### Chapter 6 Cell Decomposition - Intelligent Systems and Robotics

```Cell Decomposition
Course: Introduction to Autonomous Mobile Robotics
Prof. Jaebyung Park
Intelligent Systems & Robotics Lab.
Division of Electronic Engineering
Chonbuk National Univerisity
Presented by:
Ram Kaji Budhathoki
Student ID: 201155431
Ansu Man Singh
Student ID:201150875
Outline
•
•
•
•
•
Exact Cell Decomposition
Trapezoidal Decomposition
Boustrophedon Decomposition
Morse Decomposition
Visibility based Decomposition
Exact Cell Decomposition
• Definition
– Free space is the union of simple sub spaces called
cell
– If two cell shares common boundary, then they
Exact Cell Decomposition
– A graph where a node represent cell, and an edge
is used to connect two nodes
Path planning using cell decomposition
• Steps
– First cells containing start and goal points are
determined.
– Nodes connecting initial nodes and goal nodes is
– Path in each cell is determined
Path planning using cell decomposition
Node with grey color
represent cell covered
during the path of robot
start
Cell
goal
Cell Decomposition method
•
•
•
•
Trapezoidal decomposition
Boustrophedon decomposition
Morse cell decomposition
Visibility based decomposition
Trapezoidal Decomposition
• Shape of the cells after decomposition is
trapezoidal
• Cell can also be triangular in shape
• For the trapezoidal decomposition obstacles
should be in polygon
Trapezoidal Decomposition
• The trapezoidal decomposition is formed by extending a
vertical line at each vertex up, down, or both ways until it
intersects an obstacle boundary
Trapezoidal Decomposition
Formation of cells after
Trapezoidal decomposition
Steps in Trapezoidal decomposition
1. In order to decompose, First list of polygons
with the list of vertices are given as input to
the algorithms
2. Sort all the vertices according to the value of
X-coordinate of each vertex. This steps
require O(n*log(n)) time and O(n) memory
3. Determination of intersection of those
vertices with edges. This requires O(n^2)
time
Trapezoidal Decomposition
• However computational time in step3 can be
reduced if we use sweeping line and maintain
an efficient list.
• The list should contain the edges that will
intersect with line extended from current
vertex
• When a sweeping line touches a vertex it is
called Event
Trapezoidal Decomposition
• Some Terminology
Vertex where
event took
place
Scan line
Y – coordinates of e(upper) is always greater that e(lower)
Trapezoidal Decomposition
• There are four types of event
Trapezoidal Decomposition
Trapezoidal Decomposition
• Example
Trapezoidal Decomposition
• Example
Trapezoidal Decomposition
• Example
Trapezoidal Decomposition
• Example
Trapezoidal Decomposition
• This method requires total O(nlog(n)) time
maintaining the list
• By the help of this we can determine the
intersection point of the vertices with
boundary of other obstacle
Path planning using Trapezoidal
decomposition
• Steps
1. First cells containing start and goal points are
determined.
2. Adjacency graph is created with node is used to
represent a cell,
3. Path is searched on adjacency graph, i.e, nodes
connecting start point to goal point is searched
Path planning using Trapezoidal
decomposition
• Examples
Mid point
Path planning for coverage application
• Coverage application includes autonomous lawn
mowing, floor cleaning and snow removal
• This type of application requires robot to move
through each and every points on free spaces
• Performance is measured in terms of area
covered vs path length.
• For this type of application cell Decomposition is
very essential.
Boustrophedon Decomposition
• Boustrophedon Decomposition is similar to
Trapezoidal decomposition, however in
Boustrophedon vertical lines are formed both
upward and down ward of vertex.
Cell decomposition using Boustrophedon decomposition
Boustrophedon Decomposition
Boustrophedon decomposition
Trapezoidal decomposition
This figure proves that when there are more
Number cells, then robot have to cover more
Distance.
