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Motion Planning for Motion Planning for Car-like Robots using a Car-like Robots using a Probabilistic Learning Probabilistic Learning

ApproachApproach

--P. Svestka, M.H. Overmars. --P. Svestka, M.H. Overmars. Int. J. Robotics ReseInt. J. Robotics Researcharch, 16:119-143, 1997. , 16:119-143, 1997. Presented by: Presented by: Li YunzhenLi Yunzhen

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Paper’s Motivation & Organization

Motivationbuild a non-redundant of milestones (randomized), apply non-holonomic constraints for car-robot to do multi-query processing

Organization1.Two types of Car Robots and nonholonomic constraints2.Probabilistic Roadmap3.Application of Forest uniform Sampling in General Car-like Robot4.Application of Directed Graph uniform Sampling in Forward Car-like Robot5 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 R

)2

(max

),,( yx

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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 achievedThus, 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 RobotPath Dependent

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1. 1. Nonholonomic Car-Like RobotCar-Like Robot

yy

xx

L

q = (x,y,)q’= dq/dt = (dx/dt,dy/dt,d/dt)dx sin – dy cos = 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 dy/dt = v sin ddt = (v/L) tan

| <

dx sin – dy cos = 0

dydSdxdS

sincos

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1. 1. Nonholonomic Car-Like RobotCar-Like Robot

yy

xx

L

Upper bound turning angle=>Lower-bounded turning radius Rmin = Lctg

dx/dt = v cos dy/dt = v sin ddt = (v/L) tan

| <

dx sin – dy cos = 0

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1.Two Types of car-like Robots under Non-Holonomic ConstraintsNormal 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 RoadmapLearning 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 SamplingNon-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 walksLw: 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=10Lw=0.05So time spend on per query is bounded by 0.3 s.

Minimal turning radius: Rmin = 0.1Neighborhood size: Maxdist =0.5The 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 + directionMetric (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 ?