Robot Motion Planning Methods: An...

Robot Motion Planning Methods: An Overview 16 April 2007 1

Robot Motion Planning Methods: AnOverview

Bhaskar Dasgupta

Department of Mechanical EngineeringIndian Institute of Technology Kanpur

[email protected]

http://home.iitk.ac.in/˜ dasgupta

Bhaskar Dasgupta INDI

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http://home.iitk.ac.in/~dasgupta


Outline

1. The Basic Problem

2. Mathematical Formulation

3. Roadmap Methods

4. Cell Decomposition Methods

5. Potential Field Methods

6. Comparison of Methods

7. Extensions of the Basic Problem

8. Summary


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The Basic Problem

We have a rigid object, the robot, of known geometry, capable of freely movingin a workspace that contains a number of fixed rigid objects, calledobstacles, of known geometry and location.

Problem:

Given initial position and orientation and goal position and orientation ofthe robot in the workspace,

generate a path specifying a continuous sequence of positions andorientations of the robot

avoiding contact with the obstacles,

starting at the initial position and orientation and

terminating at the goal position and orientation.

Report failure if no such path exists.


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Mathematical Formulation

• Path Planning and Collision Avoidance → Spatial representation

– Workspace W

– Obstacle space WOi of all obstacles i

– a reference point of the robot

Workspace

Obstacle

Robot


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Def.: Configuration q of a Robot:complete specification of the position of all points of the robot

Def.: Configuration space or C-space Q:set of all possible configurations

• Configuration R(q) in the C-space is represented by a point

• Dimension of the C-space is (usually) the number of degrees of freedom


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• Example: A robot with q1 =

x1

y1

ϕ1

ϕ1y1

x1

y

x

−→

y1

x1

y

x

ϕ1

ϕ


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• Obstacles → Representation in the configuration space

Def.: C-obstacle QOi:Set of configurations, at which the robot collides with (intersects) theobstacle i

QOi = {q ∈ Q |R(q)⋂

WOi 6= ∅}

• Free space: Qfree = Q\ (⋃

i QOi)


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• Example: A robot with translatory motion in the plane

– Workspace and obstacle

Workspace

Obstacle

Robot


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– Configuration space


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– Obstacle in the configuration space


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– Free space


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• Path planning: To find a free path for the robot

– from an initial configuration qStart

– to a final configuration qGoal

– in the free space Qfree

• Mathematicallycontinuous mapping c : [0, 1] → Qfree

such that c(0) = qStart and c(1) = qGoal

• Free path: contact with obstacles not allowed

• Semi-free path: contact with obstacles, though not intersection


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Roadmap Method

• Roadmap: a network or graph of connections

• Only prescribed connections between specific pairs of end-points are de-veloped

• Path finding from Start to Goal

– Collision-free path fromStart to Roadmap

– Path along the Roadmapto the neighbourhood of the Goal

– Collision-free path fromthere to Goal


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• Representation of the Roadmap as a non-directed graph

– Nodes: Salient positions

– Edges: Paths between neighbouring nodes

Def.: Roadmap RM :Union of line segments, such that for all qStart ∈ Qfree and qGoal ∈

Qfree there exist

• a path from qStart to a q′Start ∈ RM

• a path from a q′Goal ∈ RM to qGoal

• a path from q′Start to q′Goal in RM


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• Shortest path (pathway) can be found by the usual algorithms of graph theory

• Construction of the network

– Visibility graph

– Reduced visibility graph

– Generalized Voronoi diagram

– Silhouette method


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Visibility graph

• Defined for

– two-dimensional configuration space

– polygonal C-obstacles

• Nodes vi of the visibility graph

– qStart and qGoal

– vertices of the C-obstacles

• Edges eij

– connect pairs of nodes, that are ‘visible’ from each other

– a line segment joining two such nodes does not intersect any obstacleeij 6= ∅ ⇐⇒ svi + (1 − s)vj ∈ Qfree ∀s ∈ [0, 1]

– edges of obstacles are also taken as edges of the graph


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Example: Visibility graph

Start

Goal


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Reduced visibility graph

• Not all edges of the visibility graph are included

• Only supporting edges and separating edges are used

– Supporting edge: obstacle(s) on the same side

– Separating edge: individual obstacles completely on different sides


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• Example

Start

Goal


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• Example: Reduced visibility graph

