robot lab path planning
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Robot Lab Path PlanningTRANSCRIPT
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Slide 1
Robot Lab:Robot Path Planning
William RegliDepartment of Computer Science
(and Departments of ECE and MEM)Drexel University
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Slide 2
Introduction to Motion Planning
• Applications• Overview of the Problem• Basics – Planning for Point Robot
– Visibility Graphs– Roadmap– Cell Decomposition– Potential Field
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Slide 3
Goals• Compute motion strategies, e.g.,
– Geometric paths – Time-parameterized trajectories– Sequence of sensor-based motion commands
• Achieve high-level goals, e.g.,– Go to the door and do not collide with
obstacles– Assemble/disassemble the engine– Build a map of the hallway– Find and track the target (an intruder, a
missing pet, etc.)
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Slide 4
Fundamental Question
Are two given points connected by a path?
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Slide 5
Basic Problem• Problem statement: Compute a collision-free path for a rigid or articulated
moving object among static obstacles.• Input
– Geometry of a moving object (a robot, a digital actor, or a molecule) and obstacles
– How does the robot move?– Kinematics of the robot (degrees of freedom)– Initial and goal robot configurations (positions & orientations)
• Output Continuous sequence of collision-free robot
configurations connecting the initial and goal configurations
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Slide 6
Example: Rigid Objects
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Slide 7
Example: Articulated Robot
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Slide 8
Is it easy?
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Slide 9
Hardness Results
• Several variants of the path planning problem have been proven to be PSPACE-hard.
• A complete algorithm may take exponential time.– A complete algorithm finds a path if one exists and
reports no path exists otherwise.• Examples
– Planar linkages [Hopcroft et al., 1984]– Multiple rectangles [Hopcroft et al., 1984]
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Slide 10
Tool: Configuration Space
Difficulty– Number of degrees of freedom (dimension of
configuration space)– Geometric complexity
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Slide 11
Extensions of the Basic Problem
• More complex robots– Multiple robots– Movable objects– Nonholonomic & dynamic constraints– Physical models and deformable objects– Sensorless motions (exploiting task
mechanics)– Uncertainty in control
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Slide 12
Extensions of the Basic Problem• More complex environments
– Moving obstacles– Uncertainty in sensing
• More complex objectives– Optimal motion planning– Integration of planning and control– Assembly planning– Sensing the environment
• Model building• Target finding, tracking
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Slide 13
Practical Algorithms
• A complete motion planner always returns a solution when one exists and indicates that no such solution exists otherwise.
• Most motion planning problems are hard, meaning that complete planners take exponential time in the number of degrees of freedom, moving objects, etc.
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Slide 14
Practical Algorithms• Theoretical algorithms strive for completeness
and low worst-case complexity– Difficult to implement– Not robust
• Heuristic algorithms strive for efficiency in commonly encountered situations. – No performance guarantee
• Practical algorithms with performance guarantees– Weaker forms of completeness– Simplifying assumptions on the space: “exponential
time” algorithms that work in practice
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Slide 15
Problem Formulation for Point Robot• Input
– Robot represented as a point in the plane
– Obstacles represented as polygons
– Initial and goal positions
• Output– A collision-free
path between the initial and goal positions
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Slide 16
Framework
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Slide 17
Visibility Graph Method• Observation: If there is a
collision-free path between two points, then there is a polygonal path that bends only at the obstacles vertices.
• Why? – Any collision-free path
can be transformed into a polygonal path that bends only at the obstacle vertices.
• A polygonal path is a piecewise linear curve.
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Slide 18
Visibility Graph
• A visibility graphis a graph such that– Nodes: qinit, qgoal, or an obstacle vertex.– Edges: An edge exists between nodes u and v if
the line segment between u and v is an obstacle edge or it does not intersect the obstacles.
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Slide 19
A Simple Algorithm for Building Visibility Graphs
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Slide 20
Computational Efficiency
• Simple algorithm O(n3) time• More efficient algorithms
– Rotational sweep O(n2log n) time– Optimal algorithm O(n2) time– Output sensitive algorithms
• O(n2) space
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Slide 21
Framework
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Slide 22
Breadth-First Search
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Breadth-First Search
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Slide 24
Breadth-First Search
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Breadth-First Search
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Slide 26
Breadth-First Search
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Slide 27
Breadth-First Search
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Slide 28
Breadth-First Search
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Slide 29
Breadth-First Search
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Slide 30
Breadth-First Search
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Slide 31
Breadth-First Search
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Slide 32
Other Search Algorithms
• Depth-First Search• Best-First Search, A*
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Slide 33
Framework
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Slide 34
Summary• Discretize the space by constructing
visibility graph• Search the visibility graph with
breadth-first search
Q: How to perform the intersection test?
