A Geometric Approach to Error Detection and Recovery for Robot Motion Planning with Uncertainty

1
A Geometric Approach to Error Detection and
Recovery for Robot Motion Planning with
Uncertainty

• Bruce R. Donald
• Artificial Intelligence 37 ( 1988 )

2
Presentation Outline

• Motion Planning with Uncertainty
• Generalized Configuration Space G
• Physical Axioms on G
• A Preimage Backchaining Planner
• Error Detection and Recovery
• Formulation
• An Example
• Conclusion Discussion

3
In Presence of Uncertainty

• Uncertainty is Everywhere in Real World
• Sensing Errors ( Error Ball )
• Control Errors ( Velocity Cone )
• Model Uncertainty ( Generalized Configuration
  Space )

G
A
B
?
4
Generalized Configuration Space

• G C ? J where J ? parameterizes the
  possible variations in the environment C? is the
  config. space of the possible universe indexed by
  ?.

B
y
J
G
A
x
5
Axioms for Physics in G

• Need to represent motions, forces, velocities in
  G ( as in classical mechanics )
• there is only one real universe, fixed but
  unknown
• uncertainty balls a cylinder B ? J ? C ? J
• velocity cone cone ? J ? TC ? J
• Generalized damper dynamics ( Erdmann ) and
  friction cones

6
Changing the Environment

• Generalize the framework to model the interaction
  between the robot and the environment
• Gross motion
• Compliant motion
• Pushing motion
• Interaction in terms of forces in G exerted on
  surfaces with a normal

7
Example

• Apply a force on Block B, B will start moving to
  the left
• In the generalized configuration space, ...

f
G
A
B
B
?
8
Example

• The reaction force fr normal to the surface S,
  has a component along J the robot slides on S

S
motion
y
f
J
fr
x
9
A Backchaining Planner on G

• LMT framework of preimage backchaining
• Backprojection (Erdmann) ignore recognizability
• Can we generalize preimages and backprojections
  to G?
• Yes!! The physics of G has been specified
• model uncertainty as extra degrees of freedom

10
Example

• When ? gt 0

backprojection
A
B
?
11
Example

• In G

J
12
Example

• When ? lt -size(B)

backprojection
A
13
Example

• In G

14
Error Detection and Recovery

• There may exist no guaranteed plan.
• Errors that are explicitly handled in a
  guaranteed plan is just a conditional branch.
• The only error is being in the wrong universe

preimage
G
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EDR Strategy

• Given a plan, develop an EDR strategy that might
  work, and fail in a reasonably way if cannot
• Signal failure when and only when there is
  absolutely no chance to achieve the goal
• Define bad EDR regions in which the robot
  should signal failure
• weak preimage
• forward projection
• bad region forward projection - weak preimage
  ?

16
Example
control motion
forward projection
17
Example
weak preimage
18
Example
bad region
H
H
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More EDR

• Other bad regions the robot is in the weak
  preimage but can never achieve the goal
• zero velocity
• loop motion
• n-Step EDR strategy
• one-step EDR strategy may not exist
• keep executing the plan and distinguish success
  from failure later on

20
Experiments

• EDR planner LIMIT
• approximate preimages using backprojections
• compute slice approximations to EDR regions
• Example Gear meshing problem ( G has dimension
  21 )

21
Conclusion Discussion

• Contributions
• use generalized configuration space to handle
  model uncertainty
• formalization and geometric characterization of
  error detection and recovery
• Other ways to deal with uncertainty, given
  todays technology?