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
• (Tehran Polytechnic)

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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

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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

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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
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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?
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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)

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Example of a World (and Robot)
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Configuration Space Accommodate Robot Size
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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.

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Algorithmic Approaches

• Complete Algorithms
• Probabilistic Algorithms
• Heuristic Algorithms

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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

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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

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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

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Previous Approaches
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Visibility Graphs
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Voronoi Diagrams
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Exact Cell Decomposition
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Approximate Cell Decomposition
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Potential Fields
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Probabilistic Roadmaps Method
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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

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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
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Probabilistic Roadmap (PRM) multiple queries
free space
Kavraki, Svetska, Latombe,Overmars, 96
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Probabilistic Roadmap (PRM) single query
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Multiple-Query PRM
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Classic multiple-query PRM

• Probabilistic Roadmaps for Path Planning in
  High-Dimensional Configuration Spaces, L. Kavraki
  et al., 1996.

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Assumptions

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

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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

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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

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PRM - Learning Phase
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PRM - Learning Phase
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PRM - Learning Phase
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PRM - Learning Phase
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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

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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?

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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.
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Difficulty

• Many small connected components

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Resampling (expansion)

• Failure rate
• Weight
• Resampling probability

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Resampling (expansion)
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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

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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.

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Why does it work? Intuition

• A small number of milestones almost cover the
  entire configuration space.

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Smoothing the path
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Smoothing the path
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Single-Query PRM
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Lazy PRM

• Path Planning Using Lazy PRM, R. Bohlin L.
  Kavraki, 2000.

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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

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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.

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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)

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Node enhancement

• Select nodes that close the boundary of F

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Bug algorithms
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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

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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).

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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).

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Bug1 - example
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Bug2 - example
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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
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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
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Problem
Bug 2 beats Bug 1
Bug 1 beats Bug 2
zipper world
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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
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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
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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

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Finite range sensor

• Intervals of continuity

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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.

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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)

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Tangent Bug algorithm
Motion to goal
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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

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Tangent Bug algorithm
Boundary following
Motion to goal
goal
M
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Tangent Bug - example
qgoal
qstart
Motion to goal
Boundary following
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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

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Multi-Robot Planning
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Multi-Robot Planning Examples
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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

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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

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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

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Decoupled Planning

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

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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

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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)
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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

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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,)
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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

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Decoupled Planning Failure Example

• Initial and goal configurations

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Decoupled Planning Failure Example

• Likely path generation in 1st phase

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Decoupled Planning Failure Example

• Path coordination fails in second phase

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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

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PRM Path Planner Sampling Strategy

• SBL Planner
• Single-query
• Bi-directional
• Lazy collision-checking

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Problem I Initial and goal configurations
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Problem II Initial and goal configurations
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Problem III Initial and goal configurations
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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

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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

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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

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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

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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

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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.

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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

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Sokoban
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Sample Sokoban Game