Coverage using Boustrophedon
Decomposition
• Steps in the coverage using Boustrophedon
Decomposition
– Step1: Decomposition of free space into cells
– Step2: Creation of Adjacency graph
– Step3: Visit of each node in the graph, known as
exhaustive walk
– Step4 : at each node, cell is covered using back
and forth motions
Coverage using Boustrophedon
Decomposition
Step 1
Step 2 and 3
Step 4
Boustrophedon Decomposition
• Exhaustive Walk along the graph can be
achieved using depth-first search algorithm
Depth-First search Algorithm(DFS)
• In this algorithms Each node is given 3 states.
– Node that is not visited
– Node that is in progress or process
– Node where processing is finished
Let the three states be defined by three colors (grey,
white and black)
Depth-First search Algorithm(DFS)
• Initially all nodes are white
Depth-First search Algorithm(DFS)
Depth-First search Algorithm(DFS)
• Node one has connection with node 4, so visit
node 4
Depth-First search Algorithm(DFS)
• Node 4 has connection with node 2 and 5 ,
choose one node and visit that node, e.g 2
Depth-First search Algorithm(DFS)
• Node 2 has connection with node 4 and5 ,
since 4 is already visited, Node 5
Depth-First search Algorithm(DFS)
• Similarly visit node 3
Depth-First search Algorithm(DFS)
• Since no new node are connected with node
3, so mark node 3 with black color, i.e.
processing finished status
Depth-First search Algorithm(DFS)
• Now node has connection with 5, since 5 is
already visited, Mark Node 5 with black color
Depth-First search Algorithm(DFS)
• Similarly, same applies to node 2
Depth-First search Algorithm(DFS)
• same applies to node 4 and 1
Coverage using Boustrophedon
Decomposition
• Example
Obstacles
Robot
Visibility based Decomposition
• Visibility based decomposition is based on line
of sight
• This decomposition is used for pursuit/evasion
problem
pursuers (police)
and pursuer lies in line of sight, i.e.
Visibility based decomposition
• Some Terminology
– H(Wfree):Number of pursuer required to capture an
– Contaminated space: Region in workspace where
– Clear space: Region in free space where there are no
– Re-contaminated space : Region that may be recontaminated
– Edge gap: edge of the visible polygon on free space
Visibility based decomposition
• Example
Each edge graph contains binary information, B(x) is the binary vector related to x
Edge gap
Visibility based decomposition
• In visibility based decomposition cells are
decomposed by the help of conservative
region.
• Adjacency graph is created for all cells
• Binary information is stored in each node of
Visibility based decomposition
• Conservative region is created by following
ways:
– Lines are extended from convex vertex until they
intersect other walls
– If two vertex are in line of sight, rays are extended
from each vertex to other vertex.
– E.g.
Visibility based decomposition
• Example
Visibility based algorithms to solve
pursuit/evasion problem
• Construction of Adjacency graph and
conservative region is not enough to solve
pursuit evasion problem
• For that Information graph is required
Visibility based algorithms to solve
pursuit/evasion problem
• Information graph: In this graph all possible
binary information of edge-gap is stored for
cell. Which actually becomes the nodes of
information graph
Visibility based algorithms to solve
pursuit/evasion problem
• Example
Visibility based algorithms to solve
pursuit/evasion problem
• Example
1
0
1
0
00
1
0
0
10
01
11
Information graph
1
Visibility based algorithms to solve
pursuit/evasion problem
• Example
1
0
1
0
00
1
0
0
10
01
11
1
any node with all 1 and
stop at any node
with all 0
Complexity
• In order to find the evader in simply
connected spaces with n edge,O(log(n))
pursuers are required,
• In order to find evader in spaces with ‘h’ holes
O(sqrt(h)+log(n)) pursuers are required
Morse Cell Decompositions
Drawbacks of Trapezoidal Decomposition
• Simple back-and-forth motions. No efficient paths for
coverage.
• Cells formed can be aggregated with neighboring cells to
form more efficient coverage paths.
• A polygonal workspace is required, not realistic for every
applications.
