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.

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.

INPUTS: OUTPUT:

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.

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’

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.

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.

2 types: Combinatorial planning sampling-based planning

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

2) Choose centroid in each trapezoid.3) Place one vertex at each vertical segment.

4)Connect each vertex gives obstacle free path

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.

1) Each cell should be easy to traverse2) Decomposition of cells should be easily

compatable.3) Adjacencies between cells should be straight

forward.Properties of Roadmap:1) Accessibility2) Connectivity preserving

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.

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.

1) Initialises G to contain a single vertex q02) Random config generator is used to find random

configuration qrand.3) Finds qnear between G & qrand.4) Extends the tree.

Collision detection algorithm makes sure that path is in Cfree.

Not be able to reach qrand without hitting Cobs. So new vertex is placed.

The task is to pull apart the twisted nails.

Solution is shown in above figure.

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.

Medical Applications Autonomous Vehicles Robot Navigation, Automation, Robotic surgery

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