ErrorProp
=========
TODO: writeme
Error propagation library.
Installation
$ git clone git://github.com/hepek/ErrorProp.git
$ cd ErrorProp
$ cabal install
Example 1: Two Link Planar Arm
----------------------------
Let's simulate a non-linear transformation by combining translations
and rotations of a robotic arm and evaluating error propagation at different
operation points.
Figure 1. A two link planar robotic arm
> import Math.ErrorProp
> import Math.SimpleMx
>
Coordinates
> x = defVar "x"
> y = defVar "y"
Model parameters
> a1 = defVar"a1"
> a2 = defVar"a2"
> l1 = defVar"l1"
> l2 = defVar"l2"
Couple of helper functions for defining rotations and translations
> rotGen var = fromLists
> [[cos(var), -sin(var), 0]
> ,[sin(var), cos(var), 0]
> ,[0, 0, 1]]
>
> transGen x y = fromLists
> [[1, 0, x]
> ,[0, 1, y]
> ,[0, 0, 1]]
>
Our two link planar arm can now be modeled as superposition of two rotations and two translations:
> arm1 = (rotGen a1) >< (transGen l1 0) >< (rotGen a2) >< (transGen l2 0)
To obtain our non-linear transformation we multiply arm1 with [x, y, 1] (homogenous coordinates).
> armT1 = mkTransf $ toList (arm1 >. fromList [x, y, 1])
We can view our set of resulting functions:
> r1 = print armT1
o1 = (((x * ((cos(a2) * cos(a1)) + (sin(a2) * -sin(a1)))) + (y * ((-sin(a2) * cos(a1)) + (cos(a2) * -sin(a1))))) + ((l2 * ((cos(a2) * cos(a1)) + (sin(a2) * -sin(a1)))) + (l1 * cos(a1))))
o2 = (((x * ((cos(a2) * sin(a1)) + (sin(a2) * cos(a1)))) + (y * ((-sin(a2) * sin(a1)) + (cos(a2) * cos(a1))))) + ((l2 * ((cos(a2) * sin(a1)) + (sin(a2) * cos(a1)))) + (l1 * sin(a1))))
o3 = 1.0
We could easily eliminate x and y from the equations as we know the initial coordinates (0,0).
This can be done in two ways:
1. Reevaluate our matrix-vector multiplication from above:
> armT2 = mkTransf $ toList (arm1 >. fromList [0,0,1])
2. Or use our partialEval utility function
> armT2' = (partial armT1 [(x,0), (y,0)])
> r2 = print armT2
o1 = ((l2 * ((cos(a2) * cos(a1)) + (sin(a2) * -sin(a1)))) + (l1 * cos(a1)))
o2 = ((l2 * ((cos(a2) * sin(a1)) + (sin(a2) * cos(a1)))) + (l1 * sin(a1)))
o3 = 1.0
And the order of input parameters would be:
> r3 = print (variables armT1)
[a1,a2,l1,l2,x,y]
> r4 = print (variables armT2)
[a1,a2,l1,l2]
> degrad a = a/180*pi
> r5 = print $ apply armT2 $
> measurement [ degrad 45 +- 0
> , degrad 0 +- 0
> , 10 +- 0
> , 5 +- 0]
measurement [
10.606601717798213 +- 0.0,
10.606601717798211 +- 0.0,
1.0 +- 0.0]
> r6 = apply armT1 $
> measurement [ degrad 45 +- degrad 1
> , degrad 0 +- degrad 1
> , 10 +- 0.01
> , 5 +- 0.01
> , 0 +- 0
> , 0 +- 0]
measurement [
10.606601717798213 +- 0.1953898090314297,
10.606601717798211 +- 0.1953898090314297,
1.0 +- 0.0]
> r6' = getCovariance r6
[4.241908608148729e-3,-4.219686385926508e-3,0.0]
[-4.219686385926507e-3,4.24190860814873e-3,0.0]
[0.0,0.0,0.0]
> r7 = print $ apply armT1 $
> measurement [ degrad 0 +- degrad 1
> , degrad 0 +- degrad 1
> , 10 +- 0.01
> , 5 +- 0.01
> , 0 +- 0
> , 0 +- 0]
measurement [
15.0 +- 1.4142135623730952e-2,
0.0 +- 0.27596078516100275,
1.0 +- 0.0]