Morse Function
• Cells have simple structure and can be defined in
nonpolygonal spaces.
• Critical points are nondegenerate i.e. critical points are
isolated.
• The boustrophedon decomposition is a Morse
decomposition.
Morse Decomposition Definition
• A slice is a codimension one manifold denoted by Q  .
• The slices are parameterized by λ (varying λ sweeps a slice through
the space).
• The portion of the slice in the free configuration space,Q free , is
denoted by Q free  , i.e.
Q free   Q 
Q free
• The connectivity changes that occur at critical points are used to
define cells in a cell decomposition.
• Q free  changes connectivity at critical points. Slices that contain
critical points are termed critical slices.
• Qj
is the jth open connected slice interval such that
free 
Q free  
j
Q
j
free 
Morse Decomposition Definition Contd….
It is an exact cell decomposition whose cells are the connected
components of Qfree\I*.
I
*
Is the slice intervals that contain a critical point.
Consider the boustrophedon decomposition of a nonpolygonal
environment.
• As a straight line slice is swept from left to right, its connectivity in the free
space changes first from one to two, then two to one and so forth.
• At the points where connectivity changes occur, the cell boundaries in the
free Space are located.
Morse Decomposition in terms of
Critical Points
∇h
∇m
• Slice
function: h(x,y)= x
• At a critical point x of h |M , ∇ h ( x ) =∇m ( x )
Connectedness
Connectivity of the slice in the free space changes at the critical points.
1- Connected
1-Connected
2 -Connected
2 -Connected
Robot Motion and Reeb Graph
Each cell can be covered by back and forth motions
Reeb graph represents the topology of the cellular decomposition
Examples of Morse Decomposition
• The slice may be defined as the preimage of a general realvalued function h : Q  R .
• Varying the function that defines the shape of the slice that
results in different decompositions.
• For the boustrophedon decomposition in the plane, h(x, y) = x.
Morse Decomposition
1. h(x ,y)=
x  y
2
2
Fig. Cell Decomposition for h(x,y)= x 2  y 2 and its associated spiral
coverage pattern.
• The slices are the circles that are the preimages of h.
• At the critical points, the circle-shaped slices become tangent to the
obstacles.
• The robot follows a spiral pattern until it encounters critical points.
• It maximizes the area covered per unit distance travelled in regions
sparsely populated with obstacles.
2. h(x, y)=
Morse
Decomposition
y
 
tan  
 x
 y
Fig. Decomposition for h(x, y)= tan  
 x
and a spiked pattern.
• The free space is sliced like a pie.
• At critical points, the slices are tangent to the obstacles.
• This type of pattern covers densely the region closest to the
centre of the pattern.
Morse Decomposition
3. h(x, y) = |x| + |y|
Fig. Decomposition for h(x ,y) = |x| + |y| and a coverage pattern.
• Squares are the slices.
• At the critical points the corner of a square touches an obstacle
or the side of it becomes tangent to an obstacle.
• This pattern can be used to to approximate the spiral pattern
as it is easier to move the robots along the straight line rather
than circles.
Brushfire Decomposition
• Imaginary wave fronts emanate from each obstacle and collide at
points on the GVD.
• The decomposition models the topology of the wave fronts as they
initially collide with each other and form or destroy new wave
fronts.
• The wave fronts collide with each other at the points located on the
GVD.
Fig. GVD of an environment
Brushfire Decompostion
Fig. Incremental construction of the cells of the brushfire decomposition.
Brushfire Decomposition
Coverage Path
Fig. Coverage pattern for the brushfire decomposition
• To generate this pattern, the robot follows the boundaries of
obstacles.
• It has a continuous robust reference for localization.
• The robot relies heavily on long-range sensors.
Wave-Front Decompostion
• Let h(x, y) be the length of the shortest path between a point (x,
y) and a fixed location.
• The level sets h−1(λ) foliate the free space where for a given λ.
• Wave front starts at qstart and expanding into the free space.
