximum speed at which a bird can fly indefinitely through a planar forest with probabilistic collision-avoidance guarantees.The forest-generating process specifies tree locations and sizes, while the bird is modeled as a dynamical system.Source: §I · p. 1
  • For ergodic forests, speeds below a critical value permit an infinite collision-free trajectory almost surely, while speeds above it do not.Ergodicity is essential: a stationary non-ergodic process can yield an infinite collision-free trajectory with probability strictly between zero and one.Source: §I · p. 1
  • For a homogeneous Poisson forest with equal tree radii and simple bird dynamics, the paper derives lower and upper bounds on the critical speed.The paper also reports exponentially changing probabilities for sustained flight above the upper bound and existence of an infinite trajectory below the lower bound.Source: §I · pp. 1–2
  • The paper studies a Poisson-forest special case using an equivalent percolation model and approximately constructs its phase diagram through Monte Carlo simulations.It also discusses percolation-model properties and derives the speed bounds using discrete and continuum percolation theory.Source: §I · p. 2
  • The analysis connects robot motion planning with ergodic and percolation theory, which the paper presents as a novel application of these tools to engineering design.The paper contrasts this with their use in studying physical, social, economic, and political phenomena.Source: §I · p. 2

A. The forest-generating Process

  • The forest model represents trees as disks at point-process locations, with radii assigned as marks; in the simple definition, each radius is independent of other tree locations.A more general class that accounts for such dependencies is deferred to Section III.Source: A. The forest-generating Process · p. 2

1) The dynamics of the bird:

The paper models the bird with speed-parameterized dynamics and lets a planner choose its control from a realized forest. The formal task is to fly toward new regions indefinitely while avoiding trees almost surely.

1) The dynamics of the bird:

  • The bird’s motion is represented by a speed-parameterized dynamical system with Lipschitz-continuous functions and measurable control inputs.Its output y(t) represents the bird’s position in the planar forest.Source: 1) The dynamics of the bird · p. 3

2) The planner governing the bird:

C. Problem Formulation

  • Problem 1 requires the bird to fly indefinitely toward new regions while avoiding collisions almost surely.With knowledge of all tree locations and sizes, solving the planning problem is equivalent to the almost-sure existence of an infinite collision-free trajectory.Source: C. Problem Formulation · pp. 3–4
  • For ergodic forests, planners cannot solve the task above a critical speed despite knowing all trees, but are guaranteed to exist below that threshold.The paper studies bounds on the threshold in a special case theoretically and experimentally.Source: C. Problem Formulation · p. 3

D. A Single-integrator Bird Flying in a Poisson Forest

The section specializes the bird model to a single-integrator flying through a homogeneous Poisson forest, and describes how speed determines its reachable region. It also notes that an infinite collision-free trajectory does not necessarily imply a planner when perception is limited.

D. A Single-integrator Bird Flying in a Poisson Forest

III. PHASE TRANSITIONS IN ERGODIC FORESTS

  • For ergodic forests, the existence of infinite collision-free trajectories exhibits a phase transition under mild technical assumptions on bird dynamics.Source: §III · p. 4

A. Ergodic forest-generating Processes

The paper generalizes its forest model to allow dependencies between tree radii and formalizes ergodicity for marked point processes. It then derives a zero-one law and, with further conditions on the dynamics, a critical-speed phase transition.

A. Ergodic forest-generating Processes

  • The generalized forest-generating process uses a marked point process, allowing a tree’s radius to depend on the locations of other trees.Source: A. Ergodic forest-generating Processes · p. 4
  • Ergodicity is framed for stationary marked point processes; roughly, events described by widely separated point sets are almost independent.Poisson processes are examples; the class also includes some Cox and cluster processes and allows short-range dependencies between radii.Source: A. Ergodic forest-generating Processes · p. 5

B. The Monotonic Zero-One Law of the Ergodic Forest

IV. SUB-CRITICAL FLIGHT IN A POISSON FOREST

The section applies discrete percolation theory to a single-integrator bird in a Poisson forest. It links percolation to collision-free flight, derives a lower bound on critical speed, and studies how the choice of starting point affects flight prospects.

IV. SUB-CRITICAL FLIGHT IN A POISSON FOREST

  • The section derives a lower bound on critical speed for a single-integrator bird in a Poisson forest using discrete percolation theory.Source: §IV · p. 6
  • The probability of finding a starting point within distance l that yields an infinite collision-free trajectory converges to one exponentially fast as l increases.Source: §IV · p. 6
  • In the novel percolation model, a certain type of percolation implies almost-sure existence of infinite collision-free trajectories.Source: §IV · p. 6

A. The Theory of Discrete Percolation

The section introduces discrete percolation models in which open connections form clusters, then describes critical thresholds separating regimes with and without infinite clusters. It also notes corresponding models on directed lattices.

A. The Theory of Discrete Percolation

B. The Existence of Infinite Collision-free Trajectories

For a single-integrator bird in a Poisson forest, the section links infinite collision-free flight to directed site percolation on a hexagonal lattice. An infinite open cluster implies an infinite collision-free trajectory, yielding a lower bound on the critical speed.