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Slide 35
Summary
• Represent the connectivity of the configuration space in the visibility graph
• Running time O(n3)– Compute the visibility graph– Search the graph– An optimal O(n2) time algorithm exists.
• Space O(n2)
Can we do better?
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Slide 36
Classic Path Planning Approaches• Roadmap – Represent the connectivity of
the free space by a network of 1-D curves
• Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells
• Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function
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Slide 37
Classic Path Planning Approaches
• Roadmap – Represent the connectivity of the free space by a network of 1-D curves
• Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells
• Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function
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Slide 38
Roadmap• Visibility graph Shakey Project, SRI
[Nilsson, 1969]
• Voronoi Diagram Introduced by
computational geometry researchers. Generate paths that maximizes clearance. Applicable mostly to 2-D configuration spaces.
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Slide 39
Voronoi Diagram
• Space O(n)• Run time O(n log n)
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Slide 40
Other Roadmap Methods
• Silhouette First complete general method that
applies to spaces of any dimensions and is singly exponential in the number of dimensions [Canny 1987]
• Probabilistic roadmaps
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Slide 41
Classic Path Planning Approaches• Roadmap – Represent the connectivity of
the free space by a network of 1-D curves
• Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells
• Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function
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Slide 42
Cell-decomposition Methods
• Exact cell decompositionThe free space F is represented by a collection of non-overlapping simple cells whose union is exactly F
• Examples of cells: trapezoids, triangles
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Slide 43
Trapezoidal Decomposition
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Slide 44
Computational Efficiency
• Running time O(n log n) by planar sweep
• Space O(n)• Mostly for 2-D configuration spaces
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Slide 45
Adjacency Graph
• Nodes: cells• Edges: There is an edge between every pair of
nodes whose corresponding cells are adjacent.
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Slide 46
Summary
• Discretize the space by constructing an adjacency graph of the cells
• Search the adjacency graph
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Slide 47
Cell-decomposition Methods
• Exact cell decomposition• Approximate cell decomposition
– F is represented by a collection of non-overlapping cells whose union is contained in F.
– Cells usually have simple, regular shapes, e.g., rectangles, squares.
– Facilitate hierarchical space decomposition
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Slide 48
Quadtree Decomposition
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Slide 49
Octree Decomposition
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Slide 50
Algorithm Outline
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Slide 51
Classic Path Planning Approaches• Roadmap – Represent the connectivity of
the free space by a network of 1-D curves
• Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells
• Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function
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Slide 52
Potential Fields• Initially proposed for real-time collision avoidance
[Khatib 1986]. Hundreds of papers published.• A potential field is a scalar function over the free
space.• To navigate, the robot applies a force proportional
to the negated gradient of the potential field.• A navigation function is an ideal potential field that
– has global minimum at the goal– has no local minima– grows to infinity near obstacles– is smooth
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Slide 53
Attractive & Repulsive Fields
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Slide 54
How Does It Work?
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Slide 55
Algorithm Outline
• Place a regular grid G over the configuration space
• Compute the potential field over G• Search G using a best-first algorithm
with potential field as the heuristic function
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Slide 56
Local Minima
• What can we do?– Escape from local minima by taking
random walks– Build an ideal potential field –
navigation function – that does not have local minima
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Slide 57
Question
• Can such an ideal potential field be constructed efficiently in general?
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Slide 58
Completeness
• A complete motion planner always returns a solution when one exists and indicates that no such solution exists otherwise.– Is the visibility graph algorithm complete?
Yes.– How about the exact cell decomposition
algorithm and the potential field algorithm?
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Slide 59
Why Complete Motion Planning?