• The value λ parameterizes each wave front (or level set of h).
Once the wave front crosses qgoal, the planner can backtrack a
path from qgoal to qstart .
• The shortest path-length function induces a cell
decomposition, as well.
• Critical points of this function occur when wave fronts
becomes tangent to obstacles and when wave fronts collide.
• Once the waves collide, they propagate as one wave with a
nonsmooth point that originated at the critical point.
Wave-Front Decomposition
• The non smooth point traces the set of points of equal path length to the
goal for two classes of paths, one to the right of the obstacle and one to
the left.
• This decomposition is especially useful for coverage by a tethered robot
Fig. Wave-front decomposition defined on a continuous domain.
• The wave front emanates from a point in the lower-left portion of
the figure.
Sensor Based Coverage
• If the the robot is placed in an unknown environment, but
assume it has the standard range sensor ring.
• The task is to simultaneously cover and explore the
unknown space.
• This can be reduced to concurrently and incrementally
covering each cell while constructing the adjacency graph.
• For sensor-based coverage, a Reeb graph is constructed .
• This Reeb graph is dual i.e. the nodes of the graph are the
critical points and the edges connect neighboring critical
points, i.e., correspond to cells.
.
Sensor Based Coverage
(Detect Critical Points)
Fig. A boustrophedon decomposition and its Reeb graph. At the critical points,
the surface normals and sweep direction are parallel.
Sensor Based Coverage
Fig. Incremental construction of the graph while the robot is covering the space.
While covering the space, look for critical points
Algorithm
• Cover a cell until the closing critical point is detected
• If the closing critical point has “uncleaned” cells associated with it,
chose one and cover, repeat
• If the closing critical point has no uncleaned cells,
– search reeb graph for a critical point with an uncleaned cell
– Plan a path to critical point
– Cover cell, repeat
• Else coverage is complete
Sensing Critical Points
• How does the robot sense a critical point?
The robot looks for points where the surface normals are parallel
to the sweep direction.
• How does the robot find all of the critical points?
While covering the cell, the robot looks for the other critical point
that indicates complete coverage of the cell and the next node in
the Reeb graph called the closing critical point.
It is to be guaranted for the Reeb Graph that the robot finds the
closing critical point of each cell.
The robot perform boundary-following along the ceiling in the
reverse direction so that it will sense the critical points related to
the ceiling. This motion is reverse boundary following.
Missing Closing Critical Point
Fig. Critical Points in the ceiling are missed with conventional coverage algorithms
The algorithm uses a raster scan type of motion:
move along a slice or lap to an obstacle, follow the obstacle boundary for a lateral
distance equal to interlap spacing, and repeat.
• This alternates boundary-following between the "ceiling" and "floor" of the cell.
• It can miss the closing critical point of a cell.
• The robot did not follow the boundary of the ceiling, it cannot sense the critical
points in the ceiling using the critical point sensing method.
Cycle Algorithm
Forward phase: The robot first follows the path
between points Si and 2.
Reverse phase: It follows the path between points 2
and 3.
Closing phase: The robot follows the path between
points 3 and Si.
• This algorithm is the most important part
of the incremental construction.
• It guarantees encountering the closing
critical point of a cell if it exists between
subsequent laps.
Complexity of Coverage
1. Establish a relationship among the number of critical points, cells, and
obstacles.
2. Determine an upper-bound on path length given the perimeter the obstacles
and the diameter Δ smallest disk that circumscribes the space as shown in Fig
below.
Consider coverage with the boustrophedon decomposition.
In the Reeb graph Obstacles including the outer boundary are represented with
"faces" in the graph .Euler's formula relate s the number of nodes ν, edges e and
faces f of a planar connected graph by
v- e + f =2------------(1)
Fig. To determine the complexity of the algorithm in terms of the environment size
, the diameter Δ of the "minimal" disk that fully contains the space.
Modified Euler's Formula
• In general ,the outer boundary of the space is not termed an obstacle, 1 is
subtracted from the number of faces to get the number of obstacles.
