This document discusses different path planning techniques for robot motion. It describes how configuration space represents all possible robot positions and orientations. Combinatorial planning methods decompose this space into cells to find obstacle-free paths, while sampling-based planning uses techniques like rapidly exploring random trees to quickly explore the space and find solutions without fully mapping obstacles. The document provides examples of how these methods are applied to problems in 2D and 3D worlds.

2.  getting a robot to automatically determine how to
move while avoiding collisions with
obstacles..eg:Piano mover’s problem
 getting robots to reason geometrically about their
environments and synthesize such plans.

3.  For a two-dimensional world,W═R2 and O is the
obstacle region.
 For a three-dimensional world, the only
differences are that W═R3 & O is polyhedra.
 Basic path planning: Inputs are given and
corresponding outputs are produced.

4. INPUTS: OUTPUT:

5.  set of all rigid body transformations that can be
applied to the robot is called the configuration
space or C-space.
 C-space in physics and control theory is usually
called a Lie group.
 C-space used in motion planning requires no
calculus; therefore, it is described as a topological
manifold.

6.  2-D world: A<R2 denotes polygonal robot.
• Position of robot q=(xt,yt,θ) called configuration
• Matrix used for finding co-ordinates is:
• By this C space is given by R2*S’

7.  3-D World:
• 3 translational parameters xt,yt,zt for translation
only robot.
• C space is C=R3, q=(xt,yt,zt).
• If rotation is considered θ=[0,2π),forms a sphere
• Each position corresponds to a circle.
• These circles glued together arround the sphere is
called Hopf fibration.

8.  Set of points where A(q)nC≠0
 Set of all non colliding configuration is called
free space Cfree.
 Compliment is obstacle region Cobs=C/Cfree.

9. 2 types:
 Combinatorial planning
 sampling-based planning

10.  Constructs structures in the C-space that
discretely and completely capture all information
needed to perform planning.
 Steps :
1)By vertical segments, decompose Cfree into
trapezoids

11. 2) Choose centroid in each trapezoid.
3) Place one vertex at each vertical segment.
4)Connect each vertex gives obstacle free path

12. Plane sweep principle:
 Sort polygon vertices from left to right
 At each step edge list is updated.
 If edges are on the left of vertex, they are deleted.
 If they are on the right of vertex, inserted in the
list.

13. 1) Each cell should be easy to traverse
2) Decomposition of cells should be easily
compatable.
3) Adjacencies between cells should be straight
forward.
Properties of Roadmap:
1) Accessibility
2) Connectivity preserving

14.  Translation only robot, cobs is constructed as a
polygonal boundary.
 The edge to edge contact between A & O is
connected to obtain it, if A & O are convex.
 If A & O are non convex, then convert them to
convex.

15.  Most common choice for industrial grade
problem.
 Here without completely exploring Cfree ,
finding a solution.
 Rapidly Exploring Random Trees(RRT)
algorithm is used.
 Idea is to add leaf vertex and edge between 2
configuration in each iteration.

16. 1) Initialises G to contain a single vertex q0
2) Random config generator is used to find random
configuration qrand.
3) Finds qnear between G & qrand.
4) Extends the tree.

17.  Collision detection algorithm makes sure that
path is in Cfree.
 Not be able to reach qrand without hitting Cobs.
So new vertex is placed.

18.  The task is to pull apart the twisted nails.
 Solution is shown in above figure.

19.  Produced paths are jagged as they traverse Cfree.
This makes the solution animation jumpy.
 Path smoothening is done to cleanup solution
path.
 To avoid this problem take a pair of points in path
& try to replace it with straight line.

20.  Medical Applications
 Autonomous Vehicles
 Robot Navigation, Automation, Robotic surgery

21.  Combinatorial planning solves simpler problems
in a clean, elegant way, but the running time is
too high for industrial-grade problems.
 Sampling-based planning provides practical
solutions for real-world problems but offers
weaker guarantees