B. The Existence of Infinite Collision-free Trajectories

  • The construction uses lateral inputs of either −wmax or wmax, held constant over intervals of length T = r/(ν sin(α/2)).The longitudinal and lateral distances traveled per interval are r/sin(α/2) and r/cos(α/2), respectively.Source: B. The Existence of Infinite… · p. 8, Figure 4 · p. 8
  • Theorem 9 gives an almost-sure existence condition for infinite collision-free flight in terms of tree density, tree radius, speed, and the directed hexagonal-lattice critical probability.The supplied theorem passage does not include the condition’s full expression.Source: B. The Existence of Infinite… · p. 8
  • An infinite open cluster in the directed hexagonal-lattice model implies an infinite collision-free trajectory for the single-integrator bird.The model is formed from triangular regions associated with switching points; each vertex is open when its corresponding region contains no Poisson point.Source: B. The Existence of Infinite… · pp. 8–9
  • Under Theorem 9’s conditions, Theorem 10 states that a nearby starting point can yield an infinite collision-free trajectory with very high probability.It defines the event using a start y′ within distance l of y and assumes ρr²/sin(α) < log(1/pps crit).Source: B. The Existence of Infinite… · p. 9

V. SUPER-CRITICAL FLIGHT IN A POISSON FOREST

The section studies the super-critical regime for a single-integrator bird in a Poisson forest using continuum percolation. It derives an upper bound on critical speed and characterizes survival above that bound.

V. SUPER-CRITICAL FLIGHT IN A POISSON FOREST

  • The analysis derives an upper bound on the critical speed for the single-integrator bird flying in a Poisson forest.It uses continuum percolation theory to study the super-critical regime.Source: §V · p. 9
  • Above the derived upper bound on critical speed, the probability of avoiding collision for T time units is exponentially small in T, regardless of the planner.The section uses a geometric model in which a type of percolation implies that infinite collision-free trajectories do not exist.Source: §V · p. 9
  • The section’s geometric model connects a certain type of percolation with the non-existence of infinite collision-free trajectories.Source: §V · p. 9

A. The Theory of Continuum Percolation

Continuum percolation models connected Poisson points through disks and studies when occupied or vacant regions form unbounded components. These models exhibit thresholds and provide results used in the paper’s analysis of flight through forests.

A. The Theory of Continuum Percolation

  • The Gilbert disk model has a finite non-zero critical intensity: below it the graph fails to percolate almost surely, while above it a unique infinite component exists almost surely.The graph connects each Poisson point to points within a disk of diameter D.Source: A. The Theory of Continuum Percolation · p. 9
  • Occupied and vacant percolation share the same critical intensity, with a unique unbounded vacant component in the sub-critical regime and none in the super-critical regime.Source: A. The Theory of Continuum Percolation · p. 10
  • Continuum percolation depends on intensity λ and radius r through d = λπr², and the disk model’s critical degree is estimated between 4.508 and 4.515 with 99.99% confidence.A rigorous analysis gives the wider range 2.184–10.558; the square model’s critical degree is between 4.392 and 4.398 with 99.99% confidence.Source: A. The Theory of Continuum Percolation · p. 10

B. The Primary Shadow Model

The primary shadow model maps each tree to regions of states tied to the bird’s ability to escape or reach them. For the specified single-integrator dynamics in a Poisson forest, vacant-region percolation is necessary for an infinite collision-free trajectory.

B. The Primary Shadow Model

  • The left primary shadow contains states from which the bird cannot escape collision with a tree, while the right primary shadow contains states unreachable from outside it.Their union is the tree’s primary shadow region.Source: B. The Primary Shadow Model · p. 10
  • The model’s occupied and vacant primary shadow regions are formed by taking the union of tree shadows and its complement, respectively.Percolation means the region contains an unbounded connected component.Source: B. The Primary Shadow Model · p. 10
  • If the vacant primary shadow region does not percolate, every bird trajectory eventually collides with a tree.The proof considers starts in left-shadow, vacant, and right-shadow regions; each case leads to eventual collision.Source: B. The Primary Shadow Model · pp. 10–11

C. The Non-existence of Infinite Collision-free Trajectories

For a Poisson forest and a single-integrator bird, percolation arguments yield parameter bounds under which infinite collision-free flight is impossible. Above the bound, the probability of avoiding collision for a finite duration also decays exponentially with duration and speed.

