Application of Mathematics in Robotics full report
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13-04-2010, 11:27 AM

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Robotics Geometric description of arm movement

Prepared by: Kalina Mincheva
Revi Panidha
Arbnor Hasani

Geometric Description
Simple Problems
Main parts of Robotâ„¢s arm:
Different Joints

Planar revolute joints
Prismatic joints
Ball joints
Screw joints
Types of Joints Video Presentation
General joints video
Our framework
2 segments
2 joints
Ball joints can rotate 360D

Video of our framework
The forward kinematics problem
The forward kinematic problem for a given robot arm is a systematic description of the relative positions of the segments on either side of a joint, thus determining the position and orientation of the hand from the arm.
We will consider a robot in R3, in particular the set of polynomial equations that constraint the motion of the robot arm.
The forward kinematic problem
We can consider that both segments (parts of the arm) can move in the hole space. The first segment has length 2 and the second has length 1
The forward kinematic problem
We see that the robotâ„¢s arm can move in the hole space, which is actually a sphere
The equation of a sphere is: X2 + y2 + z2= r2;
The two equations that describe the previous spheres
X2 + y2 = 22
(u “ x)2 + (v “ y)2 + (w “ z)2 = 1
Does this system of polynomial equations have real solutions If yes then:
How to solve this system of polynomial equations Difficult!
Groebner Basis is the tool we can use.
The Groebner Basis can help us solve the problem whether the robot can reach a certain point with center at (a, b, c)

In order to find the coordinates of the point we are interested in, we have to find all points the arm can reach and see if this is one of them
Points that can be reached = points that satisfy the above equations
Solve a system of polynomial equations “ find the real roots
The forward kinematic problem
Linear systems “ reduced row echelon form
In our case “ polynomial equations
Groebner basis “ the equivalent
(used to present the solutions of the equations in a reduced way)
The forward kinematic problem
Problem: We have 3 dimensions:
Looking for a point with coordinates (u, v, w)
But we have 6 variables! (x, y, z, u, v, w)
We need to get rid of (x, y, z)
The forward kinematic problem
The Groebner basis “ one of the polynomials looks like this:
u2 + v2 + w2= 5 “ equation of a sphere
Number of points presented by (x, y, z, u, v, w) is not equal to the number of points presented by the above equation
Not all points that lie in that sphere can be actually reached
The forward kinematic problem
For example seen in a plane:
The forward kinematic problem
The forward kinematic problem
Extension (another theorem J)
Check for the points excluded
If the point is not among these “ it is reachable.
The problem is solved!
Gröbner Bases

Application areas of
Gröbner Bases
Introduced by Bruno Buchberger, in 1965
Named after Wolfgang Gröbner “ Buchberger™s PhD Thesis Advisor.
The goal:
Present algorithmic solution of some of the fundamental problems in commutative algebra
(polynomial ideal theory, algebraic geometry )
The method (theory plus algorithms) of Gröbner Bases provides a uniform approach to solving a wide range of problems expressed in terms of sets of multivariate polynomials.

algebraic geometry, commutative algebra , polynomial ideal theory
invariant theory
coding theory
integer programming
partial differential equations
symbolic summation
non-commutative algebra
systems theory
compiler theory (non-commutative algebra)
Problems that can be solved by Groebner basis method
solvability and solving of polynomial systems of equations
ideal membership problem
elimination theory
effective computation in residue class rings modulo polynomial ideals
linear diophantine equations with polynomial coefficients (syzygies)
Hilbert functions

algebraic relations among polynomials
Why is Gröbner Bases Theory Attractive
The main problem solved by the theory can be explained in 5 minutes (if one knows operations addition and multiplication of polynomials).
The algorithm that solves the problem can be learned in 15 minutes
The theorem on which the algorithm is based is nontrivial to invent and to prove.
Many problems in seemingly quite different areas of mathematics can be reduced to the problem of computing Gröbner bases.
How Can Gröbner Bases Theory is Applied
Given a set F of polynomials in k[x1, ¦, xn]
We transform F into another set G of polynomials with certain nice properties (called a Gröbner Basis) such that
F and G are equivalent i.e. generate the same ideal
How Can Gröbner Bases Theory is Applied

Many problems that are difficult for general F are easy for Gröbner Bases G
There is an algorithm transforming an arbitrary F into an equivalent Gröbner basis G
The solution of the problem for G can often be easily translated back into a solution of the problem for F
Groebner Basis
Sources Symbolinen%20laskenta/Notes/Groebner/Intro.ppt
Ideal, Varieties and Algorithms
by David Cox, John Little, Donal O'Shea
Thank you for your attention
Q & A
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