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Motion Planning

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Motion Planning & Robot Planning Prof.: S. Shiry Mohsen gandomkar M.Sc Artificial Intelligence Department of Computer Eng. and IT Amirkabir Univ. of Technology – PowerPoint PPT presentation

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Title: Motion Planning


1
Motion Planning Robot Planning
  • Prof. S. Shiry
  • Mohsen gandomkar
  • M.Sc Artificial Intelligence
  • Department of Computer Eng. and IT
  • Amirkabir Univ. of Technology
  • (Tehran Polytechnic)

2
What is Motion Planning?
  • Motion planning is aimed at providing robots with
    the capability of deciding automatically which
    motions to execute in order to achieve their
    tasks without colliding with other objects in
    their work space

3
Basic Definition
  • Obstacles
  • Already occupied spaces of the world
  • In other words, robots cant go there
  • Free Space
  • Unoccupied space within the world
  • Robots might be able to go here
  • To determine where a robot can go, we need to
    discuss what a Configuration Space is

4
The Configuration Space
Configuration Space of A is the space (C) of all
possible configurations of A.
C
Cfree
qgoal
Cobs
qstart
For a point robot moving in 2-D plane, C-space is
5
The Configuration Space
C
y
Cfree
qgoal
Z
Cobs
qstart
x
For a point robot moving in 3-D, the C-space is
What is the difference between Euclidean space
and C-space?
6
The Configuration Space
  • How to create it
  • First abstract the robot as a point object.
    Then, enlarge the obstacles to account for the
    robots footprint and degrees of freedom
  • In our example, the robot was circular, so we
    simply enlarged our obstacles by the robots
    radius (note the curved vertices)

7
Example of a World (and Robot)
8
Configuration Space Accommodate Robot Size
9
Motion Planning
  • Basic problem Collision-free path planning for
    one rigid or articulated object (the robot)
    among static obstacles.
  • Inputs
  • geometric descriptions of the obstacles and the
    robot
  • kinematic and dynamic properties of the robot
  • initial and goal positions (configurations) of
    the robot
  • Output
  • Continuous sequence of collision-free
    configurations connecting the initial and goal
    configurations.

10
Algorithmic Approaches
  • Complete Algorithms
  • Probabilistic Algorithms
  • Heuristic Algorithms

11
Complete Algorithms
  • Guaranteed to find a free path between two give
    configurations when exists and report failure
    otherwise
  • Deal with connectivity of free space by capturing
    it on a graph.
  • Cell Decomposition - partition of free space
  • Roadmap Technique - network of curves

12
Probabilistic Algorithms
  • Trade-off exactness against running time
  • Dont guarantee a solution but if exists very
    likely to find it relatively quickly
  • Example Probabilistic Roadmap Algorithm
  • Experimental results show that computation takes
    less than a second

13
Heuristic Algorithms
  • Many work well in practice but offer no
    performance guarantee
  • Deal with a grid on configuration space
  • Example 1 Potential Field
  • Example 2 Approximate Cell Decomposition

14
Previous Approaches
15
Visibility Graphs
16
Voronoi Diagrams
17
Exact Cell Decomposition
18
Approximate Cell Decomposition
19
Potential Fields
20
Probabilistic Roadmaps Method
21
Problems before PRMs
  • Hard to plan for many dof robots
  • Computation complexity for high-dimensional
    configuration spaces would grow exponentially
  • Potential fields run into local minima
  • Complete, general purpose algorithms are at best
    exponential and have not been implemented

22
Difficulty with classic approaches
  • Running time increases exponentially with the
    dimension of the configuration space.
  • For a d-dimension grid with 10 grid points on
    each dimension, how many grid cells are there?
  • Several variants of the path planning problem
    have been proven to be PSPACE-hard.

10d
23
Probabilistic Roadmap (PRM) multiple queries
free space
Kavraki, Svetska, Latombe,Overmars, 96
24
Probabilistic Roadmap (PRM) single query
25
Multiple-Query PRM
26
Classic multiple-query PRM
  • Probabilistic Roadmaps for Path Planning in
    High-Dimensional Configuration Spaces, L. Kavraki
    et al., 1996.

