introduction to mobile robots motion planning prof.: s. shiry pooyan fazli m.sc computer science...
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![Page 1: Introduction to Mobile Robots Motion Planning Prof.: S. Shiry Pooyan Fazli M.Sc Computer Science Department of Computer Eng. and IT Amirkabir Univ. of](https://reader035.vdocument.in/reader035/viewer/2022062314/56649d775503460f94a58e4a/html5/thumbnails/1.jpg)
Introduction to Mobile Robots Motion Planning
Prof.: S. ShiryPooyan Fazli
M.Sc Computer ScienceDepartment of Computer Eng. and IT
Amirkabir Univ. of Technology(Tehran Polytechnic)
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What is Motion Planning?
Determining where to go
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The World consists of...
Obstacles Already occupied spaces of the world In other words, robots can’t 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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Notion of Configuration Space
Main Idea: Represent the robot as a point, called a configuration, in a parameter space, the configuration space (or C-space).
Importance: Reduce the problem of planning the motion of a robot in Euclidean Space to planning the motion of a point in C-space.
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C-space of Rigid Object
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C-space of Rigid Object
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C-space of Rigid Object
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C-space of Rigid Object
Robot Configurations Can be: 1. Free configurations: robot and obstacles do not overlap 2. Contact configurations: robot and obstacles touch 3. Blocked configurations: robot and obstacles overlap
Configuration Space partitioned into free (Cfree), contact, and blocked sets.
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Obstacles in C-space
Cspace = Cfree + Cobstacle
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Example of a World (and Robot)
Obstacles
Free Space
Robot
x,y
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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 robot’s footprint and degrees of freedom
In our example, the robot was circular, so we simply enlarged our obstacles by the robot’s radius (note the curved vertices)
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Configuration Space: Accommodate Robot Size
Obstacles
Free Space
Robot(treat as point object)x,y
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Paths in C-Space
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Motion Planning
General Goal: compute motion commands to achieve a goal arrangement of physical objects from an initial arrangement
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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Motion Planning
Path planningdesign of only geometric (kinematic) specifications of the positions and
orientations of robots
Trajectory = Path + assignment of time to points along the path
Trajectory planning path planning + design of linear and angular velocities
Motion Planning (MP), a general term, either: Path planning, or Trajectory planning
Path planning < Trajectory planning
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Extensions to the Basic Problem
movable obstacles moving obstacles multiple robots incomplete knowledge/uncertainty in geometry, sensing, etc.
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Multiple Robots,Moving/Movable Obstacles
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Classification of MP algorithms
Completeness
Exact usually computationally expensive
Heuristic aimed at generating a solution in a short time may fail to find solution or find poor one at difficult problems important in engineering applications
Probabilistically complete (probabilistic completeness 1)
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Classification of MP algorithms
Scope
Global take into account all environment information plan a motion from start to goal configuration
Local avoid obstacles in the vicinity of the robot use information about nearby obstacles only used when start and goal are close together used as component in global planner, or used as safety feature to avoid unexpected obstacles not present in
environment model, but sensed during motion
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Classification of MP Algorithms
Point-to-point
Region filling
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Point-to-point Path Planning
Point-to-point path planning looks for the best route to move an entity from point A to point B while avoiding known obstacles in its path,
not leaving the map boundaries, and not violating the entity's mobility constraints. This type of path planning is used in a large number of robotics and gaming applications such as finding routes for autonomous robots, planning the manipulator's movement of a stationary robot, or for moving entities to different locations in a map to accomplish certain goals in a gaming application.
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Region Filling Path Planning
Tasks such as vacuuming a room, plowing a field, or mowing a lawn require region filling path planning operations that are defined as follows:
1) The mobile robot must move through an entire area, i.e., the overall travel must
cover a whole region.
2) Continuous and sequential operation without any repetition of paths is required of the robot.
3) The robot must avoid all obstacles in a region.
