CS 326 A: Motion Planning robotics.stanford.edu/~latombe/cs326/2004/index.htm Configuration Space.
By Jerome Barraquand and Jean-Claude Latombe Presenter: Yubin Zou
description
Transcript of By Jerome Barraquand and Jean-Claude Latombe Presenter: Yubin Zou
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Nonholonomic Multibody Mobile Robots:Controllability and Motion Planning in the Presence of Obstacles
By Jerome Barraquand and Jean-Claude Latombe
Presenter: Yubin Zou
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What is the problem?•The motion of Multi-body mobile robot
▫See it as Multi-body mobile vehicle▫tractors towing several trailers sequentially
hooked•The controllability of its motion
▫How to control multi-body vehicle for avoiding the obstacles and reaching the specific position
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Outline•Configuration state and Configuration space•The constraint among bodies
▫Constraint equations•The structure of planner
▫Tree Searching▫Three key parameters
•Experiment Result
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Configuration State and configuration space• The simplest example: one-body vehicle
• In this case, the configuration state is (x, y,θ), so the configuration space is 3-dimension.
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Extend to two-body vehicle•Two-body vehicle
• The configuration state is (x, y,θ1,θ2), in other word, the configuration space is 4-dimension.
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Extend to n-body vehicle•N-body vehicle
The configuration state is (x, y,θ1,θ2,…, θn), so the configuration space of n-body vehicle is n+2 dimension. Therefore, velocity state is (x’, y’,θ1’,θ2’,…, θn’), they are the velocity component following different dimensions.
•what is the relationship among (x, y,θ1,θ2,…, θn) and (x’, y’,θ1’,θ2’,…, θn’)? constraint among bodies▫How other bodies move when the tractor is
moving?
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The constraint among bodies
• The simple example: one-body vehicle
• Let [x’, y’, θ1’] represent the decomposition of velocity vector V, and x’, y’, θ1’ is the velocity component following different dimensions
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Extend to n-body vehicle•Here is the distance from kth body to
(k+1)th body.
n
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Extend to n-body vehicle•For a configuration state, (x, y,θ1,θ2,…,
θn) is known, then (x’, y’,θ1’,θ2’,…, θn’) can be obtained through three equations
only when are known.
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Planner•So far, we have already constructed the
configuration space and known the constraint among parameters.
•Tree Searching▫The discretization of the continuous
configuration space (Decompose configuration space into a multiple of cells)
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Cell•The number of cells:
▫2^R(n+2) where n +2 is the number of dimension, and R is the resolution of decomposition, the bigger R make each cell smaller.
•Each cell has equal size▫(∆x, ∆y, ∆θ1, ∆θ2,…, ∆θn)
•One cell is explored▫One cell is said to be explored when it contains
a configuration state which has already been expanded.
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Tree Searching•The initial State: P (x, y,θ1,θ2,…, θn), •The goal State: P’•The successor function:
Using three functions:
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▫After getting (x’, y’,θ1’,θ2’,…, θn’), integrate them with the constant internal time ∆t.
▫We can obtain the metric distance L between the expanded state and the successor state, then adding L to the expanded state can obtain the successor state
(x, y,θ1,θ2,…, θn)
(x’, y’,θ1’,θ2’,…, θn’)
If x’, y’,θ1’,θ2’,…, θn’ are constant, then the successor state is (x + x’∆t, y + y’∆t, θ1+ θ1’∆t, θ2 + θ2’∆t,…, θn + θn’∆t)
ActionExpanded state
Tree Searching(continued)
input
parameters
output
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Tree Searching(continued)▫After getting (x’, y’,θ1’,θ2’,…, θn’), integrate
them with the constant internal time ∆t.▫We can obtain the metric distance L between
the expanded state and the successor state, then adding L to the expanded state can obtain the successor state
•Actions:▫The number of actions |V|* depends on r
Because , r is the discretization parameters given to searching. are selected from [-90,90].
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Tree Searching and planner Rules• End Searching Condition
▫The current configuration state is in the same cell with the goal configuration state (Finish Task!)
• Searching Depth▫It cuts searching at the depth H.
• Three key parameters of this planner▫R: The resolution of configuration decomposition
Decide the search space▫H: Tree Searching Depth▫∆t: the interval time between current control and
next control ▫These three parameters depend on the robot
precision
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Experiment Results•R = 8
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Experiment Results•R = 8,
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Experiment Results•R = 9,
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Thank you!