C. The Non-existence of Infinite Collision-free Trajectories

  • Vacant primary-shadow percolation is necessary for an infinite collision-free trajectory, so its almost-sure failure rules out such flight.The percolation-failure condition is obtained by comparing the shadow model to the Gilbert square model.Source: C. The Non-existence of Infinite… · p. 11
  • With α := 2 tan−1(1/ν), each primary shadow Sν,r({z}) contains a rectangle with sides r/sin(α/2) and r/cos(α/2); vacant percolation in the shadow model requires it in the corresponding rectangle model.Source: C. The Non-existence of Infinite… · p. 11
  • If ρr^2/sin(α) > d_c/2, no infinite collision-free trajectory exists almost surely, regardless of the planner.Here α = 2 tan^-1(1/ν), and d_c is the critical degree for the Gilbert square model.Source: C. The Non-existence of Infinite… · p. 11
  • When the bound holds, the probability of a collision-free trajectory lasting T time units decays exponentially with both T and speed, for any starting point.The theorem states this for the regime ρr^2/sin(α) > d_c/2.Source: C. The Non-existence of Infinite… · p. 11
  • A stationary but non-ergodic mixed Poisson forest can make the probability of infinite collision-free flight equal to q, demonstrating that the probability need not be zero or one.The mixture selects intensity ρ with probability q and intensity ρ′ with probability 1−q, where the two cases respectively permit and preclude infinite flight.Source: C. The Non-existence of Infinite… · p. 12

VI. AN EQUIVALENT GEOMETRIC MODEL

The left-shadow construction gives a geometric percolation model equivalent to infinite collision-free flight for a single-integrator bird. Its vacant region changes from unbounded below a critical speed to empty above it, with bounds on when the threshold is nonzero.

VI. AN EQUIVALENT GEOMETRIC MODEL

  • The model builds collision-inevitable shadows by adding induced regions when opposite boundaries intersect; these represent states forced into one of the existing shadows.For finite tree sets, the construction iteratively forms induced shadows until no new region appears.Source: §VI · p. 12
  • An infinite collision-free trajectory exists for a fixed tree set if and only if its vacant left shadow region is nonempty.The improved shadow model makes vacant percolation equivalent to trajectory existence for the single-integrator bird in a Poisson forest.Source: §VI · pp. 12–13
  • For any ρ, r, and ν, a finite, possibly zero, critical speed separates an empty vacant left-shadow region above it from an unbounded one below it, almost surely.The critical speed is guaranteed nonzero when ρr^2 < 2 log(1/p_s^crit), and equals zero when ρr^2 > d_c/2.Source: §VI · p. 13
  • At the critical speed, the vacant left-shadow region disappears abruptly from unbounded to empty; the authors describe this property as unique to their knowledge.They leave open whether the unbounded vacant region is unique in every regime.Source: §VI · pp. 13–14
  • Occupied and vacant shadow regions are formed from the union of primary shadows and its complement; their percolation agrees with that of the left-shadow model almost surely.The right-shadow and primary-shadow variants are also defined through the corresponding left and right shadow regions.Source: §VI · pp. 13–14

VII. COMPUTATIONAL EXPERIMENTS

Monte Carlo experiments use the equivalent shadow model to approximate the Poisson-forest phase diagram. The normalized maximum width changes rapidly near the critical speed, consistent with the theoretical bounds.

VII. COMPUTATIONAL EXPERIMENTS

  • The phase diagram fixes r = 1 without loss of generality; the same vacant-region behavior applies when ρr^2 is preserved.Thus changing tree radius can be represented by scaling tree density.Source: §VII · p. 14
  • The test treats a left-to-right collision-free crossing as equivalent to a vacant shadow region, or to the largest occupied-shadow component spanning the region’s width.The simulations use this bounded-region test as an approximation to infinite collision-free trajectories.Source: §VII · p. 14
  • Each experiment uses a 500-wide region and adjusts its length so the expected tree count is 50,000, repeating each density-speed pair 200 times.The reported normalized maximum width is the largest occupied-shadow component width divided by the region width.Source: §VII · p. 14, Figure 18 · p. 16
  • The normalized maximum width changes rapidly near the critical speed, as predicted by theory.The simulations also show similar speed-conditioned scatter distributions and suggest a sharper transition as the expected tree count in the region increases.Source: §VII · p. 15, Figure 19 · p. 16

VIII. CONCLUSION

The paper models flight through a randomly generated forest whose statistics are known but whose obstacles are not known a priori. Under ergodicity, collision-free flight has a critical-speed phase transition, with further bounds and percolation analysis for a special case.

VIII. CONCLUSION

  • The planning problem assumes known obstacle-generation statistics but unknown obstacle locations and shapes, modeling a planar forest of randomly placed disks with random radii.Bird dynamics are parametrized by speed.Source: §VIII · p. 16
  • Under an ergodic forest process, speeds above a critical value admit no infinite collision-free trajectory almost surely, while speeds below it admit at least one.An infinite trajectory diverges toward infinity.Source: §VIII · p. 16
  • For equal-radius Poisson trees and single-integrator dynamics, the paper derives lower and upper critical-speed bounds and an equivalent percolation model.Monte Carlo simulations use the model to approximate the high-speed-flight phase diagram.Source: §VIII · p. 17
  • Future work includes limited-range sensing and estimating the probability of a particular flight distance in the subcritical regime, as well as applying the theory to other spatiotemporal stochastic planning problems.Source: §VIII · p. 17

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