Path Planning Techniques

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Path Planning Techniques

• Sasi Bhushan Beera
• Shreeganesh Sudhindra

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Path Planning

• Compute motion strategies, e.g.,
• Geometric paths
• Time-parameterized trajectories
• Achieve high-level goals, e.g.,
• To build a collision free path from start point
  to the desired destination
• Assemble/disassemble the engine
• Map the environment

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Objective

• To Compute a collision-free path for a mobile
  robot among static obstacles.
• Inputs required
• Geometry of the robot and of obstacles
• Kinematics of the robot (d.o.f)
• Initial and goal robot configurations (positions
  orientations)
• Expected Result
• Continuous sequence of collision-free robot
  configurations connecting the initial and goal
  configurations

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Some of the existing Methods

• Visibility Graphs
• Roadmap
• Cell Decomposition
• Potential Field

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Visibility Graph Method

• If there is a collision-free path between two
  points, then there is a polygonal path that bends
  only at the obstacles vertices.
• A polygonal path is a piecewise linear curve.

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Visibility Graph

• A visibility graph is 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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Breadth-First Search
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Breadth-First Search
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Breadth-First Search
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Breadth-First Search
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Breadth-First Search
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Breadth-First Search
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Breadth-First Search
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Breadth-First Search
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Breadth-First Search
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A Simple Algorithm for Building Visibility Graphs
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Road mapping Technique

• Visibility graph
• Voronoi Diagram
• Introduced by computational geometry
  researchers.
• Generate paths that maximizes clearance
• Applicable mostly to 2-D configuration spaces

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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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Quadtree Decomposition
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Octree Decomposition
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Algorithm Outline
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Potential Fields

• Initially proposed for real-time collision
  avoidance Khatib 1986.
• 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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Attractive Repulsive Fields
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How Does It Work?
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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
• Note A heuristic function or simply
  a heuristic is a function that ranks
  alternatives in various search algorithms at each
  branching step basing on an available information
  in order to make a decision which branch is to be
  followed during a search.

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Use the Local Minima Information

• Identify the local minima
• Build an ideal potential field navigation
  function that does not have local minima
• Using the calculations done so far, we can create
  a PATH PLANNER which would give us the optimum
  path for a set of inputs.

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Active Sensing

• The main question to answerWhere to move
  next?
• Given a current knowledge about the robot state
  and the environment, how to select the next
  sensing action or sequence of actions. A vehicle
  is moving autonomously through an environment
  gathering information from sensors. The sensor
  data are used. to generate the robot actions
• Beginning from a starting configuration
  (xs,ys,?s) to a goal configuration (xg,yg,?g) in
  the presence of a reference trajectory and
  without it With and without obstacles Taking
  into account the constraints on the velocity,
  steering angle, the obstacles, and other
  constraints

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Active Sensing of a WMR
Robot model
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Trajectory optimization

• Between two points there are an infinite number
  of possible trajectories. But not each trajectory
  from the configuration space represents a
  feasible trajectory for the robot.
• How to move in the best way according to a
  criterion from the starting to a goal
  configuration?
• The key idea is to use some parameterized family
  of possible trajectories and thus to reduce the
  infinite-dimensional problem to a finitely
  parametrized optimization problem. To
  characterize the robot motion and to process the
  sensor information in efficient way, an
  appropriate criterion is need. So, active sensing
  is a decision making, global optimization problem
  subject to constraints.

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Trajectory Optimization

• Let Q is a class of smooth functions. The
  problem of determining the best trajectory q
  with respect to a criterion J can be then
  formulated as
• q argmin(J)
• where the optimization criterion is chosen of
  the form
• information part losses (time,
  traveled distance)
• subject to constraints
• l lateral deviation, v WMR
  velocity steering angle d distance
• to obstacle

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Trajectory Optimization

• The class Q of harmonic functions is chosen,
• Q Q(p),
• p vector of parameters obeying to preset
  constraints
• Given N number of harmonic functions, the new
  modified robot trajectory is generated on the
  basis of the reference trajectory by a lateral
  deviation as a linear superposition

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Why harmonic functions?

• They are smooth periodic functions
• Gives the possibility to move easily the robot to
  the desired final point
• Easy to implement
• Multisinusoidal signals are reach excitation
  signals and often used in the experimental
  identification. They have proved advantages for
  control generation of nonholonomic WMR (assure
  smooth stabilization). For canonical chained
  systems Brockett (1981) showed that optimal
  inputs are sinusoids at integrally related
  frequencies, namely 2?, 2. 2?, , m/2. 2?.

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Optimality Criterion

• I trace(WP),
• I is computed at the goal configuration or on the
  the whole trajectory (part of it, e.g. in an
  interval )
• where W MN
• M scaling matrix N normalizing matrix
• P estimation error covariance matrix
  (information matrix or entropy) from a filter
  (EKF)

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Trajectory (N2 sinusoids)
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Point-to-point optimization
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Implementation

• Using Optimization Toolbox of MATLAB,
• fmincon finds the constrained minimum of a
  function of several variables
• With small number of sinusoids (Nlt5) the
  computational complexity is such that it is
  easily implemented on-line. With more sinusoidal
  terms (Ngt10), the complexity (time, number of
  computations) is growing up and a powerful
  computer is required or off-line computation. All
  the performed experiments prove that the
  trajectories generated even with N3 sinusoidal
  terms respond to the imposed requirements.

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Conclusions

• An effective approach for trajectories
  optimization has been considered
• Appropriate optimality criteria are defined. The
  influence of the different factors is decoupled
• The approach is applicable in the presence of
  and without obstacles.

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Few Topics with related videos of work done by
people around the world in the field Robot
navigation

• 3-D path planning and target trajectory
  prediction http//www.youtube.com/watch?vpr1Y21
  mexzsfeaturerelated
• Path Planning http//www.youtube.com/watch?vd0P
  luQz5IuQ
• Potential Function Method By Leng Feng Lee
  http//www.youtube.com/watch?vLf7_ve83UhE
• Real-Time Scalable Motion Planning for Crowds
  -http//www.youtube.com/watch?vifimWFs5-hcNR1
• Robot Potential with local minimum avoidance
  -http//www.youtube.com/watch?vCr7PSr6SHTIfeatur
  erelated

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References

• Tracking, Motion Generation and Active Sensing of
  Nonholonomic Wheeled Mobile Robots -Lyudmila
  Mihaylova Katholieke Universiteit Leuven
• Robot Path Planning - By William RegliDepartment
  of Computer Science(and Departments of ECE and
  MEM)Drexel University
• Part II-Motion Planning- by Steven M. LaValle
  (University of Illinois)
• Wikipedia
• www.mathworks.com