4) An "optimal" path is desired under the available conditions.
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Motion Planning Statement
The Problem: Given an initial position and orientation POinit
Given a goal position and orientation POgoal
Generate: continuous path t from POinit to POgoal
t is a continuous sequence of Pos’
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The Wavefront Planner
A common algorithm used to determine the shortest paths between two points In essence, a breadth first search of a graph
For simplification, we’ll present the world as a two-dimensional grid
Setup: Label free space with 0 Label C-Obstacle as 1 Label the destination as 2
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Representations: A Grid
Distance is reduced to discrete steps For simplicity, we’ll assume distance is uniform
Direction is now limited from one adjacent cell to another
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Representations: Connectivity
8-Point Connectivity 4-Point Connectivity
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The Wavefront Planner: Setup
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The Wavefront in Action (Part 1)
Starting with the goal, set all adjacent cells with “0” to the current cell + 1 4-Point Connectivity or 8-Point Connectivity? Your Choice. We’ll use 8-Point Connectivity in our
example
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The Wavefront in Action (Part 2)
Now repeat with the modified cells This will be repeated until no 0’s are adjacent to cells with values >=
2 0’s will only remain when regions are unreachable
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The Wavefront in Action (Part 3)
Repeat again...
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The Wavefront in Action (Part 4)
And again...
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The Wavefront in Action (Part 5)
And again until...
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The Wavefront in Action (Done)
You’re done Remember, 0’s should only remain if unreachable regions exist
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The Wavefront, Now What?
To find the shortest path, according to your metric, simply always move toward a cell with a lower number
The numbers generated by the Wavefront planner are roughly proportional to their distance from the goal
Twopossibleshortest
pathsshown
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Map-Based Approaches: Roadmap Theory
Idea: capture the connectivity of Cfree with a roadmap (graph or network) of one-dimensional curves
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Roadmap Methods
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Roadmap Methods
Properties of a roadmap: Accessibility: there exists a
collision-free path from the start to the road map
Departability: there exists a collision-free path from the roadmap to the goal.
Connectivity: there exists a collision-free path from the start to the goal (on the roadmap).
Examples of Roadmaps– Visibility Graph – Generalized Voronoi Graph (GVG)
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Roadmap: Visibility Graph
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Visibility Graph of C-Space
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The Visibility Graph in Action (Part 1)
First, draw lines of sight from the start and goal to all “visible” vertices and corners of the world.
start
goal
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The Visibility Graph in Action (Part 2)
Second, draw lines of sight from every vertex of every obstacle like before. Remember lines along edges are also lines of sight.
start
goal
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The Visibility Graph in Action (Part 3)
Second, draw lines of sight from every vertex of every obstacle like before. Remember lines along edges are also lines of sight.
start
goal
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The Visibility Graph in Action (Part 4)
Second, draw lines of sight from every vertex of every obstacle like before. Remember lines along edges are also lines of sight.
start
goal
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The Visibility Graph (Done)
Repeat until you’re done.
start
goal
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Reduced Visibility Graphs
Idea: we don't really need all the edges in the visibility graph (even if we want shortest paths)
Definition: Let L be the line passing through an edge (x,y) in the visibility graph G. The segment (x,y) is a tangent segment iff L is tangent to CB at both x and y.
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Reduced Visibility Graphs
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Roadmaps:the Retraction Approach
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Roadmaps:the Retraction Approach
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Retraction Example: Generalized Voronoi Diagrams
A GVG is formed by paths equidistant from the two closest objects
This generates a very safe roadmap which avoids obstacles as much as possible
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Retraction Example: Generalized Voronoi Diagrams
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Retraction Example: Generalized Voronoi Diagrams
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Generalized Voronoi Diagrams
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Generalized Voronoi Diagrams
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Generalized Voronoi Diagrams
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Generalized Voronoi Diagrams
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Cell Decomposition Methods
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Cell Decomposition Methods
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Exact Cell Decomposition Method
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Exact Cell Decomposition Method
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Path Planning with a Convex Polygonal Decomposition
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Exact Cell Decompositions: Trapezoidal Decomposition
A way to divide the world into smaller regions Assume a polygonal world
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Exact Cell Decompositions: Trapezoidal Decomposition
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Applications: Coverage
By reducing the world to cells, we’ve essentially abstracted the world to a graph.
