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
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.
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
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 translational parameters xt,yt,zt for translation
• 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.
Constructs structures in the C-space that
discretely and completely capture all information
needed to perform planning.
1)By vertical segments, decompose Cfree into
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
1) Each cell should be easy to traverse
2) Decomposition of cells should be easily
3) Adjacencies between cells should be straight
Properties of Roadmap:
2) Connectivity preserving
Translation only robot, cobs is constructed as a
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
Most common choice for industrial grade
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 q0
2) Random config generator is used to find random
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
To avoid this problem take a pair of points in path
& try to replace it with straight line.
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