Tichakorn Wongpiromsarn

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Two Approaches to Dynamic Refinement in
Hierarchical Motion Planning
AIAA Guidance, Navigation, and Control
Conference August 16, 2005

• Tichakorn Wongpiromsarn
• Undergraduate
• Cornell University

Venkatesh G. Rao Postdoctoral Associate Cornell
University
Raffaelo DAndrea Associate Professor Cornell
University
With input from Thientu Ho
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Motion Planning why is it important?
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Motion Planning why is it important?
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In many applications, time is critical.So we ask
Can we make a formula 1 driver an automatic robot
driver?
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Hierarchical Approach
Abstract planning layer - Decomposes the map
- Decides on a destination - Plans a coarse path
Geometric planning layer - Searches for a
collision-free path based on the information
given by the abstract layer - Repair
dynamically unfeasible path
Dynamic planning layer - Generates a feasible
trajectory for the vehicle based on the
collision- free path
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Dynamic Planning Layer
Given
General task
Compute a feasible trajectory, satisfying certain
requirements, and attempting to optimize some
performance measures.
For this project
Requirements A trajectory must be fully
contained within the same cells as the given
polyline Performance measure Time
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Motivation
How does a formula 1 driver make decisions to
brake and speed up based on his knowledge of the
curves ahead and the behavior of his vehicle?
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Problem Formulation
Given a collision-free path in the form of a
polyline, construct a feasible trajectory, with
an objective of minimizing time, for certain
vehicles such as car-like or omni-directional
vehicles
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Path Refinement with Circular-Arc Motion
Primitives

• Previous work
• Paths made up of line segments connected with
  tangential circular arcs of minimum radius
• Jacobs and Canny 1989 minimize path length
• Chandler et al. 2000 minimize a function of the
  deviation of the refined path from the polygonal
  path
• Beard et al. 2002
• Circular turning arcs to switch from one
  straight-line segment to another in minimum time
  at constant maximum velocity
• The failure risk due to limited deceleration
  capability, as well as the maximum deviation
  constraint is not addressed

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Path Refinement with Circular-Arc Motion
Primitives

• Our Contributions
• Vary the radii of the arcs so as to reduce travel
  time, while respecting a specified maximum
  deviation constraint with respect to the
  polygonal path
• Employ a scheme which permits trading off of
  failure risk against performance important in
  demanding applications for agile vehicles
• A simple repair method which can significantly
  decrease failure risk without significant loss in
  performance.

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Challenge
Given
Compute
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VS

• Big radius
• Potentially faster
• More risk of failure

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Failure Risk VS Speed Tradeoff

• Linear tuning equation
• Discounted horizon scheme
• Tune dependence on future
• Allow trading off of failure risk against
  performance

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Failure Risk VS Speed Tradeoff
Assume that the vehicle always travels as fast as
it can along straight paths and arcs, since this
is consistent with our goal of minimizing time.
General trend the higher the discounting factor
(?), the lower the time and the higher the
failure risk.
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Results

• Example

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Compared to previous work

• Case study (Comparison to Chandlers approach)

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Repair Method

• Reduce the turning radius at the failed vertex to

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So circular arcs are simple to use and give
fairly satisfying results.
BUT
Can the performance be improved given some
advantages such as the omni-directional drive?
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Motivation
How do we take advantage of the dynamically
simpler omni-directional drive to construct a
trajectory for the vehicle in pre-defined
channels?
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Path Refinement with Minimum-Time Motion
Primitives

• Previous work
• Kalmár-Nagy et al. 2001 free-space minimum-time
  maneuvers, with a specified initial velocity and
  zero final velocity
• Frazzoli 2001 minimum-time segments in the
  bottom-up maneuver-automaton method in
  conjunction with randomized road-map path
  planners
• Our contributions
• Use minimum-time segments in a top-down
  architecture, using polygonal paths computed by a
  geometric planner
• Offer greater high-level control (an important
  capability in multiagent domains where vehicles
  must move in precisely coordinated ways)

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Global Problem
D

C
A
B
v01
v02

• Given a polyline connecting a sequence of cells
• Compute a feasible trajectory by joining a
  minimum-time prmitive for each cell chosen so as
  to minimize time of the overall trajectory
• Requirements
• The continuity conditions at the boundary are
  satisfied
• Each primitive is fully contained within the
  corresponding cell

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Local Problem
Given the width and height of a cell and the
starting and terminating points, determine the
allowable set of minimum-time motion primitives
under this condition.
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Preliminary Results

• A safe velocity at entrance
• The initial velocity such that the resulting
  minimum-time trajectory did not hit the boundary
  of the cell
• Can be pre-computed and employed to determine the
  set of minimum-time primitives for each cell.

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Modified Minimum-Time Primitives

• Apply the maximum control effort for some
  specified transfer time in the direction away
  from the side with maximum heuristic estimate of
  probability of being hit by the minimum-time
  trajectory before applying the bang-bang control.

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Modified Minimum-Time Primitives

• Case study
• Increases the area of the safe velocity at
  entrance region by approximately 5

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Conclusions

• Circular-arc motion primitives
• Can be implemented on both car-like and
  omni-directional vehicles.
• The discounted finite horizon scheme allows
  trading off of failure risk against speed
• Faster trajectory and a smoother velocity profile
  compared to that of the previous implementations
• The repair method reduces the risk of failure by
  as much as 40 while adjusting at most 4.5 of
  the turning arcs

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Conclusions (contd.)

• Minimum-time motion primitives
• For omni-directional vehicles only
• A preliminary investigation of the use of these
  minimum-time primitives in a top-down
  architecture
• Safe-velocity curves to determine the set of the
  allowable primitives for each cell
• Our modified approach increases the area of the
  safe velocity region by approximately 5.

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Proposed Future Work

• Circular-arc motion primitives
• Take into account the interdependence between the
  maximum velocity and maximum acceleration
  according to the actuator characteristics.
• A scheme that handles the cooperation of a team
  of vehicles instead of only minimizing time of a
  certain vehicle.
• Minimum-time motion primitives
• A scheme to compute a primitive with non-zero
  terminal velocity to enable its use in a full
  refinement technique comparable to circular arc
  refinement.
• A scheme for choosing a primitive for each cell
  so as to minimize time of the overall trajectory

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Questions?