• Let Ncp be the number of critical points, Nce be the number of cells and Nob
be the number of obstacles . Then from equation (1)
Nce=Ncp+Nob-1.------------------(2)
Example:
There are 21 critical points (nodes in the graph).i.e , Ncp = 21,
and two obstacles (faces f2, f3 in the graph; f1 is the outer boundary), Nob = 2.
Using the modified Euler's formula Nce = Ncp + Nob − 1, we get , Nce = 22.
Therefore, the number of cells increases linearly as the robot discovers new
critical points.
Extra Lap
•
•
•
•
•
•
Analyzing lapping, boundary following and backtracking motions separately.
The space is fully contained within a Δ diameter disk, the length of each lapping
path can be at most Δ.
  
There must be at least   lapping paths where 2r is the interlap spacing.
 2r 
There is an additional lap associated with starting the coverage operation within a
cell as shown in Fig. below.
  
The maximum number of lapping paths is  2 r   N ce .
The length of each lapping path is bounded above by Δ, the total path length of the
lapping motions is bounded above by       N ------------------(A)
 2r 


ce
Fig. When the robot starts to cover a new cell, it performs an "extra" lap starting
from a critical point on one of its boundaries.
•
•
•
•
•
Length
of
Boundary
Following
Path
P :Length of the floor and ceiling of a cell.
cell
The length of boundary-following paths in a cell is at least Pcell.
Considering an undo-reverse boundary-following motion to get to the start point,
the lower bound is 1. 5Pcell.
In the worst case, the upper bound becomes 2Pcell as shown in Fig. below.
The total length of the boundary-following paths is less than 2Ptotal ------(B)
where Ptotal is the length of the perimeter of all of the obstacles and the outer
boundary.
Fig. The total perimeter of the cell is x + y ; y is the length of the floor and x is the
length of the ceiling and x >> y. The total path length traveled along the boundary of
the cell is bounded above by 2x + y. As x >> y, this value is equal to 2(x + y).
Backtracking to Closing Critical Point
In the worst case, the length of this backtracking path is Pcell + Δ (where the
robot follows every boundary and the longest slice).
When we consider all the backtracking paths, the upper bound becomes
Ptotal + ΔNce. -……………………………..(C)
The extra boundary-following path followed by the robot to discover the critical
point is bounded above by Pcell.
Hence, the total extra boundary-following path length is bounded above by
Ptotal……………………………(D)
Fig. After discovering the closing critical point of a cell, the robot backtracks to the
closing critical point of a cell with uncovered cells by boundary-following and (if
necessary) lapping .
DFS on Reeb Graph
When the robot finishes covering a cell, it performs a depth-first search on the
Reeb graph to choose an uncovered cell (if any are left).
Figure 6.29: The robot starts to cover the space from Cp1. Whenever the robot finishes
covering a cell, a depth-first search is performed on the graph to choose a new cell to
cover.
Since we perform a depth-first search on the graph, each cell is traversed at most once ,
and therefore the backtracking path length is bounded by .
N
ce

Pc e l l i  2 N
ce

i 1
o r , Pt o t a l  2 N
ce

---------------------------------(E)
Length of Coverage Path
Fig. After finishing covering cells A and B, the robot needs to travel from Cp1 to Cp2
to start to cover cell C. The robot simply follows the boundary of the obstacle either
along the ceiling or floor of the cell. In the worst case, the boundary-following path
length is bounded above by the length of the perimeter of the obstacles that form
the boundary of the cell.
Length of Coverage Path Contd….
Combining the above upper bounds, the length of the coverage
path is less than
2

 4  N ce  5 Ptotal
2r
Using Modified Eulers Formula,

2
2r
 4  ( N cp  N ob )  5 Ptotal  4 
Therefore, the total coverage path length is bounded linearly by
the area of the space, the number of critical points, and the
length of the perimeter of the obstacles and the outer boundary.
```