• Probabilistic roadmap motion planning– Efficient– Work for complex problems
with many DOF
– Difficult for narrow passages
– May not terminate when no path exists
• Complete motion planning– Always terminate
– Not efficient– Not robust even for
low DOF
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Slide 60
Path Non-existence Problem
ObstacleObstacle
GoalInitial
Robot
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Slide 61
Main Challenge
Obstacle
GoalInitial
Robot
• Exponential complexity: nDOF
– Degree of freedom: DOF– Geometric complexity: n
• More difficult than finding a path– To check all possible paths
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Slide 62
Approaches• Exact Motion Planning
– Based on exact representation of free space
• Approximation Cell Decomposition (ACD)
• A Hybrid planner
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Slide 63
Configuration Space: 2D TranslationWorkspace Configuration Space
xyRobot
Start
Goal
Free
Obstacle C-obstacle
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Slide 64
Configuration Space Computation• [Varadhan et al, ICRA 2006]• 2 Translation + 1 Rotation• 215 seconds
Obstacle
Robot
x
y
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Slide 65
Exact Motion Planning• Approaches
– Exact cell decomposition [Schwartz et al. 83]– Roadmap [Canny 88]– Criticality based method [Latombe 99]– Voronoi Diagram– Star-shaped roadmap [Varadhan et al. 06]
• Not practical– Due to free space computation– Limit for special and simple objects
• Ladders, sphere, convex shapes• 3DOF
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Slide 66
Approaches• Exact Motion Planning
– Based on exact representation of free space
• Approximation Cell Decomposition (ACD)
• A Hybrid Planner Combing ACD and PRM
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Slide 67
Approximation Cell Decomposition (ACD)
• Not compute the free space exactly at once• But compute it incrementally
• Relatively easy to implement– [Lozano-Pérez 83]– [Zhu et al. 91]– [Latombe 91]– [Zhang et al. 06]
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Slide 68
full mixed
empty
Approximation Cell Decomposition
• Full cell
• Empty cell
• Mixed cell– Mixed– Uncertain
Configuration Space
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Slide 69
Connectivity GraphGf : Free Connectivity Graph G: Connectivity Graph
Gf is a subgraph of G
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Slide 70
Finding a Path by ACD
Goal
Initial Gf : Free Connectivity Graph
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Slide 71
Finding a Path by ACDL: Guiding Path• First Graph Cut Algorithm
– Guiding path in connectivity graph G
– Only subdivide along this path
– Update the graphs G and Gf
Described in Latombe’s book
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Slide 72
First Graph Cut Algorithm
Only subdivide along L
L
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Slide 73
Finding a Path by ACD
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Slide 74
ACD for Path Non-existence
C-space
Goal
Initial
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Slide 75
Connectivity Graph Guiding PathACD for Path Non-existence
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Slide 76
ACD for Path Non-existence
Connectivity graph is not connected
No path!
Sufficient condition for deciding path non-existence
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Slide 77
Two-gear Example
no path!
Cells in C-obstacle
Initial
Goal
Roadmap in F
Video 3.356s
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Slide 78
Cell Labeling
• Free Cell Query– Whether a cell
completely lies in free space?
• C-obstacle Cell Query– Whether a cell
completely lies in C-obstacle?
full mixed
empty
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Slide 79
Free Cell Query A Collision Detection Problem
•Does the cell lie inside free space?
• Do robot and obstacle separate at all configurations?
Obstacle
WorkspaceConfiguration space
?
Robot
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Slide 80
Clearance
• Separation distance– A well studied geometric problem
• Determine a volume in C-space which are completely free
d
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Slide 81
C-obstacle QueryAnother Collision Detection Problem
•Does the cell lie inside C-obstacle?
• Do robot and obstacle intersect at all configurations?
Obstacle
WorkspaceConfiguration space
?Robot
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Slide 82
‘Forbiddance’
• ‘Forbiddance’: dual to clearance• Penetration Depth
– A geometric computation problem less investigated
• [Zhang et al. ACM SPM 2006]
PD
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Slide 83
Limitation of ACD
• Combinatorial complexity of cell decomposition
• Limited for low DOF problem– 3-DOF robots
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Slide 84
Approaches• Exact Motion Planning
– Based on exact representation of free space
• Approximation Cell Decomposition (ACD)
• A Hybrid Planner Combing ACD and PRM
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Slide 85
Hybrid Planning• Probabilistic roadmap
motion planning+ Efficient+ Many DOFs
- Narrow passages- Path non-existence
• Complete Motion Planning+ Complete
- Not efficient
Can we combine them together?
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Slide 86
Hybrid Approach for Complete Motion Planning • Use Probabilistic Roadmap
(PRM): – Capture the connectivity for
mixed cells– Avoid substantial subdivision
• Use Approximation Cell Decomposition (ACD)– Completeness – Improve the sampling on
narrow passages
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Slide 87
Connectivity GraphGf : Free Connectivity Graph G: Connectivity Graph
Gf is a subgraph of G
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Slide 88
Pseudo-free edges
Pseudo free edge for two adjacent cells
Goal
Initial
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Slide 89
Pseudo-free Connectivity Graph: Gsf
Goal
Initial
Gsf = Gf + Pseudo-edges
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Slide 90
Algorithm
• Gf
• Gsf
• G
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Slide 91
Results of Hybrid Planning
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Slide 92
Results of Hybrid Planning
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Slide 93
Results of Hybrid Planning• 2.5 - 10 times speedup
3 DOF 4 DOF 4 DOF
timing cells # timing cells # timing cells #
Hybrid 34s 50K 16s 48K 102s 164K
ACD 85s 168K ? ? ? ?Speedup 2.5 3.3 ≥10 ? ≥10 ?
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Slide 94
Summary • Difficult for Exact Motion Planning
– Due to the difficulty of free space configuration computation
• ACD is more practical– Explore the free space incrementally
• Hybrid Planning– Combine the completeness of ACD and efficiency
of PRM