Robot Motion Planning Using Topological Robotics Chintan Patel, Alexander Rios Wilbur Wright College
April 19, 2017
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
What is Topology?
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Topology is a branch of mathematics which is similar to geometry. It studiesshapes, but distances or angles do not matter. In topological representation of the reality, only connections matter, in particular holes and loops.
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It has been used in many different fields like Robotics, Molecular Biology, Physics, and Computer Science.
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
Topological Robotics
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Topological robotics deals with the problem of robotic motion planning.
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A robot is a machine capable of performing a task autonously.
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Topological Robotics plans the motion algorithms for the robot to perform its task.
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
Motion Planning Problem Let’s imagine you are on a driver-less car and you say ”Go Home”. The car will use the ’Google Maps’ and will drive you home. Here, the ’Google Maps’ uses a motion planner.
Figure: Google Maps showing the path from Wrigley Field to Margie’s Candies Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
Path in Space X 1
Mathematically, it is a function from an interval I to the space X α:I →X
α(0) = a Initial Position α(1) = b Final Position 1
The set of all paths will be called Path Space PX . Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
Cartesian Product
In set theory, Cartesian product is the set of pairs of x and y, where x belongs to set A and y belongs to set B. A×B
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
Evaluation Function
It is the function which takes a path from the Path Space PX as an input and gives the pair of initial and final points in the Cartesian product X × X as an output. ev : PX → X × X
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
Section of the Evaluation
A section is a function that takes a pair on a Cartesian product as an input, and gives a path between those two points as an output. s : X × X → PX To solve the motion planning problem means to find this section.
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
Continuity Given (a, b) on the Cartesian product and consider the path α between them. Let’s take another point (a0 , b 0 ) in the neighborhood of (a, b). The path α0 between them should be close to α for the section to be continuous.
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
One Robot on a Circle S 1
S : S 1 × S 1 → PS 1
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
One Robot on a Circle S 1
S : S 1 × S 1 → PS 1
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
One Robot on a Circle S 1
S : S 1 × S 1 → PS 1
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
One Robot on a Circle S 1
S : S 1 × S 1 → PS 1
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
One Robot on a Circle S 1
S : S 1 × S 1 → PS 1
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
One Robot on a Circle S 1
S : S 1 × S 1 → PS 1
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
Farber’s Theorem
Theorem A continuous motion planning s : X × X → PX exists if and only if the configuration space X is contractible.
⇐⇒
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
Topological Complexity
Definition Given a path-connected space X , the topological complexity of X is the minimum number TC(X )=k, such that the Cartesian product X × X may be covered by k open subsets X × X = U1 ∪ U2 ∪ · · · ∪ Uk such that for any i = 1, 2, ..., k there exists a continuous motion planning si : Ui → PX , over Ui . If no such k exists we will set TC(X ) = ∞
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
TC of one robot moving on the circle S 1 is 2
The two instructions are 1
Go counterclockwise.
2
Go shortest path.
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
Physical and Configuration space
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Physical space Γ is the space where the actual motion of the robot occurs. Configuration space C n (Γ) is the space which shows the combined position of all robots. C n (Γ) = Γ × Γ × Γ × · · · × Γ − ∆ where, ∆ = {(x1 , x2 , · · · , xn ) | ∃i 6= j with xi = xj }
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
Robot motion planning of two robots moving on a track T The configuration space is C 2 (T ) = T × T − ∆, where ∆ = {(x, y ) ∈ T × T | x = y }
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
Robot motion planning of two robots moving on a track T The configuration space is C 2 (T ) = T × T − ∆, where ∆ = {(x, y ) ∈ T × T | x = y }
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
Robot motion planning of two robots moving on a track T The configuration space is C 2 (T ) = T × T − ∆, where ∆ = {(x, y ) ∈ T × T | x = y }
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
Robot motion planning of two robots moving on a track T The configuration space is C 2 (T ) = T × T − ∆, where ∆ = {(x, y ) ∈ T × T | x = y }
Chintan Patel, Alexander Rios
Robot Motion Planning Using Topological Robotics
