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Motion Planning in Dynamic Environments

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Motion Planning in Dynamic Environments. Jur van den Berg. Outline ... Frogger. 6 DOF Articulated Robot. Configuration-Time Space. One additional dimension: time ... – PowerPoint PPT presentation

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Title: Motion Planning in Dynamic Environments


1
Motion Planning in Dynamic Environments
  • Jur van den Berg

2
Outline
  • Recap Configuration Spaces, PRM
  • Moving obstacles
  • Configuration-time space
  • Time constraints
  • Exact methods
  • PRM?
  • RRT
  • Roadmap based
  • Multi-Robot Motion Planning

3
Configuration Space
  • Static environment
  • Dimension DOF
  • Translating in 2D
  • Minkowski Sums

Workspace
Configuration Space
4
Configuration Space
  • Articulated Robots (2 Rotating DOF)
  • Hard to compute explicitly

Workspace
Configuration Space
5
Probabilistic Roadmap Method
  • Collision-test
  • Preprocessing create roadmap
  • Query find path in roadmap (multiple-shot)

Preprocessing
Query
6
Probabilistic Roadmap Method
  • High-dimensional Configuration Spaces
  • Probabilistically complete

6 DOF Articulated Robot
6 DOF (3T 3R) Freeflying Robot
7
Dynamic Environments
  • Moving Obstacles Static Obstacles

Frogger
6 DOF Articulated Robot
8
Configuration-Time Space
  • One additional dimension time
  • Obstacles are stationary in CT-space

Configuration Space
Configuration-Time Space
9
Path Constraints
  • Cannot go backward in time
  • Maximum velocity

2D Configuration-Time Space
3D Configuration-Time Space
10
Goal Specification
  • Specific configuration and moment in time
  • Specific configuration, as fast as possible

g (x, y, t)
g (x, y)
11
Exact Algorithms
  • Cell decomposition (2D Translation time)
  • Piecewise linear motions of obstacles
  • No bound on velocity of robot

12
Exact Algorithms
  • Asteroid Avoidance (2D Translation time)
  • Constant linear motions of obstacles
  • Constant number of obstacles
  • Bound on velocity of robot
  • Time-minimal path
  • Polynomial time algorithm

13
Exact Algorithms
  • Robots with more degrees of freedom
  • Rotating obstacles
  • At least exponential running time
  • General problem PSPACE-hard

14
Other Approaches
  • Path-velocity decomposition
  • First plan path in configuration space
  • Then tune velocity along path

Workspace
2D Configuration-Time Space
15
Path-Velocity Decomposition
  • Reduces problem to 2D
  • Cell decomposition, visibility graph

Cell decomposition
(Adapted) Visibility Graph
16
Probabilistic Approaches
  • PRM?

17
Probabilistic Approaches
  • PRM?
  • Directed Edges

18
Probabilistic Approaches
  • PRM?
  • Directed Edges
  • Transitory Configuration Space
  • Multiple-shot paradigm does not hold

19
Probabilistic Approaches
  • Repetitive dynamic environments
  • Specific class of problems
  • Period is least common multiple of obstacles

Roadmap in Configuration-Time Space
Resulting Path
20
Probabilistic Approaches
  • (Rapid Random Trees) RRT
  • Single-shot
  • Build tree oriented along time-axis

21
Probabilistic Approaches
  • Advantages
  • Any dimensional configuration-spaces
  • Any behavior of obstacles
  • Only requirement is robot configured at c
    collision-free at time t ?
  • Disadvantages
  • Narrow passages
  • All effort in query phase

22
Roadmap-based Approaches
  • Roadmap-velocity decomposition
  • First build roadmap in configuration space
  • Then find trajectory on roadmap avoiding moving
    obstacles

Roadmap in Workspace
Roadmap-Time Space
23
Roadmap-based Approaches
  • Discretize Roadmap-time space
  • Select time step Dt
  • Constrain velocity to be -vmax, 0, vmax
  • Find shortest path using A

24
Roadmap-based Approaches
25
Multi-Robot Motion Planning
  • Environment static obstacles (possibly dynamic
    obstacles as well)
  • N robots with start and goal position

12 Robots
24 Robots
26
Composite Configuration Space
  • Configuration spaceC C1 ? C2 ? ? CN
  • Dimension is sum of DOFs of all robots
  • Very high-dimensional
  • Cylindrical obstacles

