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Landmark-Based Robot NavigationA. Lazanas, J.-C. Latombe

Presented by Tim Bretl

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Main Idea

• Use backchaining of omnidirectional backprojections to create a plan with guaranteed success despite uncertainty in control and sensing.

Problem—Such planning is intractable in general.Solution—Make assumptions about the environment, resulting in a polynomial-time algorithm.

The backprojection of a robot state S is the set of all states from which the robot is guaranteed to be able to get to S, despite uncertainty.

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Outline

• Assumptions

• Example Problem

• Directional Backchaining

• Omnidirectional Backchaining

• Obstacles

• Extensions (Geometry, Landmark Types)

• Summary and Problems

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Assumptions (1)

• Landmark areas exist in which both control and sensing are perfect. (Outside these areas, control is noisy and sensing is null.)– Otherwise, recursive dependence between a given

state and the backprojection of the state.

A landmark is a workspace feature recognizable from anywhere within its area, or field of influence.

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Assumptions (2)

• Point robot

• Landmark areas are circles or unions of circles

Robot

Landmark Areas

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Assumptions (3)

• Motion commands are of two types:– Perfect (P-Command), inside landmark areas– Imperfect (I-Command), outside landmark areas

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P-Commands

• Inside landmark areas only

• Followed exactly, without errors

Robot

Landmark AreaP-Command

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I-Commands (1)

• Outside landmark areas

• Of the form (d,TC)d = direction of motion

TC = termination condition

• Uncertainty is parameterized by θθ = directional uncertainty of motion

Do not consider time or uncertainty in velocity.

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θ

θ

I-Commands (2)

• Example– (d,TC) = (d1,L2)

Robot

Landmark Area L1

d1

I-Command

Landmark Area L2

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Assumptions (4)

• A kernel of the goal exists that can be recognized independently of how it is achieved.– Otherwise, recursive dependence between goal

recognition and goal preimage (Lozano-Pérez et al., 1984)

• The kernel of the goal is a circle (or a union of circles.)

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Assumptions (5)

• Circular obstacles

• No obstacle collisions are allowed

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Example Problem

StartGoal

I-Command

P-Command

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Planning Algorithm (1)

• Main Idea– Iteratively expand a set of landmark areas from

which the goal is guaranteed to be reachable until their backprojection completely covers the set of possible initial conditions.

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Planning Algorithm (2)

• Iterate…– Given a set of landmark areas E(G) from which the

goal is guaranteed to be reachable.– Expand these areas to their backprojection.– If a landmark area not in E(G) intersects the

backprojection, add it to the set E(G) and record the associated motion commands.

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Planning Algorithm (3)

• Until…– The set of possible initial conditions is completely

contained in E(G) => SUCCESS!– No new landmark areas intersect the

backprojection => FAILURE!

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Planning Algorithm (4)

• Complexity Bounds– Number s of landmark areas – Number l of landmark circles (l ≥ s)

Complete algorithm takes O(sl3logl) time.

Each iteration takes O(l3logl) time.

Maximum number of iterations is s.

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Directional Backprojection (1)

• Project goal regions “backwards in time” along some direction according to a control noise θ.

• If the backprojection intersects a landmark area, the robot is guaranteed to be able to reach the goal from that area.

I-Command

P-Command

Start

Goal

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Directional Backprojection (2)

• The extension E(G) of a goal region G is the largest set of landmark areas intersecting G (so G can be attained with a single P-command.

• The directional backprojection B(G,d) of E(G) is the largest subset of the workspace such that if the I-command (d, E(G) ) is executed from any point in B(G,d), the robot is guaranteed to reach and terminate in the goal G.

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Directional Backchaining

• Backchaining is iterative backprojection.

• Use the backprojection Bi(Gi,d) as the goal Gi+1

for the next iteration, giving Bi+1(Gi+1,d).

Start

G0

G1

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Omnidirectional Backprojection (1)

• Same as before, but now along any direction.

I-Command

P-Command

Start

Goal

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Omnidirectional Backprojection (2)

• Problem—The direction of the backprojection is now arbitrary, so searching through the total omnidirectional backprojection space is hard.

• Solution—Identify critical directions at which the backprojection changes. The total backprojection can be represented using a finite number of directions, one between each consecutive pair of critical directions.

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Critical Directions

• Identifying critical directions by events.

Goal

d1 (L-Left Touch)

d2 (L-Right Exit)

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Events

• E-events: involve extension disks (landmark areas in current E(G))

• I-Events: involve initial-region disks

• L-Events: involve intermediate goal disks (landmark areas that haven’t already been visited)

Result in O(l3) critical directions, where l is the number of landmark disks.

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Summary of Planning Algorithm

• At each iteration…– Calculate critical directions for backprojection.– Compute directional backprojection between each

consecutive pair of critical directions.– Compute new extension set of landmark areas

intersecting the new goal.

Complete algorithm takes O(sl3logl) time.

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More Examples… (1)

θ = 0.1 rad

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More Examples… (2)

θ = 0.2 rad

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More Examples… (3)

θ = 0.3 rad

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Obstacles (1)

StartGoal

The start area no longer intersects the backprojection of the goal area!

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Obstacles (2)

• Changes the backprojection

• Add O-events– O-Left-Touch– O-Right-Exit– …

• Complexity remains O(sl3logl)

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Obstacles (3)

θ = 0.1 rad

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Extensions

• Geometry– With small adaptations, extends to polygonal

landmark areas and obstacles. (Can also add compliant motion.)

• Type of Landmarks– Some uncertainty inside landmarks is ok– Regions that enable reliable navigation to certain

other regions (wall and a corner)

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Summary

• Use backchaining of omnidirectional backprojections to create a plan with guaranteed success despite uncertainty in control and sensing.

• Assume that landmark areas exist where control and sensing are perfect.

• Resulting planning algorithm takes polynomial time.

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Problems

• Relies on the existence of recognizable landmarks.– Engineering the environment.

• Conic model of uncertainty may not be realistic.

• How does GPS change the situation?