Start

Goal


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Generalized Voronoi diagram (GVD)

• Retraction of Qfree: GVD

• Paths maintain clearance from the obstacles

• Two-dimensional C-space

• GVD V of Qfree

– Set of points from Qfree

– Each point of the set at equaldistance from both of itsnearest C-obstacles

• Every point q ∈ Qfree possesses animage ρ(q) ∈ V

ρ ( )1qρ ( )2q1

q1

q2

v

2e


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Silhouette method

• In n-dimensional C-space

– An (n − 1)-dimensional hyperplane is swept through Qalong the remaining dimension,e.g. in xi-direction, andproduces slices

– Extreme points inxj -direction (xj⊥xi) on theslices are marked

– Set of the marked pointsconstitutes the silhouette curves

– Silhouette curves, in general,not connected

x 2

x 1

Slic

e

Extreme points


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Silhouette method

• In n-dimensional C-space

– An (n − 1)-dimensional hyperplane is swept through Qalong the remaining dimension,e.g. in xi-direction, andproduces slices

– Extreme points inxj -direction (xj⊥xi) on theslices are marked

– Set of the marked pointsconstitutes the silhouette curves

– Silhouette curves, in general,not connected

2

1

x

x


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– Slices, at which the number ofsilhouette curves changes: critical slices∗ new (recursive) application of the algorithm with the slice

as (n − 1)-dimensional C-space∗ (n − 2)-dimensional hyperplane sweeps across the critical slice∗ one-dimensional slice→ slice is the silhouetteand recursion terminates

• qStart and qGoal are consideredas extreme points

2

1

x

x


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– Slices, at which the number ofsilhouette curves changes: critical slices∗ new (recursive) application of the algorithm with the slice

as (n − 1)-dimensional C-space∗ (n − 2)-dimensional hyperplane sweeps across the critical slice∗ one-dimensional slice→ slice is the silhouetteand recursion terminates

• qStart and qGoal are consideredas extreme points

2

1

Start

x

x

Goal


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Cell Decomposition Method

• Cell decomposition

• Segmentation of the free space Qfree into cells

• Crossing over possibilities between cells are ascertained based on adjacency

– neighbouring cells: having common boundary

– crossing over only between neighbouring cells

• Representation of the crossing over possibilities as a graph

• Different methods

– Exact cell decomposition

– Approximate cell decomposition


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Exact cell decomposition

• Union of the cells describes the free space Qfree exactly

• Assumptions

– two-dimensional C-space

– polygonal boundary of C-space

– polygonal C-obstacles

• Several possible strategies, e.g.

– Trapezoidal

– Polygonal


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• Example: trapezoidal decomposition, vertical trapeziums

– from all vertices, vertical lines

– example: xy-coordinate system→ Lines parallel to y-axis

– end-points: boundary of C-space or C-obstacle

– enumeration of the cells


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• Example: trapezoidal decomposition, vertical trapeziums

– from all vertices, vertical lines

– example: xy-coordinate system→ Lines parallel to y-axis

– end-points: boundary of C-space or C-obstacle

– enumeration of the cells

c 15

c 2

c 1 c 10

c 14

c 7c 5

c 4

c 3

c 6

c 11

c 13

c 9 c 12

c 8


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– Graph development based on adjacency

– Sequene of cells: channel

– Path finding: line segments between mid-points of the vertical boundari-es

c 15

c 2

c 1 c 10

c 14

c 5c 4

c 3

c 6

c 11

c 13

c 9

c 7

c 12

c 8


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Approximate cell decomposition

• Fundamental segmentation

• Entire free space is segmented into

– simple and

– similar

cells of specified shape

• Cells must be non-overlapping

• Characterization of the cells: covered by C-obstacles

– completely: filled cell

– partly: Mixed cell

– not at all: empty cell


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• Example: identical rectangles as cells

qstart

qgoal

• Only approximate description of the free space possible


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• Maximum error: dependent on the cell size

– too large cell size: bottlenecks may remain unmanoueverable

– too small cell size: huge number of cells, inefficiency

• Hierarchical methods, e.g. Divide-and-label

– Gradual reduction of error

– mixed cells → further subdivision

– Continuation of the method through several levels of hierarchy till the requiredaccuracy is achieved