Composite Configuration Space 3 Robots, 1 DOF
each
27
Prioritized Multi-Robot Planning
  • Assign priorities to robots
  • Plan path for robot in order of priorities
  • Treat previously planned robots as moving
    obstacles

24 Robots
Problematic Case
28
Next Class
  • Unknown obstacle motions
  • Online planning
  • Continuous push by real-world time

29
References
  • Introduction
  • M. de Berg, M. van Kreveld, M. Overmars, O.
    Schwarzkopf. Computational Geometry, Algorithms
    and Applications (book)
  • S. LaValle. Planning Algorithms (book freely
    available on-line)
  • J. van den Berg. Path Planning in Dynamic
    Environments (PhD thesis)
  • Exact Planning with Moving Obstacles
  • J. Reif, M. Sharir. Motion Planning in the
    Presence of Moving Obstacles
  • K. Fujimura, H. Samet. Planning a Time-Minimal
    Motion among Moving Obstacles
  • Path-Velocity Decomposition
  • K. Kant, S. Zucker. Toward Efficient Trajectory
    Planning the Path-Velocity Decomposition
  • K. Fujimura. Time-Minimum Routes in
    Time-Dependent Networks
  • Probabilistic Approaches
  • D. Hsu, R. Kindel, J.-C. Latombe, S. Rock.
    Randomized Kinodynamic Motion Planning with
    Moving Obstacles
  • J. van den Berg, M. Overmars. Path Planning in
    Repetitive Environments
  • Roadmap-Based Approaches
  • J. van den Berg, M. Overmars. Roadmap-Based
    Motion Planning in Dynamic Environments
  • Prioritized Multi-Robot Motion Planning
  • J. van den Berg, M. Overmars. Prioritized Motion
    Planning for Multiple Robots

30
Motion Planning in Dynamic Environments 2
  • Jur van den Berg

31
Outline
  • Recap Configuration-Time space
  • Offline vs. Online (real-time)
  • Anytime (partial) planning
  • Known vs. Unknown Trajectories
  • Continuous cycle of sensing and planning
  • Static vs. Dynamic
  • Estimating future trajectories

32
Configuration-Time Space
  • Natural space for planning problem
  • Time as additional dimension

Configuration Space
Configuration-Time Space
33
Offline vs. Online Planning
  • Offline enough time for planning
  • Online plan while the world is changing
  • Real world time and time modeled in CT space are
    related
  • Applications offline
  • Multi-robot motion planning
  • Applications online
  • Real-time vehicle navigation

34
Online Real-Time Planning
  • t 0

Configuration-Time Space
Configuration Space
35
Online Real-Time Planning
  • t 0.5

Configuration-Time Space
Configuration Space
36
Online Real-Time Planning
  • t 1

Configuration-Time Space
Configuration Space
37
Online Real-Time Planning
  • t 1.5

Configuration-Time Space
Configuration Space
38
Online Real-Time Planning
  • t 2

Configuration-Time Space
Configuration Space
39
Online Real-Time Planning
  • t 2.5 Query! Move from s to g asap. v 1

Configuration-Time Space
Configuration Space
40
Online Real-Time Planning
  • t 3 Planning in progress

Configuration-Time Space
Configuration Space
41
Online Real-Time Planning
  • t 3.5 Planning in progress

Configuration-Time Space
Configuration Space
42
Online Real-Time Planning
  • t 4 Planning ready!

Configuration-Time Space
Configuration Space
43
Online Real-Time Planning
  • t 4.5

Configuration-Time Space
Configuration Space
44
Online Real-Time Planning
  • t 5

Configuration-Time Space
Configuration Space
45
Online Planning
  • Planning takes time!
  • If query (s, g) is posed at real-world time tw
  • Reserve t time for planning
  • Find path in CT beginning at (s, tw t)
  • Start planning at real-world time tw
  • Requirement planning must finish before
    real-world time tw t has come

46
Problem!
  • Planners do not guarantee to finish within a
    preset amount of time t
  • RRT
  • Roadmap-based

47
Anytime or Partial Planning
  • Not necessarily plan all the way to the goal
  • Returns best initial path when time runs out
  • Continue planning while executing path
  • Planning must always be ahead of execution

t t
t gt t
t gtgt t
48
Issue
  • How to choose a good value for t
  • Large t large latency
  • Small t risky small look-ahead

49
Example
  • (James Kuffner, CMU)

50
Unknown Trajectories
  • Obstacles have unknown future trajectories
  • Available information
  • Past trajectories
  • Current observations (velocity)