27
Assumptions
  • Static obstacles
  • Many queries to be processed in the same
    environment
  • Examples
  • Navigation in static virtual environments
  • Robot manipulator arm in a workcell

28
Enter PRMs
  • PRMs use fast collision checking techniques
  • PRMs avoid computing an explicit representation
    of the configuration space
  • Two Phases
  • A Learning Phase
  • A Query Phase

29
The Learning Phase
  • Construct a probabilistic roadmap by generating
    random free configurations of the robot and
    connecting them using a simple, but very fast
    motion planer, also know as a local planner
  • Store as a graph whose nodes are the
    configurations and whose edges are the paths
    computed by the local planner

30
PRM - Learning Phase
31
PRM - Learning Phase
32
PRM - Learning Phase
33
PRM - Learning Phase
34
The Query Phase
  • Find a path from the start and goal
    configurations to two nodes of the roadmap
  • Search the graph to find a sequence of edges
    connecting those nodes in the roadmap
  • Concatenating the successive segments gives a
    feasible path for the robot

35
Two geometric primitives in configuration space
  • CLEAR(q)Is configuration q collision free or
    not?
  • LINK(q, q) Is the straight-line path between q
    and q collision-free?

36
Uniform sampling
Input geometry of the moving object
obstacles Output roadmap G (V, E) 1 V ? ?
and E ? ?. 2 repeat 3 q ? a configuration
sampled uniformly at random from C. 4 if
CLEAR(q)then 5 Add q to V. 6 Nq ? a set
of nodes in V that are close to q. 6 for
each q? Nq, in order of increasing d(q,q) 7
if LINK(q,q)then 8 Add an edge
between q and q to E.
37
Difficulty
  • Many small connected components

38
Resampling (expansion)
  • Failure rate
  • Weight
  • Resampling probability

39
Resampling (expansion)
40
Query processing
  • Connect qinit and qgoal to the roadmap
  • Start at qinit and qgoal, perform a random walk,
    and try to connect with one of the milestones
    nearby
  • Try multiple times

41
Error
  • If a path is returned, the answer is always
    correct.
  • If no path is found, the answer may or may not be
    correct. We hope it is correct with high
    probability.

42
Why does it work? Intuition
  • A small number of milestones almost cover the
    entire configuration space.

43
Smoothing the path
44
Smoothing the path
45
Single-Query PRM
46
Lazy PRM
  • Path Planning Using Lazy PRM, R. Bohlin L.
    Kavraki, 2000.

47
Precomputation roadmap construction
  • Nodes
  • Randomly chosen configurations, which may or may
    not be collision-free
  • No call to CLEAR
  • Edges
  • an edge between two nodes if the corresponding
    configurations are close according to a suitable
    metric
  • no call to LINK

48
Query processing overview
  • Find a shortest path in the roadmap
  • Check whether the nodes and edges in the path are
    collision.
  • If yes, then done. Otherwise, remove the nodes or
    edges in violation. Go to (1).
  • We either find a collision-free path, or exhaust
    all paths in the roadmap and declare failure.

49
Query processing details
  • Find the shortest path in the roadmap
  • A algorithm
  • Dijkstras algorithm
  • Check whether nodes and edges are collisions free
  • CLEAR(q)
  • LINK(q0, q1)

50
Node enhancement
  • Select nodes that close the boundary of F

51
Bug algorithms
52
Bug algorithms
  • Assumptions
  • Point robot
  • Contact sensor (Bug1,Bug2) or finite range sensor
    (Tangent Bug)
  • Bounded environment
  • Robot position is perfectly known
  • Robot can measure the distance between two points

53
Bug algorithms
  • Algorithm consists of two behaviors
  • 1. Motion to goal move toward the goal
  • Bug1 move along the line that connects an
    initial point to the goal until you reach the
    goal or an obstacle (hit point).
  • Bug2 move along the line that connects the start
    point to the goal until you reach the goal or an
    obstacle (hit point).

