Application of Mathematics in Robotics full report
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13042010, 11:27 AM
Application of Mathematics in Robotics.ppt (Size: 147.5 KB / Downloads: 176) Robotics Geometric description of arm movement Prepared by: Kalina Mincheva Revi Panidha Arbnor Hasani Content Introduction Geometric Description Simple Problems Conclusion Main parts of Robotâ„¢s arm: Different Joints Segments 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 â€œ x)2 + (v â€œ y)2 + (w â€œ 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) Solution: 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 â€œ find the real roots The forward kinematic problem Linear systems â€œ reduced row echelon form In our case â€œ polynomial equations Groebner basis â€œ 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) Solution: Elimination The forward kinematic problem The Groebner basis â€œ one of the polynomials looks like this: u2 + v2 + w2= 5 â€œ equation of a sphere Problem: 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 â€œ it is reachable. The problem is solved! GrÃƒÂ¶bner Bases Application areas of GrÃƒÂ¶bner Bases Introduced by Bruno Buchberger, in 1965 Named after Wolfgang GrÃƒÂ¶bner â€œ Buchbergerâ„¢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 GrÃƒÂ¶bner 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 robotics coding theory integer programming partial differential equations symbolic summation statistics noncommutative algebra systems theory compiler theory (noncommutative algebra) Problems that can be solved by Groebner basis method solvability and solving of polynomial systems of equations ideal membership problem elimination theory implicitization effective computation in residue class rings modulo polynomial ideals linear diophantine equations with polynomial coefficients (syzygies) Hilbert functions algebraic relations among polynomials Why is GrÃƒÂ¶bner 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 GrÃƒÂ¶bner bases. How Can GrÃƒÂ¶bner Bases Theory is Applied Given a set F of polynomials in k[x1, Â¦, xn] We transform F into another set G of polynomials with certain nice properties (called a GrÃƒÂ¶bner Basis) such that F and G are equivalent i.e. generate the same ideal How Can GrÃƒÂ¶bner Bases Theory is Applied Many problems that are difficult for general F are easy for GrÃƒÂ¶bner Bases G There is an algorithm transforming an arbitrary F into an equivalent GrÃƒÂ¶bner 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 http://www.energid.com/site/actin_movies.htm mark.math.helsinki.fi/ 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 Use Search at http://topicideas.net/search.php wisely To Get Information About Project Topic and Seminar ideas with report/source code along pdf and ppt presenaion



