This document presents a probabilistic learning approach to motion planning for car-like robots using a probabilistic roadmap method. It describes two types of car-like robots that are subject to non-holonomic constraints, and how a probabilistic roadmap is constructed to capture the connectivity of the robot's configuration space. The approach is evaluated experimentally for both general and forward-moving car-like robots, demonstrating its effectiveness at solving multi-query motion planning problems for these systems.

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Motion Planning for Car-Motion Planning for Car-
like Robots using alike Robots using a
Probabilistic LearningProbabilistic Learning
ApproachApproach
--P. Svestka, M.H. Overmars.--P. Svestka, M.H. Overmars. Int. J. RoboticsInt. J. Robotics
ResearchResearch, 16:119-143, 1997., 16:119-143, 1997.
Presented by:Presented by:
Li YunzhenLi Yunzhen

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Paper’s Motivation & Organization
Motivation
build a non-redundant of milestones (randomized), apply non-holonomic
constraints for car-robot to do multi-query processing
Organization
1.Two types of Car Robots and nonholonomic constraints
2.Probabilistic Roadmap
3.Application of Forest uniform Sampling in General Car-like Robot
4.Application of Directed Graph uniform Sampling in Forward Car-
like Robot
5 Summary

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1.Car-Like Robots: Configuration
 Configuration Space:
 Front point F
 Rear point R
 Maximal steering angle
 configuration
]2,0[2
π×R
)
2
(max
π
ψ <
),,( θyx

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1.Car-Like Robots
 Translational motion: along main axis
 Rotational motion: around a point on A’s
perpendicular axis. Rotational angle is decided
by forward and backward motion

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1. Holonomic Constraints--Free flying
robot
Its motions are of a holonomic nature
infinitesimal motion in
Cfree-space can be
achieved
Thus, path independent

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1 Nonholonomic Constraints
 the number of degrees of freedom of motion
is less than the dimension of the
configuration space
 Path dependent (collision-free path not
always feasible)

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1.Nonholonomic Constraints
—Forward car-like Robot
Start
Not possible for forward
Car-like Robot
Path Dependent

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1.1. Nonholonomic Car-Like RobotCar-Like Robot
yy
xx
θθ
φ
φ
L
q = (x,y,θ)
q’= dq/dt = (dx/dt,dy/dt,dθ/dt)
dx sinθ – dy cosθ = 0 is a particular form of f(q,q’)=0
A robot is nonholonomic if its motion is constrained by a non-
integrable equation of the form f(q,q’) = 0
dx/dt = v cos θ
dy/dt = v sin θ
dθ/dt = (v/L) tan φ
||φ| < Φ
dx sinθ – dy cosθ = 0
dydS
dxdS
=×
=×
θ
θ
sin
cos

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1.1. Nonholonomic Car-Like RobotCar-Like Robot
yy
xx
φ
φ
L
Upper bound turning angle
=>Lower-bounded turning radius
Rmin = Lctg
dx/dt = v cos θ
dy/dt = v sin θ
dθ/dt = (v/L) tan φ
||φ| < Φ
dx sinθ – dy cosθ = 0
φ
θθ

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1.Two Types of car-like Robots under
Non-Holonomic Constraints
Normal Car-like Robot:
Move Forwards &
Backwards, (Bounded) turn,
cannot move sidewise
Forwards Car-like Robot:
Move Forwards , (Bounded)
turn, cannot move sidewise

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2. Probabilistic Roadmap
Learning Phase:
Local Method: used to compute a feasible path for
connection of 2 nodes. deterministic & terminative
Metric: determine the distance of 2 nodes
Edge adding Methods:
Cycle detection & try to connect nodes within maximum
dist to avoid failure
Query Phase: start from start position and goal position, do
random walk
For Holonomic Constraints, Local method can return any path as
long as it does not intersects with obstacles. (Local method
returns line-segments in Lecture notes)

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2.Forest Uniform Sampling
Non-redundant Property:
From one node to another
node, there is only one or
no path

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2. Directed Graph uniform sampling
Similar to Forest Sampling.
Redundant Checking: An edge e=(a,b) in a Graph
G=(V,E) is redundant iff there is a directed path from a
to b in the graph G=(V,E-e).

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3.Apply Undirected graph to general
car-like robot
 Link method: constructs a path connecting its
argument configurations in the absence of obstacles,
and then test whether this path intersects any
obstacles.
 RTR path: concatenation of an extreme rotational path,
a translational path, and another extreme rotational
path.

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3.Apply Undirected graph to general
car-like robot
Two RTR paths for a triangular car-like robot,
connecting configurations a,b
RTR link method: given two
argument configurations a and
b, if the shortest RTR path
connecting a to b intersects no
obstacles, return the path, else
return failure.
RTR metric (DRTR): distance
between two configurations is
defined as the length of the
shortest RTR path connecting them.

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3.Apply Undirected graph to general
car-like robot---Query phase
Nw: maximal number of walks
Lw: maximal length of the walk( used for upper bound
of RTR metric)
Use these two constraints to upper-bound the random
walk

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3.General car-like robot: Node Adding
Strategy
 Random Node Adding
 Non-Random Node Adding: guiding the node adding by
the geometry of the workspace

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3.General car-like robot: guiding the node adding by
the geometry of the workspace
 Random Node adding strategy
 1.Compute Geometry Configurations at important
position, e.g. along edges, next to vertices of obstacles.
Each edge and convex vertex defines two such geo-
configurations.

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3.General car-like robot: guiding the node adding by
the geometry of the workspace
 2. Add configurations from Geo-Configuration set (just
computed) in a random order to the graph, but discard
those are not free.
 3. Learning Process can be continued by adding random
nodes.

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3.General car-like robot:
Experiments(1)
Experimental Set up:
Random Walk parameter:
Nw=10
Lw=0.05
So time spend on per query is bounded by 0.3 s.
Minimal turning radius: Rmin = 0.1
Neighborhood size: Maxdist =0.5
The percentage number in the table shows how many percent of
trials of query is solved.

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3.General car-like robot:
Experiments(1)
The lower left table gives results for geometric node adding, the
table at the lower right for random node adding.

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3.General car-like robot:
Experiments(2)
The lower left table gives results for geometric node adding, the
table at the lower right for random node adding

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3.General car-like robot:
Experiments(3)
The lower left table gives results for geometric node adding, the
table at the lower right for random node adding

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3.General car-like robot:
Experiments(4)
Parking with large minimal turning radii. In the left case rmin is
0.25 and in the right case 0.5

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4.Forward car-like robot
RTR forward path: the concatenation of extreme
forward rotational path, a forward translational
path and another extreme forward rotational path.
RTR forward link method: RTR link method +
direction
Metric (RTR forward metric): RTR
metric+direction

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4.Forward car-like robot
Why do we need to build directed graph?
The red RTR path does not suitable for forward
car-like. So directed edge refers to directed RTR
path.

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4.Forward car-like robot
The table gives result for random node adding

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4.Forward car-like robot
The table gives result for geometric adding

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5.Summary
Apply Non-redundant Graph roadmap for the
motion of car-like robots.
Why not build redundant graph roadmap?
--After smoothing, redundant graph and non-
redundant graph will general similar results.

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Q&A
 ?