54
Bug algorithms
  • 2. Boundary following obstacle handeling
  • Bug1 circumnavigate the entire perimeter of the
    obstacle, find the closest point to the goal on
    the perimeter (leave point), move to that point .
  • Bug2 circumnavigate the obstacle until you reach
    a new point on the line connecting start and
    goal, that is closer to the goal (leave point).

55
Bug1 - example
56
Bug2 - example
57
head-to-head comparison
What are worlds in which Bug 2 does better than
Bug 1 (and vice versa) ?
Bug 2 beats Bug 1
Bug 1 beats Bug 2
Start
58
head-to-head comparison
What are worlds in which Bug 2 does better than
Bug 1 (and vice versa) ?
Bug 2 beats Bug 1
Bug 1 beats Bug 2
zipper world
Start
59
Problem
Bug 2 beats Bug 1
Bug 1 beats Bug 2
zipper world
60
Problem
  • use Bug2 until the robot finds itself on the
    S-line farther from the goal than it started
  • if it does, revert to to Bug1 for that obstacle

Bug M1
Adjusted bug algorithm
61
Problem
  • use Bug2 until the robot finds itself on the
    S-line farther from the goal than it started
  • if it does, revert to to Bug1 for that obstacle

Bug M1
Adjusted bug algorithm
62
Bug1 vs. Bug2
  • Bug1
  • Exhaustive search
  • Optimal leave point
  • Performs better with complex obstacles
  • Path length
  • n of obstacles
  • Pi perimeter of obstacle i
  • Bug2
  • Opportunistic (greedy) search
  • Performs better with simple obstacles
  • Path length
  • ni of times the start-goal line intersects
    obstacle i

63
Finite range sensor
  • Intervals of continuity

64
Tangent Bug algorithm
  • Improvement to the Bug2 algorithm
  • Assumptions
  • All assumptions of Bug1/Bug2 except for contact
    sensor.
  • Finite range sensor with 360? infinite
    orientation resolution.

65
Tangent Bug algorithm
  • Like Bug1/Bug2, iterates between two behaviors
  • motion to goal consists of two parts
  • Move in a straight line towards the goal until
    you sense an obstacle directly between you and
    the goal
  • Move toward an intermediate point Oj according
    to some heuristic distance until you reach the
    goal or until you reach a local minimum Mi in
    which case, switch to boundary following
  • Ojs are end points of an interval of
    continuity
  • For example d(x, Oj) d(Oj,goal)

66
Tangent Bug algorithm
Motion to goal
67
Tangent Bug algorithm
  • boundary following define two distances
  • dfollowing The shortest distance between the
    sensed boundary and the goal
  • dreach The distance between the point on the
    boundary that has a line of sight to the goal,
    and the goal
  • continue moving around the obstacle in the same
    direction until dreach lt dfollowing then switch
    to motion to goal

68
Tangent Bug algorithm
Boundary following
Motion to goal
goal
M
69
Tangent Bug - example
qgoal
qstart
Motion to goal
Boundary following
70
Bug algorithms
  • Simple and intuitive
  • Straightforward to implement
  • Success guaranteed (when possible)
  • Assumes perfect positioning and sensing
  • Sensor based planning has to be incremental and
    reactive

71
Multi-Robot Planning
72
Multi-Robot Planning Examples
73
Multi-Robot Planning
  • An initial and a goal configuration are given as
    input for each robot
  • Result is a coordinated path between the two
    configurations
  • A coordinated path is one that indicates the
    configuration of every robot at each instant
  • Collisions must be avoided between each pair of
    robot and obstacles, and between each pair of
    robots

74
Centralized Planning
  • Paths for all robots are planned simultaneously
    by searching the c-space of the multi-arm robot
  • Collisions between robots are self-collisions of
    the multi-arm robot
  • For spot-welding example, 6 robots each with 6
    dofs, so C will have 36-D

75
Centralized Planning
  • Advantages
  • Complete guaranteed to find a solution if one
    exists (if the underlying planner is complete)
  • Disadvantages
  • Potentially expensive typically requires
    searching high-dimensional spaces
  • Requires knowledge of goals and states of all
    robots