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• Example: Quadtree

– Begin with root cell, containing the entire free space

– Subdivide the mixed cells hierarchically∗ each dimension of a cell is halved∗ till the required accuracy


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– Efficient storage as a tree: “Quadtree”

∗ Nodes of the tree: cells∗ Root of the tree: the root cell∗ Every mixed cell has four child nodes: NW, NE, SW, SE

– 3-dimensional: Octree

– n-dimensional: 2n-tree


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Potential Field Methods

• Robot (as point in C-space Q) under influence of artificial field forces

– attractive force of the goal

– repulsive force of the C-obstacles

• Artificial forces generated through artificial potential fields


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• C-space described throughpotential function U : Q → R

for every point q ∈ Q

• Example: attractive potential Uatt of the Goal

– Potential strength increases with distance!

– quadratic variation with distance

– Uatt(q) = 1

2ζ ||q − qGoal||

2

– with ζ ∈ R as scale factor


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• Example: repulsive potential Urep of the C-obstacle

– infinite inside the obstacle

– decays fast to zero at a threshold distance d0

– Urep(q) =

1

2η

(

D−1(q) − d−1

0

)2

for D(q) ≤ d0

0 for D(q) > d0

– with D(q) as clearance from nearest obstacle

– with η as scale factor


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• Total potential field: U(q) = Uatt(q) + Urep(q)

• With U differentiable,Force vector F (q) = −∇U(q)

• Robot follows negative gradient −∇U(q)

• Motion of the Robot

– explicit solution of the dynamic equations, or

– incremental: steps in the negative gradient direction→ a real-time process

• Local minima

– navigation around the neighbouring obstacle

– definition of a new potential function

– random deviations

– incorporation of global considerations


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Potential field variants

• Combination with cell decomposition

– operate along pre-planned path based on a priori crude knowledge of theC-space

– take corrective action from potential field based on sensory input

• Off-line applications: global (computation-intensive) considerations

– Variational formulation

– Global navigation function

– Domain mapping


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Probabilistic Roadmaps

• PRM: similar to conventional roadmaps

• Nodes of the graph randomly selected from Qfree

• Every node is connected with “neighbours”

– as long as the connection intersects no obstacle

– number of neighbours: implementation-dependent

• Roadmap can be successivelyrefined/enhanced

• Several strategies to

– locate nodes

– connect them


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Comparison of the Methods

Roadmap Cell decomposition Potential field

VG / GVD / Sil exact / approx local / global

Dimensions 2 / 2 / arbitrary 2 / arbitrary arbitrary

Completeness yes yes / no no / (yes)

Optimality yes / no? / no no no / YES

Complexity strongly dependent on dimension of Q and

obstacles (number, complexity)


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Extensions of the Basic Problem

• Moving obstacles

– a priori knowledge ?

– time as another variable in C-space, or CT-space

– multiple robots

– articulated manipulation robots

• Movable obstacles

– C-space can be modified by the robot

• Kinematic constraints

– Holonomic constraints: reduction of the dimension of Q

– Non-holonomic constraints: car-like robots

• Uncertainities


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Summary

• Motion planning in Robotics

• Classes of methods

– Roadmap methods

– Cell decomposition

– Potential field

• Comparison of methods

• Extensions of the basic problem


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Literature

• Howie Choset, Kevin M. Lynch, Seth Hutchinson, George Kantor, Wolfram Burgard,Lydia E. Kavraki and Sebastian Thrun. Principles of Robot Motion. MIT Press, 2005.

• Jean-Claude Latombe. Robot Motion Planning. Kluwer Academic Publishers, 3.Auflage, 1993.

• S. M. LaValle. Planning Algorithms. Cambridge University Press(http://msl.cs.uiuc.edu/planning/), 2006.

• Gregory Dudek and Michael Jenkin. Computational Principles of Mobile Robotics.Cambridge University Press, 2000.

• Richard M. Murray, Zexiang Li and S. Shankar Sastry. A Mathematical Introduction toRobotic Manipulation. CRC Press, 1994.

• Robert J. Schilling. Fundamentals of Robotics. Prentice Hall, 1990.


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Acknowledgements

Thanks to

• audience of my classrooms of2001, 2003, 2003 (summer), 2005, 2005 (Berlin) and 2006

• Dr.-Ing. habil. Dietmar Tutsch of TU Berlin.

Thank you!


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