51
Trajectory Estimation
  • Estimate future trajectory based on past
    trajectory and current observation

known
assume static
worst case
extrapolation
52
Trajectory Estimation
  • Estimated trajectories are less valuable further
    away in the future
  • Continuously updating estimations

53
Continuous cycle of sensing and planning
  • Receive query (s, g) at time tw
  • Read sensor input at time tw
  • Reserve t time for planning
  • Plan path with start configuration-time (s, tw
    t) based on estimations acquired at time tw
  • Start planning at time tw

54
Continuous cycle
  • Finish planning at time tw t
  • Resulting path p0 T ? C
  • Start executing path p0 at time tw t
  • Read sensor input at time tw t
  • Reserve t time for planning
  • Plan path with start configuration-time (p0(tw
    2t), tw 2t) based on estimations acquired at
    time tw t
  • Start planning at time tw t

55
Continuous cycle
  • Finish planning at time tw 2t
  • Resulting path p1 T ? C
  • Start executing path p1 at time tw 2t
  • Read sensor input at time tw 2t
  • Reserve t time for planning
  • Plan path with start configuration-time (p1(tw
    3t), tw 3t) based on estimations acquired at
    time tw 2t
  • Start planning at time tw 2t

56
Continuous cycle
  • Finish planning at time tw kt
  • Resulting path pk 1 T ? C
  • Start executing path pk 1 at time tw kt
  • Read sensor input at time tw kt
  • Reserve t time for planning
  • Plan path with start configuration-time (pk
    1(tw (k 1)t), tw (k 1)t) based on
    estimations acquired at time tw kt
  • Start planning at time tw kt

57
Continuous cycle
  • Until the goal has been reached
  • A path whose planning starts at time t is used
    between t t and t 2t
  • Paths must be valid for at least t time
  • Estimates must be valid for at least 2t time

58
Example
  • Linear extrapolation of obstacles trajectories

Configuration space
Configuration-time space
59
A good value for t
  • Large t
  • Large look-ahead
  • Heavy reliance on estimates
  • Slow reaction on changes in environment (may be
    unsafe)
  • Spends much time on (portions of) path that will
    not be used
  • Important decisions in near-future based on
    unreliable estimates of far future
  • Small t
  • Small look-ahead (may be unsafe)
  • Hard to bias the search in the direction of the
    goal
  • Current works
  • Choose t based on dynamicity of the environment

60
Worst-case estimates
  • Obstacles have known maximum velocity
  • Region containing them is a growing disc, or a
    cone in CT-space

61
Planning amidst growing discs
  • Paths avoiding the growing discs are inherently
    safe, regardless of value of t

62
Planning amidst growing discs
  • Space fills up quickly continuous replanning
  • Robot must move faster than obstacles
  • O(n3 log n) algorithm very fast in practice

63
Advanced trajectory estimation
  • Based on statistical data

64
Obstacle avoidance
  • Highly reactive
  • No planning (no look-ahead)
  • Velocity Obstacles

65
Obstacle avoidance
  • Example

66
Next Class
  • Kinodynamic Motion Planning
  • State Space
  • Planning in Dynamic Environments with Kinodynamic
    Constraints

67
References
  • Introduction
  • J. van den Berg. Path Planning in Dynamic
    Environments (PhD thesis)
  • Online Motion Planning in Unknown Environments
  • S. Petty, T. Fraichard. Safe Motion Planning in
    Dynamic Environments
  • D. Hsu, R. Kindel, J.-C. Latombe, S. Rock.
    Randomized Kinodynamic Motion Planning with
    Moving Obstacles
  • J. van den Berg, D. Ferguson, J. Kuffner. Anytime
    Path Planning and Replanning in Dynamic
    Environments
  • Planning among Growing Discs
  • J. van den Berg, M. Overmars. Planning the
    Shortest Safe Path amidst Unpredictably Moving
    Obstacles
  • Trajectory Estimation
  • D. Vasquez, F. Large, T. Fraichard, C. Laugier.
    Moving obstacles motion prediction for
    autonomous navigation.
  • Velocity Obstacles
  • P. Fiorini, Z. Shiller. Motion Planning in
    Dynamic Environments using Velocity Obstacles
  • F. Large, S. Sckhavat, Z. Shiller, C. Laugier.
    Using non-linear velocity obstacles to plan
    motions in a dynamic environment
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