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Find a path
By reducing the world to cells, we’ve essentially abstracted the world to a graph.
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Find a path
With an adjacency graph, a path from start to goal can be found by simple traversal
start goal
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Find a path
With an adjacency graph, a path from start to goal can be found by simple traversal
start goal
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Find a path
With an adjacency graph, a path from start to goal can be found by simple traversal
start goal
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Find a path
With an adjacency graph, a path from start to goal can be found by simple traversal
start goal
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Find a path With an adjacency graph, a path from start to goal
can be found by simple traversal
start goal
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Find a path
With an adjacency graph, a path from start to goal can be found by simple traversal
start goal
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Find a path
With an adjacency graph, a path from start to goal can be found by simple traversal
start goal
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Find a path
With an adjacency graph, a path from start to goal can be found by simple traversal
start goal
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Find a path
With an adjacency graph, a path from start to goal can be found by simple traversal
start goal
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Find a path
With an adjacency graph, a path from start to goal can be found by simple traversal
start goal
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Find a path
With an adjacency graph, a path from start to goal can be found by simple traversal
start goal
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Approximate Cell Decomposition
Idea: construct a collection K of non-overlapping cells such that the union of all the cells approximately covers the free C-space
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Approximate Decomposition into Rectangloids
A rectangloid decomposition K of C is a finite collection of interior disjoint rectangloids (cells) such that
Cells and are adjacent if they share a boundry
rKKK ,...,, 21
iK jK
ii kC
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Approximate Decomposition into Rectangloids
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Channels and Paths in Rectangloid Decomposition
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Path Planning with a Rectangloid Decomposition
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Hierarchical Path Planning
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Potential Field Method
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Potential Field Methods
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Potential Field
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The Field of Artificial Forces
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The Attractive Potential
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The Attractive Potential
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The Attractive Potential
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The Attractive Potential
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The Repulsive Potential
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The Repulsive Potential
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The Repulsive Potential
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Local Minimum Problem with the Charge Analogy
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Escaping Local Minima - Random Motions
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Gradient Descent Planning
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RRT
Rapidly Exploring Random Tree (LaValle 1998) • Explores continuous spaces efficiently • Can form the basis for a probabilistically complete path planner
Basic RRT Algorithm: (1) Initially, start with the initial configuration as root of tree (2) Pick a random state in the configuration space (3) Find the closest node in the tree (4) Extend that node toward the state if possible (5) Goto (2)
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Basic RRT Example
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RRT Demo
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CMDragon Small Size Team
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ERRT: RRT + Waypoint Cache
Plain RRT planner has little bias toward plan quality, and replanning is
naive (all previous information is thrown out before replanning).
Waypoints: • Previously successful plans can guide new search • Biases can be encoded by modifying the target point distribution
Waypoint Cache: • When a plan is found, store nodes into a bin with random replacement • During target point selection, choose a random item from waypoint cache with probability q
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RRT-GoalBias Algorithm
(1) Initially, start with the initial configuration as root of tree
(2) Pick a target state
- goal configuration with probability p
- random state from waypoint cache with probability q
- random configuration with probability 1 - p - q
(3) Find the closest node in the tree
(4) Extend that node toward the state if possible
(5) Goto (2)
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Demos
Movie 1 Movie 2 Movie 3
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References
Seth Hutchinson at the University of Illinois at Urbana-Champaign
Leo Joskowicz at Hebrew University, Jean-Claude Latombe at Stanford University, Nancy Amato at Texas A&M University.