76
Decoupled Planning
  • First Phase - a collision-free path ti is
    generated for each robot considering only
    obstacles (ignoring other robots) in its space

77
Decoupled Planning
  • Second Phase (Velocity Tuning) coordination of
    the robots velocities along their pre-generated
    paths to prevent collisions between robots. Two
    coordination methods discussed
  • Pairwise Coordination
  • Global Coordination
  • Each robot is restricted to motion in its
    pre-generated path although it may stop, retreat
    or change velocity to allow coordination with
    other robots

78
Decoupled Planning with Pairwise Coordination
  • The paths t1 and t2 of the first two robots are
    coordinated in their 2-dimensional coordination
    space
  • Results in a collision-free coordinated patht1,2

Done by using planning a pathbetween (0,0) and
(1,1)
79
Decoupled Planning with Pairwise Coordination
  • The process is repeated for paths t1,2 and t3
    resulting in a coordinated path t1,2,3
  • Eventually a collision-free coordinate path
    t1,2,,m is generated that defines a valid
    coordination of all m robots

80
Decoupled Planning with Global Coordination
  • The paths of all m robots are coordinated in an
    m-dimensional coordination space
  • Results in a collision-free path t1,2,.m

Done by planning a path from (0,0,0,) to
(1,1,1,)
81
Decoupled Planning
  • Advantages
  • Less expensive than centralized planning because
    lower dimensional spaces are searched
  • Disadvantages
  • Incomplete Failures usually occur in the second
    phase as it might not be possible to coordinate
    the paths generated in the first phase without
    collision between robots

82
Decoupled Planning Failure Example
  • Initial and goal configurations

83
Decoupled Planning Failure Example
  • Likely path generation in 1st phase

84
Decoupled Planning Failure Example
  • Path coordination fails in second phase

85
Implemented Planners
  • C-SBL Centralized Planning
  • DG-SBL Decoupled Planning with Global
    Coordination
  • DP-SBL Decoupled Planning with Pairwise
    Coordination
  • Experiments conducted with groups of 2, 4 and 6
    robots on 3 separate sets of initial/goal
    configurations

86
PRM Path Planner Sampling Strategy
  • SBL Planner
  • Single-query
  • Bi-directional
  • Lazy collision-checking

87
Problem I Initial and goal configurations
88
Problem II Initial and goal configurations
89
Problem III Initial and goal configurations
90
Experimental Results
  • T average running time (seconds)
  • DG-SBL and DP-SBL - 20 runs per experiment
  • C-SBL 100 runs per experiment
  • F number of failures
  • Maximum of 50,000 milestones allowed per call to
    SBL

91
Experimental Results
  • Centralized planning had no failures
  • At least one failure suffered in each experiment
    with decoupled planning
  • Failure rate increased as problems became more
    complex

92
Experimental Results
  • pairwise coordination more unreliable than
    global coordination
  • Failure always occurred in the 2nd stage during
    path coordination, a result of wrong path choices
    made in the 1st stage

93
Experimental Results
  • Similar running times for both planners in most
    experiments
  • However, centralized planning required a lot more
    time than decoupled planning in 3rd problem with
    6 robots

94
Conclusions
  • Reliability Decoupled planning can be quite
    unreliable particularly in tight robot
    coordination. Centralized planning appears to
    have perfect reliability.
  • Planning Time Using SBL, there is not a huge
    difference between the two methods

95
Conclusions Contd.
  • Results invalidate the assumptions that loss of
    incompleteness with decoupled planning is fairly
    insignificant and can be ignored in practice.
  • SBL makes usage of centralized planning for
    multi-robot systems practical.
  • But centralized planning still requires knowledge
    of all robot states, which may be impossible in
    some settings.

96
Sokoban
  • Objective of Robot
  • To push boxes into their storage locations
    without getting himself or boxes stuck.
  • Rules Cannot pull, can push only one box at a
    time

97
Sokoban
98
Sample Sokoban Game
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