by chris wan jessica gima the secrets of mr. roboto

BY CHRIS WANJESSICA GIMA

The Secrets of Mr. Roboto

Goals of Project

To develop an understanding of how a robot arm moves (in 2-D)

To cover the linear systems topics of transformations (rotation and translation matrix) and homogeneous coordinates.

Canadarm

Applications of Robotics

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•Used to place satellites into their proper orbit and for repair.

Transformations

So… we’ve covered this topic in class already. But let’s refresh your memories!

The transformation matrix can either translate or rotate a vector on the 2-D plane.

This matrix causes the vector to move tx over and ty up.

Let A be an m x n matrix. Then the matrix transformation

TA(x) = Ax (for x in Rn)

Is a linear transformation.

Rotational Matrices

•In two dimensions, rotation can be defined by one single θ •Positive angles represent counterclockwise rotation

•For rotation by an angle θ counterclockwise about the origin, the functional form is x' = xcosθ − ysinθ and y' = xsinθ + ycosθ. Written in matrix form,this becomes :

Translation

Translations are not a linear transformation because T(0) ≠ 0 If x = [x ; y] & v = [a ; b], then a translation along v is

the transformationT(x) = x + v T[x ; y] = [(x+a) ; (y+b)]

Homogeneous Coordinates

Affine transformation – any transformation that preserves co-linearity. It can be a combo of translation and rotation matrices (this pertains to our project). Homogeneous coordinates allow affine transformations

to be easily represented by a matrix. Homogeneous coordinates - points in 2D space that

may be viewed as projections of points in 3D space, all being multiples of each other. The coordinates of any of these points are the homogeneous coordinates of the given point.

More Homogeneous Coordinates…

Since translation is not a linear transformation we need to represent vector x as the vector [x ; y ; 1] in R3. This is called representing x in homogeneous

coordinates. And that gives us…

X = [(x+a) ; (y+b) ; (1)]

Rotation in Homogeneous Coordinates

X= [(xcosγ – ysin γ) ; (xsin γ + ycos γ) ; 1]= R(x)

Combining Translation and Rotation

T º R = =

Robotic Arms

Give each link its own coordinate systemΘ is the angle between the links

θ

Frames.

Bibliography

Weisstein, Eric W. "Rotation Matrix." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RotationMatrix.html

Poole, David. Linear Algebra. 2nd ed. Ontario: Thompson Brooks/Cole, 2006. 224-227.

Farin, Gerald, and Dianne Hansford. Practical Linear Algebra - a Geometry Toolbook. 2nd ed. Haverford: AK Peters, Ltd., 2005. 1-4.

"Transformation Matrix." Wikipedia. Wikipedia Foundation Inc., 2006. Wikipedia. 26 Apr. 2007. http://en.wikipedia.org/wiki/Transformation_Matrices